On certain subspaces of p for 0 < p ≤ 1 and their applications to conditional quasi-greedy bases in p-Banach spaces

We construct for each 0<p≤1 an infinite collection of subspaces of ℓp that extend the example of Lindenstrauss (Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 12, 539–542, 1964) of a subspace of ℓ1 with no unconditional basis. The structure of this new class of p-Banach spaces is analyzed and some applications to the general theory of Lp-spaces for 0<p<1 are provided. The introduction of these spaces serves the purpose to develop the theory of conditional quasi-greedy bases in p-Banach spaces for p<1. Among the topics we consider are the existence of infinitely many conditional quasi-greedy bases in the spaces ℓp for p≤1 and the careful examination of the conditionality constants of the 'natural basis' of these spaces.F. Albiac acknowledges the support of the Spanish Ministry for Economy and Competitivity under Grant MTM2016-76808-P as well as the Spanish Ministry for Science and Innovation under Grant PID2019-1077701GB-I00 for Operators, lattices, and structure of Banach spaces. F. Albiac and J. L. Ansorena acknowledge the support of the Spanish Ministry for Science, Innovation, and Universities under Grant PGC2018-095366-B-I00 for Análisis Vectorial, Multilineal y Aproximación. P. Wojtaszczyk was supported by National Science Centre, Poland Grant UMO-2016/21/B/ST1/00241

Screening for impact of popular herbs improving mental abilities on the transcriptional level of brain transporters

There are a number of compounds that can modify the activity of ATP-binding cassette (ABC) and solute carrier (SLC) transporters in the blood-brain barrier (BBB). The aim of this study was to investigate the effect of natural and synthetic substances on the expression level of genes encoding transporters present in the BBB (mdr1a, mdr1b, mrp1, mrp2, oatp1a4, oatp1a5 and oatp1c1). Our results showed that verapamil caused the greatest reduction in the mRNA level while other synthetic (piracetam, phenobarbital) and natural (codeine, cyclosporine A, quercetin) substances showed a selective inhibitory effect. Moreover, extract from roots of Panax ginseng C. A. Meyer exhibited a decrease of transcription against selected transporters whereas extract from Ginkgo biloba L. leaves resulted in an increase of the expression level of tested genes except for mrp2. Extract from aerial parts of Hypericum perforatum L. was the only one to cause an increased mRNA level for mdr1 and oatp1c1. These findings suggest that herbs can play an important role in overcoming the BBB and multidrug resistance against pharmacotherapy of brain cancer and mental disorders, based on the activity of selected drug-metabolizing enzymes and transporters located in the BBB