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.

The Effects of Commercially Available Preservative-Free FDA-Approved Triamcinolone (Triesence (R)) on Retinal Cells in Culture

Purpose: To evaluate the effects of Triesence (R) (TRI), a new preservative-free triamcinolone approved by the U. S. Food and Drug Administration (FDA) for intraocular use, on human retina pigment epithelial (ARPE-19) and rat neurosensory (R28) cells in culture. Methods: ARPE-19 and R28 cell cultures were treated 24 h with 1,000, 500, 200, or 100 mu g/mL of crystalline (cTRI) or 1,000, 500, or 200 mu g/mL of solubilized (sTRI). TRI was solubilized by centrifuging the drug, discarding the supernatant containing the vehicle and then resuspending the drug pellet in an equivalent amount of Dimethyl sulfoxide to achieve the same concentration as the commercial preparation. Percentage of cell viability (CV) was evaluated by a trypan blue dye-exclusion assay. The mitochondrial membrane potential (Delta Psi m) was analyzed with the JC-1 assay. The caspase-3/7 activity was measured by a fluorochrome assay. Results: In the ARPE-19 cultures, the cTRI caused a decrease in CV at 1,000 mg/mL (13.03 +/- 6.51; P < 0.001), 500 mu g/mL (28.87 +/- 9.3; P < 0.001), 200 mu g/mL (54.93 +/- 5.61; P < 0.001), and 100 mu g/mL (82.53 +/- 0.65; P < 0.005) compared with the untreated controls (96.98 +/- 0.16). In R28 cultures, the cTRI treatment also reduced CV values significantly (P < 0.001) for the 1,000 mu g/mL (22.73 +/- 2.44), 500 mu g/mL (34.63 +/- 1.91), 200 mu g/mL (58.70 +/- 1.39), and 100 mu g/m (75.33 +/- 2.47) compared with the untreated controls (86.08 +/- 3.54). Once the TRI was solubilized (sTRI), the CV and Delta Psi m remained similar to the untreated controls for both ARPE-19 and R28 cells. The sTRI treatment with 1,000, 500, and 200 mu g/mL increased in caspase-3/7 activity in ARPE-19 cells (P < 0.01) and in R28 cells (P < 0.05) compared with dimethyl sulfoxide equivalent controls. Conclusion: The crystalline form of TRI (cTRI) can cause a significant decrease in CV to cultured retinal cells. Once the TRI is solubilized (sTRI), at the same concentrations, the cells remain viable with no decrease in CV or Delta Psi m. The sTRI can, however, increase caspase-3/7 activity, thus suggesting some degree of apoptosis.

AFFINE INTERVAL EXCHANGE TRANSFORMATIONS WITH FLIPS AND WANDERING INTERVALS

There exist uniquely ergodic affine interval exchange transformations of [0,1] with flips which have wandering intervals and are such that the support of the invariant measure is a Cantor set.

Process Capability Indexes Determination on Tabletting Performance during the Process Validation for Metamizol Tablets

The aim of the present study was to provide a numerical measure, through the process capability indexes (PCIs), C(p) and C(pk), on whether or not the manufacturing process can be considered capable of producing metamizol (500 mg) tablets. They were also used as statistical tool in order to prove the consistency of the tabletting process, making sure that the tablet weight and the content uniformity of metamizol are able to comply with the preset requirements. Besides that, the ANOVA, the t-test and the test for equal variances were applied to this study, allowing additional knowledge of the tabletting phase. Therefore, the proposed statistical approach intended to assure more safety, precision and accuracy on the process validation analysis.

Optimization of preservative system in ophthalmic suspension with dexamethasone and polymyxin B

A simplex-lattice statistical project was employed to study an optimization method for a preservative system in an ophthalmic suspension of dexametasone and polymyxin B. The assay matrix generated 17 formulas which were differentiated by the preservatives and EDTA (disodium ethylene diamine-tetraacetate), being the independent variable: X-1 = chlorhexidine digluconate (0.010 % w/v); X-2 = phenylethanol (0.500 % w/v); X-3 = EDTA (0.100 % w/v). The dependent variable was the Dvalue obtained from the microbial challenge of the formulas and calculated when the microbial killing process was modeled by an exponential function. The analysis of the dependent variable, performed using the software Design Expert/W, originated cubic equations with terms derived from stepwise adjustment method for the challenging microorganisms: Pseudomonas aeruginosa, Burkholderia cepacia, Staphylococcus aureus, Candida albicans and Aspergillus niger. Besides the mathematical expressions, the response surfaces and the contour graphics were obtained for each assay. The contour graphs obtained were overlaid in order to permit the identification of a region containing the most adequate formulas (graphic strategy), having as representatives: X-1 = 0.10 ( 0.001 % w/v); X-2 = 0.80 (0.400 % w/v); X-3 = 0.10 (0.010 % w/v). Additionally, in order to minimize responses (Dvalue), a numerical strategy corresponding to the use of the desirability function was used, which resulted in the following independent variables combinations: X-1 = 0.25 (0.0025 % w/v); X-2 = 0.75 (0.375 % w/v); X-3 = 0. These formulas, derived from the two strategies (graphic and numerical), were submitted to microbial challenge, and the experimental Dvalue obtained was compared to the theoretical Dvalue calculated from the cubic equation. Both Dvalues were similar to all the assays except that related to Staphylococcus aureus. This microorganism, as well as Pseudomonas aeruginosa, presented intense susceptibility to the formulas independently from the preservative and EDTA concentrations. Both formulas derived from graphic and numerical strategies attained the recommended criteria adopted by the official method. It was concluded that the model proposed allowed the optimization of the formulas in their preservation aspect.

Bioequivalence Evaluation of Two Different Tablet Formulations of Tinidazole in Healthy Volunteers

The bioequivalence of two different tablet formulations of tirtidazole (CAS 19387-91-8) was determined in healthy volunteers after a single dose in a randomized crossover study, with a 1-week washout period between the doses. Reference and test products were administered to 24 volunteers with 240 mL water after overnight fasting. Plasma concentrations of tinidazole were monitored by a high-performance liquid chromatographic method (HPLC) over a period of 72 h after the administration. The pharmacokinetic parameters AUC(0-t), AUC(0-infinity), C(max), T(max), T((1/2)el) and beta were determined from plasma concentration time profile of both formulations and found to be in good agreement with previously reported values. The calculated pharmacokinetic parameters were compared statistically to evaluate bioequivalence between the two brands. The analysis of variance (ANOVA) did not show any significant difference between the two formulations and 90% confidence intervals for the ratio of C(max) (93.9 - 102.6%), AUC(0-t), (94.9-101.1%) and AUC(0-infinity) (94.6-100.8%) values for the test and reference products were within the 80 - 125% interval, satisfying bioequivalence criteria of the European Committee for Proprietary Medicinal Products and the US Food and Drug Administration Guidelines. These results indicate that the test and the reference products of tinidazole are bioequivalent and, thus, may be prescribed interchangeably.

Bioequivalence study of two oral formulations of cefadroxil in healthy volunteers

Two different cefadroxil (CAS 50370-12-2) formulations were evaluated for their relative bioavailability in 24 healthy volunteers who received a single 500 mg oral dose of each preparation. An open, randomized clinical trial designed as a two-period crossover study with a 7-day washout period between doses was employed. Plasma samples for assessments of their cefadroxil concentration by HPLC-UV were obtained over 8 h after administration. Values of 48.94 +/- 10.18 mu g . h/ml for test, and 48.51 +/- 9.02 mu g . h/ml for the reference preparation AUC(0-t) demonstrate a nearly identical extend of drug absorption. Maximum plasma concentration C-max of 16.04 +/- 4.94 mu g/ml and 16.01 +/- 4.02 mu g/ml achieved for the test and reference preparations did not differ significantly. The parametric 90% confidence intervals (CI) of the mean of the difference (test-reference) between log-transformed values of the two formulations were 96.80% to 104.51% and 92.01% to 107.00% for AUC(0-t) and C-max, respectively. Since for both AUC(0-t) or C-max the 90% CI values are within the interval proposed by the Food and Drug Administration, the test product is bioequivalent to the reference product for both the rate and extent of absorption after single dose administration.

Treatment of advanced hepatocellular carcinoma with very low levels of amplitude-modulated electromagnetic fields

BACKGROUND: Therapeutic options for patients with advanced hepatocellular carcinoma (HCC) are limited. There is emerging evidence that the growth of cancer cells may be altered by very low levels of electromagnetic fields modulated at specific frequencies. METHODS: A single-group, open-label, phase I/II study was performed to assess the safety and effectiveness of the intrabuccal administration of very low levels of electromagnetic fields amplitude modulated at HCC-specific frequencies in 41 patients with advanced HCC and limited therapeutic options. Three-daily 60-min outpatient treatments were administered until disease progression or death. Imaging studies were performed every 8 weeks. The primary efficacy end point was progression-free survival >= 6 months. Secondary efficacy end points were progression-free survival and overall survival. RESULTS: Treatment was well tolerated and there were no NCI grade 2, 3 or 4 toxicities. In all, 14 patients (34.1%) had stable disease for more than 6 months. Median progression-free survival was 4.4 months (95% CI 2.1-5.3) and median overall survival was 6.7 months (95% CI 3.0-10.2). There were three partial and one near complete responses. CONCLUSION: Treatment with intrabuccally administered amplitude-modulated electromagnetic fields is safe, well tolerated, and shows evidence of antitumour effects in patients with advanced HCC. British Journal of Cancer (2011) 105, 640-648. doi:10.1038/bjc.2011.292 www.bjcancer.com Published online 9 August 2011 (C) 2011 Cancer Research UK

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.