A remark on totally smooth renormings

[EN] E. Oja, T. Viil, andD. Werner showed, in Totally smooth renormings, Archiv der Mathematik, 112, 3, (2019), 269-281, that a weakly compactly generated Banach space ( X, center dot) with the property that every linear functional on X has a unique Hahn-Banach extension to the bidual X ** (the so-called Phelps' property U in X **, also known as the Hahn-Banach smoothness property) can be renormed to have the stronger property that for every subspace Y of X, every linear functional on Y has a unique Hahn-Banach extension to X ** (the so-called total smoothness property of the space). We mention here that this result holds in full generality -without any restriction on the space- and in a stronger form, thanks to a result ofM. Raja, On dual locally uniformly rotund norms, Israel Journal of Mathematics 129 (2002), 77-91.Supported by AEI/FEDER (project MTM2017-83262-C2-2-P of Ministerio de Economia y Competitividad), by Fundacion Seneca, Region de Murcia (Grant 19368/PI/14), and Universitat Politecnica de Valencia (A. J. Guirao). Supported by AEI/FEDER (project MTM2017-83262-C2-1-P of Ministerio de Economia y Competitividad) and Universitat Politecnica de Valencia (V. Montesinos). We thank the referees for their work, that neatly improved the original version of this note to its final form.Cobollo, C.; Guirao Sánchez, AJ.; Montesinos Santalucia, V. (2020). A remark on totally smooth renormings. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 114(2):1-4. https://doi.org/10.1007/s13398-020-00831-5S141142Fabian, M., Habala, P., Hájek, P., Montesinos, V., Zizler, V.: Banach space theory: the basis of linear and nonlinear analysis. Springer, New York (2011)Fabian, M., Montesinos, V., Zizler, V.: Smoothness in Banach spaces. Selected problems. Rev. R. Acad. Cien. Ser. A. Mat. RACSAM. 100(1–2), 101–125 (2006)Ferrari, S., Orihuela, J., Raja, M.: Generalized metric properties of spheres and renorming of Banach spaces. Rev. R. Acad. Cienc. Exactas Fis. Natl. Ser. A Math. RACSAM. 113, 2655–2663 (2019)Foguel, S.R.: On a theorem by A. E. Taylor. Proc. Amer. Math. Soc. 9, 325 (1958)Godefroy, G.: Points de Namioka, espaces normants, applications à la théorie isométrique de la dualité. Israel J. Math. 38, 209–220 (1981)Guirao, A.J., Montesinos, V., Zizler, V.: Open Problems in the geometry and analysis of Banach spaces. Springer International Pub, Switzerland (2016)Harmand, P., Werner, D., Werner, W.: M-ideals in Banach spaces and Banach algebras. Lecture notes in math, vol. 1547. Springer, Berlin (1993)Haydon, R.: Locally uniformly rotund norms in Banach spaces and their duals. J. Funct. Anal. 254, 2023–2039 (2008)Oja, E., Viil, T., Werner, D.: Totally smooth renormings. Archiv. der. Mathematik. 112(3), 269–281 (2019)Phelps, R.R.: Uniqueness of Hahn–Banach extensions and unique best approximation. Trans. Amer. Math. Soc. 95, 238–255 (1960)Raja, M.: On dual locally uniformly rotund norms. Israel J. Math. 129, 77–91 (2002)Smith, R.J., Troyanski, S.L.: Renormings of $$C(K)$$ spaces. Rev. R. Acad. Cienc. Exactas Fís. Natl. Ser. A Math. RACSAM 104(2), 375–412 (2010)Sullivan, F.: Geometrical properties determined by the higher duals of a Banach space. Illinois J. Math. 21, 315–331 (1977)Taylor, A.E.: The extension of linear functionals. Duke Math. J. 5, 538–547 (1939

Geometric properties and continuity of the pre-duality mapping in Banach space

We use the preduality mapping in proving characterizations of some geometric properties of Banach spaces. In particular, those include nearly strongly convexity, nearly uniform convexity-a property introduced by K. Goebel and T. Sekowski-, and nearly very convexity.We thank a referee for the careful reading of the manuscript. His/her observations substantially improved the overall aspect of the present work, detected several misprints and made some convenient changes. This work was supported by: (1) The National Natural Science Foundation of China (Grant no. 11271248). (2) Specific Academic Discipline Project of Shanghai Municipal Education Commission (Grant no. B-8932-13-0136). (3) Project MTM2011-22417, Ministerio de Ciencia e Innovacion, Spain (V. Montesinos).Zhang, ZH.; Montesinos Santalucia, V.; Liu, CY.; Gong, WZ. (2015). Geometric properties and continuity of the pre-duality mapping in Banach space. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas. 109(2):407-416. https://doi.org/10.1007/s13398-014-0190-6S4074161092Bandyopadhyay, P., Huang, D., Lin, B.L., Troyanski, S.L.: Some generalizations of local uniform rotundity. J. Math. Anal. Appl. 252, 906–916 (2000)Bandyopadhyay, P., Li, Y., Lin, B., Narayana, D.: Proximinality in Banach spaces. J. Math. Anal. Appl. 341, 309–317 (2008)Diestel, J.: Geometry of Banach Spaces. Selected Topics, LNM, vol. 485. Springer, Berlin (1975)Fabian, M., Habala, P., Hájek, P., Montesinos, V., Zizler, V.: Banach Space Theory. The Basis for Linear and Nonlinear Analysis, CMS Books in Mathematics. Springer, Berlin (2011)Giles, J.R., Gregory, D.A., Sims, B.: Geometrical implications of upper semi-continuity of the duality mapping on a Banach space. Pacific J. Math. 79(1), 99–109 (1978)Goebel, K., Sekowski, T.: The modulus of non-compact convexity. Ann. Univ. M. Curie-Sklodowska, Sect. A 38, 41–48 (1984)Guirao, A.J., Montesinos, V.: A note in approximative compactness and continuity of metric projections in Banach spaces. J. Convex Anal. 18, 397–401 (2011)Huff, R.: Banach spaces which are nearly uniformly convex. Rocky Mountain J. Math. 10(4), 743–749 (1980)Kutzarova, D., Rolewicz, S.: On nearly uniformly convex sets. Arch. Math. 57, 385–394 (1991)Kutzarova, D., Lin, B.L., Zhang, W.: Some geometrical properties of Banach spaces related to nearly uniform convexity. Contemp. Math. 144, 165–171 (1993)Kutzarova, D., Prus, S.: Operators which factor through nearly uniformly convex spaces. Boll. Un. Mat. Ital. B (7) 9, 2, 479–494 (1995)Montesinos, V.: Drop property equals reflexivity. Studia Math. 87, 93–100 (1987)Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability, LNM, vol. 1364, 2nd edn. Springer, Berlin (1993)Rolewicz, S.: On drop property. Studia Math. 85, 27–37 (1986)Rolewicz, S.: On $$\Delta$$ Δ -uniform convexity and drop property. Studia Math. 87, 181–191 (1987)Wu, C.X., Li, Y.J.: Strong convexity in Banach spaces. J. Math. Wuhan Univ. 13(1), 105–108 (1993)Wang, J.H., Nan, C.X.: The continuity of subdifferential mapping. J. Math. Anal. Appl. 210, 206–214 (1997)Wang, J.H., Zhang, Z.H.: Characterization of the property (C-K). Acta Math. Sci. Ser. A Chin. Ed. 17(A)(3), 280–284 (1997)Zhang, Z.H., Liu, C.Y.: Some generalization of locally and weakly locally uniformly convex space. Nonlinear Anal. 74(12), 3896–3902 (2011)Zhang, Z.H., Liu, C.Y.: Convexity and proximinality in Banach spaces. J. Funct. Spaces Appl. 2012, 11 (2012). doi: 10.1155/2012/724120 . Article ID 724120Zhang, Z.H., Liu, C.Y.: Convexity and existence of the farthest point. Abstract Appl. Anal. 2011, 9 (2011). doi: 10.1155/2011/139597 . Article ID 139597Zhang, Z.H., Shi, Z.R.: Convexities and approximative compactness and continuity of the metric projection in Banach spaces. J. Approx. Theory 161(2), 802–812 (2009)Zhang, Z.H., Zhang, C.J.: On very rotund Banach spaces. Appl. Math. Mech. (English Ed.) 21(8), 965–970 (2000

On the compact operators case of the Bishop-Phelps-Bollobás property for numerical radius

The authors would like to thank Bill Johnson for kindly answering several inquiries.We study the Bishop–Phelps–Bollobás property for numerical radius restricted to the case of compact operators (BPBp-nu for compact operators in short). We show that C0(L) spaces have the BPBp-nu for compact operators for every Hausdorff topological locally compact space L. To this end, on the one hand, we provide some techniques allowing to pass the BPBp-nu for compact operators from subspaces to the whole space and, on the other hand, we prove some strong approximation property of C0(L) spaces and their duals. Besides, we also show that real Hilbert spaces and isometric preduals of ℓ1 have the BPBp-nu for compact operators

Patterns of nucleotide diversity at the regions encompassing the Drosophila insulin-like peptide (dilp) genes: demography vs positive selection in Drosophila melanogaster.

In Drosophila, the insulin-signaling pathway controls some life history traits, such as fertility and lifespan, and it is considered to be the main metabolic pathway involved in establishing adult body size. Several observations concerning variation in body size in the Drosophila genus are suggestive of its adaptive character. Genes encoding proteins in this pathway are, therefore, good candidates to have experienced adaptive changes and to reveal the footprint of positive selection. The Drosophila insulin-like peptides (DILPs) are the ligands that trigger the insulin-signaling cascade. In Drosophila melanogaster, there are several peptides that are structurally similar to the single mammalian insulin peptide. The footprint of recent adaptive changes on nucleotide variation can be unveiled through the analysis of polymorphism and divergence. With this aim, we have surveyed nucleotide sequence variation at the dilp1-7 genes in a natural population of D. melanogaster. The comparison of polymorphism in D. melanogaster and divergence from D. simulans at different functional classes of the dilp genes provided no evidence of adaptive protein evolution after the split of the D. melanogaster and D. simulans lineages. However, our survey of polymorphism at the dilp gene regions of D. melanogaster has provided some evidence for the action of positive selection at or near these genes. The regions encompassing the dilp1-4 genes and the dilp6 gene stand out as likely affected by recent adaptive events

Depth-specific fluctuations of gene expression and protein abundance modulate the photophysiology in the seagrass Posidonia oceanica

Here we present the results of a multiple organizational level analysis conceived to identify acclimative/adaptive strategies exhibited by the seagrass Posidonia oceanica to the daily fluctuations in the light environment, at contrasting depths. We assessed changes in photophysiological parameters, leaf respiration, pigments, and protein and mRNA expression levels. The results show that the diel oscillations of P. oceanica photophysiological and respiratory responses were related to transcripts and proteins expression of the genes involved in those processes and that there was a response asynchrony between shallow and deep plants probably caused by the strong differences in the light environment. The photochemical pathway of energy use was more effective in shallow plants due to higher light availability, but these plants needed more investment in photoprotection and photorepair, requiring higher translation and protein synthesis than deep plants. The genetic differentiation between deep and shallow stands suggests the existence of locally adapted genotypes to contrasting light environments. The depth-specific diel rhythms of photosynthetic and respiratory processes, from molecular to physiological levels, must be considered in the management and conservation of these key coastal ecosystems.Portuguese funds from FCT - Foundation for Science and Technology [UID/Multi/04326/2013]; SZN PhD fellowship via the Open University; ESF COST Action Seagrass Productivity: From Genes to Ecosystem Management [ES0906]info:eu-repo/semantics/publishedVersio

Drosophila Evolution over Space and Time (DEST): A New Population Genomics Resource

Drosophila melanogaster is a leading model in population genetics and genomics, and a growing number of whole-genome data sets from natural populations of this species have been published over the last years. A major challenge is the integration of disparate data sets, often generated using different sequencing technologies and bioinformatic pipelines, which hampers our ability to address questions about the evolution of this species. Here we address these issues by developing a bioinformatics pipeline that maps pooled sequencing (Pool-Seq) reads from D. melanogaster to a hologenome consisting of fly and symbiont genomes and estimates allele frequencies using either a heuristic (PoolSNP) or a probabilistic variant caller (SNAPE-pooled). We use this pipeline to generate the largest data repository of genomic data available for D. melanogaster to date, encompassing 271 previously published and unpublished population samples from over 100 locations in >20 countries on four continents. Several of these locations have been sampled at different seasons across multiple years. This data set, which we call Drosophila Evolution over Space and Time (DEST), is coupled with sampling and environmental metadata. A web-based genome browser and web portal provide easy access to the SNP data set. We further provide guidelines on how to use Pool-Seq data for model-based demographic inference. Our aim is to provide this scalable platform as a community resource which can be easily extended via future efforts for an even more extensive cosmopolitan data set. Our resource will enable population geneticists to analyze spatiotemporal genetic patterns and evolutionary dynamics of D. melanogaster populations in unprecedented detail.We thank four reviewers and the handling editor for helpful comments on previous versions of our manuscript. We are grateful to the members of the DrosEU and DrosRTEC consortia for their long-standing support, collaboration, and for discussion. DrosEU was funded by a Special Topic Networks (STN) grant from the European Society for Evolutionary Biology (ESEB). M.K. was supported by the Austrian Science Foundation (grant no. FWF P32275); J.G. by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (H2020-ERC-2014-CoG-647900) and by the Spanish Ministry of Science and Innovation (BFU-2011-24397); T.F. by the Swiss National Science Foundation (SNSF grants PP00P3_133641, PP00P3_165836, and 31003A_182262) and a Mercator Fellowship from the German Research Foundation (DFG), held as a EvoPAD Visiting Professor at the Institute for Evolution and Biodiversity, University of Münster; AOB by the National Institutes of Health (R35 GM119686); M.K. by Academy of Finland grant 322980; V.L. by Danish Natural Science Research Council (FNU) (grant no. 4002-00113B); FS Deutsche Forschungsgemeinschaft (DFG) (grant no. STA1154/4-1), Project 408908608; J.P. by the Deutsche Forschungsgemeinschaft Projects 274388701 and 347368302; A.U. by FPI fellowship (BES-2012-052999); ET Israel Science Foundation (ISF) (grant no. 1737/17); M.S.V., M.S.R. and M.J. by a grant from the Ministry of Education, Science and Technological Development of the Republic of Serbia (451-03-68/2020-14/200178); A.P., K.E. and M.T. by a grant from the Ministry of Education, Science and Technological Development of the Republic of Serbia (451-03-68/2020-14/200007); and TM NSERC grant RGPIN-2018-05551. The authors acknowledge Research Computing at The University of Virginia for providing computational resources and technical support that have contributed to the results reported within this publication (https://rc.virginia.edu, last accessed September 6, 2021)

Population genetics of sexual conflict in the genomic era

Sexual conflict occurs when selection acts in opposing directions on males and females. Case studies in both vertebrates and invertebrates indicate that sexual conflict maintains genetic diversity through balancing selection, which might explain why many populations show more genetic variation than expected. Recent population genomic approaches based on different measures of balancing selection have suggested that sexual conflict can arise over survival, not just reproductive fitness as previously thought. A fuller understanding of sexual conflict will provide insight into its contribution to adaptive evolution and will reveal the constraints it might impose on populations