Weighted p-regular kernels for reproducing kernel Hilbert spaces and Mercer Theorem

[EN] Let (X, Sigma, mu) be a finite measure space and consider a Banach function space Y(mu). Motivated by some previous papers and current applications, we provide a general framework for representing reproducing kernel Hilbert spaces as subsets of Kothe Bochner (vectorvalued) function spaces. We analyze operator-valued kernels Gamma that define integration maps L-Gamma between Kothe-Bochner spaces of Hilbert-valued functions Y(mu; kappa). We show a reduction procedure which allows to find a factorization of the corresponding kernel operator through weighted Bochner spaces L-P(gd mu; kappa) and L-P (hd mu; kappa) - where 1/p + 1/p' = 1 - under the assumption of p-concavity of Y(mu). Equivalently, a new kernel obtained by multiplying Gamma by scalar functions can be given in such a way that the kernel operator is defined from L-P (mu; kappa) to L-P (mu; kappa) in a natural way. As an application, we prove a new version of Mercer Theorem for matrix-valued weighted kernels.The second author acknowledges the support of the Ministerio de Economia y Competitividad (Spain), under project MTM2014-53009-P (Spain). The third author acknowledges the support of the Ministerio de Ciencia, Innovacion y Universidades (Spain), Agencia Estatal de Investigacion, and FEDER under project MTM2016-77054-C2-1-P (Spain).Agud Albesa, L.; Calabuig, JM.; Sánchez Pérez, EA. (2020). Weighted p-regular kernels for reproducing kernel Hilbert spaces and Mercer Theorem. Analysis and Applications. 18(3):359-383. https://doi.org/10.1142/S0219530519500179S359383183Agud, L., Calabuig, J. M., & Sánchez Pérez, E. A. (2011). The weak topology on q-convex Banach function spaces. Mathematische Nachrichten, 285(2-3), 136-149. doi:10.1002/mana.201000030CARMELI, C., DE VITO, E., & TOIGO, A. (2006). VECTOR VALUED REPRODUCING KERNEL HILBERT SPACES OF INTEGRABLE FUNCTIONS AND MERCER THEOREM. Analysis and Applications, 04(04), 377-408. doi:10.1142/s0219530506000838CARMELI, C., DE VITO, E., TOIGO, A., & UMANITÀ, V. (2010). VECTOR VALUED REPRODUCING KERNEL HILBERT SPACES AND UNIVERSALITY. Analysis and Applications, 08(01), 19-61. doi:10.1142/s0219530510001503Cerdà, J., Hudzik, H., & Mastyło, M. (1996). Geometric properties of Köthe–Bochner spaces. Mathematical Proceedings of the Cambridge Philosophical Society, 120(3), 521-533. doi:10.1017/s0305004100075058Chavan, S., Podder, S., & Trivedi, S. (2018). Commutants and reflexivity of multiplication tuples on vector-valued reproducing kernel Hilbert spaces. Journal of Mathematical Analysis and Applications, 466(2), 1337-1358. doi:10.1016/j.jmaa.2018.06.062Christmann, A., Dumpert, F., & Xiang, D.-H. (2016). On extension theorems and their connection to universal consistency in machine learning. Analysis and Applications, 14(06), 795-808. doi:10.1142/s0219530516400029Defant, A. (2001). Positivity, 5(2), 153-175. doi:10.1023/a:1011466509838Defant, A., & Sánchez Pérez, E. A. (2004). Maurey–Rosenthal factorization of positive operators and convexity. Journal of Mathematical Analysis and Applications, 297(2), 771-790. doi:10.1016/j.jmaa.2004.04.047De Vito, E., Umanità, V., & Villa, S. (2013). An extension of Mercer theorem to matrix-valued measurable kernels. Applied and Computational Harmonic Analysis, 34(3), 339-351. doi:10.1016/j.acha.2012.06.001Eigel, M., & Sturm, K. (2017). Reproducing kernel Hilbert spaces and variable metric algorithms in PDE-constrained shape optimization. Optimization Methods and Software, 33(2), 268-296. doi:10.1080/10556788.2017.1314471Fasshauer, G. E., Hickernell, F. J., & Ye, Q. (2015). Solving support vector machines in reproducing kernel Banach spaces with positive definite functions. Applied and Computational Harmonic Analysis, 38(1), 115-139. doi:10.1016/j.acha.2014.03.007Galdames Bravo, O. (2014). Generalized Kӧthe $p$-dual spaces. Bulletin of the Belgian Mathematical Society - Simon Stevin, 21(2). doi:10.36045/bbms/1400592625Lin, P.-K. (2004). Köthe-Bochner Function Spaces. doi:10.1007/978-0-8176-8188-3Lindenstrauss, J., & Tzafriri, L. (1979). Classical Banach Spaces II. doi:10.1007/978-3-662-35347-9Meyer-Nieberg, P. (1991). Banach Lattices. Universitext. doi:10.1007/978-3-642-76724-1Okada, S., Ricker, W. J., & Sánchez Pérez, E. A. (2008). Optimal Domain and Integral Extension of Operators. doi:10.1007/978-3-7643-8648-1Zhang, H., & Zhang, J. (2013). Vector-valued reproducing kernel Banach spaces with applications to multi-task learning. Journal of Complexity, 29(2), 195-215. doi:10.1016/j.jco.2012.09.00

Banach Lattice Structures and Concavifications in Banach Spaces

[EN] Let (Omega,sigma,mu) be a finite measure space and consider a Banach function space Y(mu). We say that a Banach space E is representable by Y(mu) if there is a continuous bijection I:Y(mu)-> E. In this case, it is possible to define an order and, consequently, a lattice structure for E in such a way that we can identify it as a Banach function space, at least regarding some local properties. General and concrete applications are shown, including the study of the notion of the pth power of a Banach space, the characterization of spaces of operators that are isomorphic to Banach lattices of multiplication operators, and the representation of certain spaces of homogeneous polynomials on Banach spaces as operators acting in function spaces.The authors would like to thank the referees for their valuable comments, which helped to improve the manuscript. The work of the second author was supported by the Ministerio de Ciencia e Innovacion, Agencia Estatal del Investigacion (Spain) and FEDER under project #PGC2018-095366-B-100. The work of the fourth author was supported by the Ministerio de Ciencia e Innovacion, Agencia Estatal del Investigacion (Spain) and FEDER under project #MTM2016 77054-C2-1-P. We did not receive any funds for covering the costs of publishing in open access.Agud Albesa, L.; Calabuig, JM.; Juan, MA.; Sánchez Pérez, EA. (2020). Banach Lattice Structures and Concavifications in Banach Spaces. Mathematics. 8(1):1-20. https://doi.org/10.3390/math8010127S12081Bu, Q., Buskes, G., Popov, A. I., Tcaciuc, A., & Troitsky, V. G. (2012). The 2-concavification of a Banach lattice equals the diagonal of the Fremlin tensor square. Positivity, 17(2), 283-298. doi:10.1007/s11117-012-0166-8Buskes, G., & van Rooij, A. (2001). Squares of Riesz Spaces. Rocky Mountain Journal of Mathematics, 31(1). doi:10.1216/rmjm/1008959667Troitsky, V. G., & Zabeti, O. (2013). Fremlin tensor products of concavifications of Banach lattices. Positivity, 18(1), 191-200. doi:10.1007/s11117-013-0239-3Defant, A. (2001). Positivity, 5(2), 153-175. doi:10.1023/a:1011466509838Defant, A., & Sánchez Pérez, E. A. (2004). Maurey–Rosenthal factorization of positive operators and convexity. Journal of Mathematical Analysis and Applications, 297(2), 771-790. doi:10.1016/j.jmaa.2004.04.047Berezhnoĭ, E. I., & Maligranda, L. (2005). Representation of Banach Ideal Spaces and Factorization of Operators. Canadian Journal of Mathematics, 57(5), 897-940. doi:10.4153/cjm-2005-035-0Reisner, S. (1981). A factorization theorem in Banach lattices and its application to Lorentz spaces. Annales de l’institut Fourier, 31(1), 239-255. doi:10.5802/aif.825Curbera, G. P. (1992). Operators intoL 1 of a vector measure and applications to Banach lattices. Mathematische Annalen, 293(1), 317-330. doi:10.1007/bf01444717Curbera, G. P., Okada, S., & Ricker, W. J. (2019). Extension and integral representation of the finite Hilbert transform in rearrangement invariant spaces. Quaestiones Mathematicae, 43(5-6), 783-812. doi:10.2989/16073606.2019.1605423Delgado, O., & Soria, J. (2007). Optimal domain for the Hardy operator. Journal of Functional Analysis, 244(1), 119-133. doi:10.1016/j.jfa.2006.12.011Bravo, O. G. (2016). On the Optimal Domain of the Laplace Transform. Bulletin of the Malaysian Mathematical Sciences Society, 40(1), 389-408. doi:10.1007/s40840-016-0402-7Mockenhaupt, G., & Ricker, W. J. (2008). Optimal extension of the Hausdorff-Young inequality. Journal für die reine und angewandte Mathematik (Crelles Journal), 2008(620). doi:10.1515/crelle.2008.054Leśnik, K., & Tomaszewski, J. (2017). Pointwise mutipliers of Orlicz function spaces and factorization. Positivity, 21(4), 1563-1573. doi:10.1007/s11117-017-0485-xDe Jager, P., & Labuschagne, L. E. (2019). Multiplication operators on non-commutative spaces. Journal of Mathematical Analysis and Applications, 475(1), 874-894. doi:10.1016/j.jmaa.2019.03.001Bartle, R. G., Dunford, N., & Schwartz, J. (1955). Weak Compactness and Vector Measures. Canadian Journal of Mathematics, 7, 289-305. doi:10.4153/cjm-1955-032-1Calabuig, J. M., Delgado, O., & Sánchez Pérez, E. A. (2008). Generalized perfect spaces. Indagationes Mathematicae, 19(3), 359-378. doi:10.1016/s0019-3577(09)00008-1Maligranda, L., & Persson, L. E. (1989). Generalized duality of some Banach function spaces. Indagationes Mathematicae (Proceedings), 92(3), 323-338. doi:10.1016/s1385-7258(89)80007-1Schep, A. R. (2009). Products and factors of Banach function spaces. Positivity, 14(2), 301-319. doi:10.1007/s11117-009-0019-2Delgado, O., & Sánchez Pérez, E. A. (2010). Summability Properties for Multiplication Operators on Banach Function Spaces. Integral Equations and Operator Theory, 66(2), 197-214. doi:10.1007/s00020-010-1741-7Sánchez Pérez, E. A. (2015). Product spaces generated by bilinear maps and duality. Czechoslovak Mathematical Journal, 65(3), 801-817. doi:10.1007/s10587-015-0209-yGillespie, T. A. (1981). Factorization in Banach function spaces. Indagationes Mathematicae (Proceedings), 84(3), 287-300. doi:10.1016/1385-7258(81)90040-8Calderón, A. (1964). Intermediate spaces and interpolation, the complex method. Studia Mathematica, 24(2), 113-190. doi:10.4064/sm-24-2-113-190Calabuig, J. M., Delgado, O., & Sánchez Pérez, E. A. (2010). Factorizing operators on Banach function spaces through spaces of multiplication operators. Journal of Mathematical Analysis and Applications, 364(1), 88-103. doi:10.1016/j.jmaa.2009.10.034Calabuig, J. M., Gregori, P., & Sánchez Pérez, E. A. (2008). Radon–Nikodým derivatives for vector measures belonging to Köthe function spaces. Journal of Mathematical Analysis and Applications, 348(1), 469-479. doi:10.1016/j.jmaa.2008.07.024Bu, Q., & Buskes, G. (2012). Polynomials on Banach lattices and positive tensor products. Journal of Mathematical Analysis and Applications, 388(2), 845-862. doi:10.1016/j.jmaa.2011.10.001Kusraeva, Z. A. (2018). Powers of quasi-Banach lattices and orthogonally additive polynomials. Journal of Mathematical Analysis and Applications, 458(1), 767-780. doi:10.1016/j.jmaa.2017.09.01

New trends on the numerical representability of semiordered structures

[EN] We introduce a survey, including the historical back-ground, on different techniques that have recently been issued in the search for a characterization of the representability of semiordered structures, in the sense of Scott and Suppes, by means of a real-valued function and a strictly positive threshold of discrimination.This work has been supported by the research projects MTM2007-62499, ECO2008-01297, MTM2009-12872-C02-02 and MTM2010-17844 (Spain)Abrísqueta, F.; Campión, M.; Catalán, R.; De Miguel, J.; Estevan, A.; Induráin, E.; Zudaire, M.... (2012). New trends on the numerical representability of semiordered structures. Mathware & Soft Computing Magazine. 19(1):25-37. http://hdl.handle.net/10251/57632S253719

Loss of the sphingolipid desaturase DEGS1 causes hypomyelinating leukodystrophy.

Sphingolipid imbalance is the culprit in a variety of neurological diseases, some affecting the myelin sheath. We have used whole-exome sequencing in patients with undetermined leukoencephalopathies to uncover the endoplasmic reticulum lipid desaturase DEGS1 as the causative gene in 19 patients from 13 unrelated families. Shared features among the cases include severe motor arrest, early nystagmus, dystonia, spasticity, and profound failure to thrive. MRI showed hypomyelination, thinning of the corpus callosum, and progressive thalamic and cerebellar atrophy, suggesting a critical role of DEGS1 in myelin development and maintenance. This enzyme converts dihydroceramide (DhCer) into ceramide (Cer) in the final step of the de novo biosynthesis pathway. We detected a marked increase of the substrate DhCer and DhCer/Cer ratios in patients' fibroblasts and muscle. Further, we used a knockdown approach for disease modeling in Danio rerio, followed by a preclinical test with the first-line treatment for multiple sclerosis, fingolimod (FTY720, Gilenya). The enzymatic inhibition of Cer synthase by fingolimod, 1 step prior to DEGS1 in the pathway, reduced the critical DhCer/Cer imbalance and the severe locomotor disability, increasing the number of myelinating oligodendrocytes in a zebrafish model. These proof-of-concept results pave the way to clinical translation

Ghrelin and its receptors in Gglthead Sea bream: nutritional regulation

Ghrelin is involved in the regulation of growth in vertebrates through controlling different functions, such as feed intake, metabolism, intestinal activity or growth hormone (Gh) secretion. The aim of this work was to identify the sequences of preproghrelin and Ghrelin receptors (ghsrs), and to study their responses to different nutritional conditions in gilthead sea bream (Sparus aurata) juveniles. The structure and phylogeny of S. aurata preproghrelin was analyzed, and a tissue screening was performed. The effects of 21 days of fasting and 2, 5, 24 h, and 7 days of refeeding on plasma levels of Ghrelin, Gh and Igf-1, and the gene expression of preproghrelin, ghsrs and members of the Gh/Igf-1 system were determined in key tissues. preproghrelin and the receptors are well conserved, being expressed mainly in stomach, and in the pituitary and brain, respectively. Twenty-one days of fasting resulted in a decrease in growth while Ghrelin plasma levels were elevated to decrease at 5 h post-prandial when pituitary ghsrs expression was minimum. Gh in plasma increased during fasting and slowly felt upon refeeding, while plasma Igf-1 showed an inverse profile. Pituitary gh expression augmented during fasting reaching maximum levels at 1 day post-feeding while liver igf-1 expression and that of its splice variants decreased to lowest levels. Liver Gh receptors expression was down-regulated during fasting and recovered after refeeding. This study demonstrates the important role of Ghrelin during fasting, its acute down-regulation in the post-prandial stage and its interaction with pituitary Ghsrs and Gh/Igf-1 axis

Voltage-Gated Ion Channel Dysfunction Precedes Cardiomyopathy Development in the Dystrophic Heart

Duchenne muscular dystrophy (DMD), caused by mutations in the dystrophin gene, is associated with severe cardiac complications including cardiomyopathy and cardiac arrhythmias. Recent research suggests that impaired voltage-gated ion channels in dystrophic cardiomyocytes accompany cardiac pathology. It is, however, unknown if the ion channel defects are primary effects of dystrophic gene mutations, or secondary effects of the developing cardiac pathology.To address this question, we first investigated sodium channel impairments in cardiomyocytes derived from dystrophic neonatal mice prior to cardiomyopahty development, by using the whole cell patch clamp technique. Besides the most common model for DMD, the dystrophin-deficient mdx mouse, we also used mice additionally carrying an utrophin mutation. In neonatal cardiomyocytes, dystrophin-deficiency generated a 25% reduction in sodium current density. In addition, extra utrophin-deficiency significantly altered sodium channel gating parameters. Moreover, also calcium channel inactivation was considerably reduced in dystrophic neonatal cardiomyocytes, suggesting that ion channel abnormalities are universal primary effects of dystrophic gene mutations. To assess developmental changes, we also studied sodium channel impairments in cardiomyocytes derived from dystrophic adult mice, and compared them with the respective abnormalities in dystrophic neonatal cells. Here, we found a much stronger sodium current reduction in adult cardiomyocytes. The described sodium channel impairments slowed the upstroke of the action potential in adult cardiomyocytes, and only in dystrophic adult mice, the QRS interval of the electrocardiogram was prolonged.Ion channel impairments precede pathology development in the dystrophic heart, and may thus be considered potential cardiomyopathy triggers

SNi from SN2: a front-face mechanism ‘synthase’ engineered from a retaining hydrolase

SNi or SNi-like mechanisms, in which leaving group departure and nucleophile approach occur on the same ‘front’ face, have been observed previously experimentally and computationally in both the chemical and enzymatic (glycosyltransferase) substitution reactions of α-glycosyl electrophiles. Given the availability of often energetically comparable competing pathways for substitution (SNi vs SN1 vs SN2) the precise modulation of this archetypal reaction type should be feasible. Here, we show that the drastic engineering of a protein that catalyzes substitution, a retaining β-glycosidase (from Sulfolobus solfataricus SSβG), apparently changes the mode of reaction from “SN2” to “SNi”. Destruction of the nucleophilic Glu387 of SSβG-WT through Glu387Tyr mutation (E387Y) created a catalyst (SSβG-E387Y) with lowered but clear transglycosylation substitution activity with activated substrates, altered substrate and reaction preferences and hence useful synthetic (‘synthase’) utility by virtue of its low hydrolytic activity with unactivated substrates. Strikingly, the catalyst still displayed retaining β stereoselectivity, despite lacking a suitable nucleophile; pH-activity profile, mechanism-based inactivators and mutational analyses suggest that SSβG-E387Y operates without either the use of nucleophile or general acid/base residues, consistent with a SNi or SNi-like mechanism. An x-ray structure of SSβG-E387Y and subsequent metadynamics simulation suggest recruitment of substrates aided by a π-sugar interaction with the introduced Tyr387 and reveal a QM/MM free energy landscape for the substitution reaction catalyzed by this unnatural enzyme similar to those of known natural, SNi-like glycosyltransferase (GT) enzymes. Proton flight from the putative hydroxyl nucleophile to the developing p-nitrophenoxide leaving group of the substituted molecule in the reactant complex creates a hydrogen bond that appears to crucially facilitate the mechanism, mimicking the natural mechanism of SNi-GTs. An oxocarbenium ion-pair minimum along the reaction pathway suggests a step-wise SNi-like DN*ANss rather than a concerted SNi DNAN mechanism. This first observation of a front face mechanism in a β-retaining glycosyl transfer enzyme highlights, not only that unusual SNi reaction pathways may be accessed through direct engineering of catalysts with suitable environments, but also suggests that ‘β-SNi’ reactions are also feasible for glycosyl transfer enzymes and the more widespread existence of SNi or SNi-like mechanism in nature

Clinical rating scale for pantothenate kinase-associated neurodegeneration: A pilot study

Pantothenate kinase-associated neurodegeneration is a progressive neurological disorder occurring in both childhood and adulthood. The objective of this study was to design and pilot-test a disease-specific clinical rating scale for the assessment of patients with pantothenate kinase-associated neurodegeneration.info:eu-repo/semantics/publishedVersio

Variations in TcdB Activity and the Hypervirulence of Emerging Strains of Clostridium difficile

Hypervirulent strains of Clostridium difficile have emerged over the past decade, increasing the morbidity and mortality of patients infected by this opportunistic pathogen. Recent work suggested the major C. difficile virulence factor, TcdB, from hypervirulent strains (TcdBHV) was more cytotoxic in vitro than TcdB from historical strains (TcdBHIST). The current study investigated the in vivo impact of altered TcdB tropism, and the underlying mechanism responsible for the differences in activity between the two forms of this toxin. A combination of protein sequence analyses, in vivo studies using a Danio rerio model system, and cell entry combined with fluorescence assays were used to define the critical differences between TcdBHV and TcdBHIST. Sequence analysis found that TcdB was the most variable protein expressed from the pathogenicity locus of C. difficile. In line with these sequence differences, the in vivo effects of TcdBHV were found to be substantially broader and more pronounced than those caused by TcdBHIST. The increased toxicity of TcdBHV was related to the toxin's ability to enter cells more rapidly and at an earlier stage in endocytosis than TcdBHIST. The underlying biochemical mechanism for more rapid cell entry was identified in experiments demonstrating that TcdBHV undergoes acid-induced conformational changes at a pH much higher than that of TcdBHIST. Such pH-related conformational changes are known to be the inciting step in membrane insertion and translocation for TcdB. These data provide insight into a critical change in TcdB activity that contributes to the emerging hypervirulence of C. difficile