Maharam-type kernel representation for operators with a trigonometric domination

[EN] Consider a linear and continuous operator T between Banach function spaces. We prove that under certain requirements an integral inequality for T is equivalent to a factorization of T through a specific kernel operator: in other words, the operator T has what we call a Maharam-type kernel representation. In the case that the inequality provides a domination involving trigonometric functions, a special factorization through the Fourier operator is given. We apply this result to study the problem that motivates the paper: the approximation of functions in L2[0, 1] by means of trigonometric series whose Fourier coefficients are given by weighted trigonometric integrals.This research has been supported by MTM2016-77054-C2-1-P (Ministerio de Economia, Industria y Competitividad, Spain).Sánchez Pérez, EA. (2017). Maharam-type kernel representation for operators with a trigonometric domination. Aequationes Mathematicae. 91(6):1073-1091. https://doi.org/10.1007/s00010-017-0507-6S10731091916Calabuig, J.M., Delgado, O., Sánchez Pérez, E.A.: Generalized perfect spaces. Indag. Math. 19(3), 359–378 (2008)Calabuig, J.M., Delgado, O., Sánchez Pérez, E.A.: Factorizing operators on Banach function spaces through spaces of multiplication operators. J. Math. Anal. Appl. 364, 88–103 (2010)Delgado, O., Sánchez Pérez, E.A.: Strong factorizations between couples of operators on Banach function spaces. J. Convex Anal. 20(3), 599–616 (2013)Dodds, P.G., Huijsmans, C.B., de Pagter, B.: Characterizations of conditional expectation type operators. Pacific J. Math. 141(1), 55–77 (1990)Flores, J., Hernández, F.L., Tradacete, P.: Domination problems for strictly singular operators and other related classes. Positivity 15(4), 595–616 (2011). 2011Fremlin, D.H.: Tensor products of Banach lattices. Math. Ann. 211, 87–106 (1974)Hu, G.: Weighted norm inequalities for bilinear Fourier multiplier operators. Math. Ineq. Appl. 18(4), 1409–1425 (2015)Halmos, P., Sunder, V.: Bounded Integral Operators on $$L^2$$ L 2 Spaces. Springer, Berlin (1978)Kantorovitch, L., Vulich, B.: Sur la représentation des opérations linéaires. Compositio Math. 5, 119–165 (1938)Kolwicz, P., Leśnik, K., Maligranda, L.: Pointwise multipliers of Calderón- Lozanovskii spaces. Math. Nachr. 286, 876–907 (2013)Kolwicz, P., Leśnik, K., Maligranda, L.: Pointwise products of some Banach function spaces and factorization. J. Funct. Anal. 266(2), 616–659 (2014)Kuo, W.-C., Labuschagne, C.C.A., Watson, B.A.: Conditional expectations on Riesz spaces. J. Math. Anal. Appl. 303, 509–521 (2005)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1979)Maharam, D.: The representation of abstract integrals. Trans. Am. Math. Soc. 75, 154–184 (1953)Maharam, D.: On kernel representation of linear operators. Trans. Am. Math. Soc. 79, 229–255 (1955)Maligranda, L., Persson, L.E.: Generalized duality of some Banach function spaces. Indag. Math. 51, 323–338 (1989)Neugebauer, C.J.: Weighted norm inequalities for averaging operators of monotone functions. Publ. Mat. 35, 429–447 (1991)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal Domain and Integral Extension of Operators Acting in Function Spaces. Operator Theory: Adv. Appl., vol. 180. Birkhäuser, Basel (2008)Rota, G.C.: On the representation of averaging operators. Rend. Sem. Mat. Univ. Padova. 30, 52–64 (1960)Sánchez Pérez, E.A.: Factorization theorems for multiplication operators on Banach function spaces. Integr. Equ. Oper. Theory 80(1), 117–135 (2014)Schep, A.R.: Factorization of positive multilinear maps. Ill. J. Math. 28(4), 579–591 (1984)Schep, A.R.: Products and factors of Banach function spaces. Positivity 14(2), 301–319 (2010

NucTools: analysis of chromatin feature occupancy profiles from high-throughput sequencing data

Background: Biomedical applications of high-throughput sequencing methods generate a vast amount of data in which numerous chromatin features are mapped along the genome. The results are frequently analysed by creating binary data sets that link the presence/absence of a given feature to specific genomic loci. However, the nucleosome occupancy or chromatin accessibility landscape is essentially continuous. It is currently a challenge in the field to cope with continuous distributions of deep sequencing chromatin readouts and to integrate the different types of discrete chromatin features to reveal linkages between them. Results: Here we introduce the NucTools suite of Perl scripts as well as MATLAB- and R-based visualization programs for a nucleosome-centred downstream analysis of deep sequencing data. NucTools accounts for the continuous distribution of nucleosome occupancy. It allows calculations of nucleosome occupancy profiles averaged over several replicates, comparisons of nucleosome occupancy landscapes between different experimental conditions, and the estimation of the changes of integral chromatin properties such as the nucleosome repeat length. Furthermore, NucTools facilitates the annotation of nucleosome occupancy with other chromatin features like binding of transcription factors or architectural proteins, and epigenetic marks like histone modifications or DNA methylation. The applications of NucTools are demonstrated for the comparison of several datasets for nucleosome occupancy in mouse embryonic stem cells (ESCs) and mouse embryonic fibroblasts (MEFs). Conclusions: The typical workflows of data processing and integrative analysis with NucTools reveal information on the interplay of nucleosome positioning with other features such as for example binding of a transcription factor CTCF, regions with stable and unstable nucleosomes, and domains of large organized chromatin K9me2 modifications (LOCKs). As potential limitations and problems we discuss how inter-replicate variability of MNase-seq experiments can be addressed

Product factorability of integral bilinear operators on Banach function spaces

[EN] This paper deals with bilinear operators acting in pairs of Banach function spaces that factor through the pointwise product. We find similar situations in different contexts of the functional analysis, including abstract vector lattices¿orthosymmetric maps, C¿-algebras¿zero product preserving operators, and classical and harmonic analysis¿integral bilinear operators. Bringing together the ideas of these areas, we show new factorization theorems and characterizations by means of norm inequalities. The objective of the paper is to apply these tools to provide new descriptions of some classes of bilinear integral operators, and to obtain integral representations for abstract classes of bilinear maps satisfying certain domination properties.The first author was supported by TUBITAK-The Scientific and Technological Research Council of Turkey, Grant No. 2211/E. The second author was supported by Ministerio de Economia y Competitividad (Spain) and FEDER, Grant MTM2016-77054-C2-1-P.Erdogan, E.; Sánchez Pérez, EA.; Gok, O. (2019). Product factorability of integral bilinear operators on Banach function spaces. Positivity. 23(3):671-696. https://doi.org/10.1007/s11117-018-0632-zS671696233Abramovich, Y.A., Kitover, A.K.: Inverses of Disjointness Preserving Operators. American Mathematical Society, Providence (2000)Abramovich, Y.A., Wickstead, A.W.: When each continuous operator is regular II. Indag. Math. (N.S.) 8(3), 281–294 (1997)Alaminos, J., Brešar, M., Extremera, J., Villena, A.R.: Maps preserving zero products. Studia Math. 193(2), 131–159 (2009)Alaminos, J., Brešar, M., Extremera, J., Villena, A.R.: On bilinear maps determined by rank one idempotents. Linear Algebra Appl. 432, 738–743 (2010)Alaminos, J., Extremera, J., Villena, A.R.: Orthogonality preserving linear maps on group algebras. Math. Proc. Camb. Philos. 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Displaced midshaft fractures of the clavicle: non-operative treatment versus plate fixation (Sleutel-TRIAL). A multicentre randomised controlled trial

Contains fulltext : 96826.pdf (publisher's version ) (Open Access)BACKGROUND: The traditional view that the vast majority of midshaft clavicular fractures heal with good functional outcomes following non-operative treatment may be no longer valid for all midshaft clavicular fractures. Recent studies have presented a relatively high incidence of non-union and identified speciic limitations of the shoulder function in subgroups of patients with these injuries. AIM: A prospective, multicentre randomised controlled trial (RCT) will be conducted in 21 hospitals in the Netherlands, comparing fracture consolidation and shoulder function after either non-operative treatment with a sling or a plate fixation. METHODS/DESIGN: A total of 350 patients will be included, between 18 and 60 years of age, with a dislocated midshaft clavicular fracture. The primary outcome is the incidence of non-union, which will be determined with standardised X-rays (Antero-Posterior and 30 degrees caudocephalad view). Secondary outcome will be the functional outcome, measured using the Constant Score. Strength of the shoulder muscles will be measured with a handheld dynamometer (MicroFET2). Furthermore, the health-related Quality of Life score (ShortForm-36) and the Disabilities of Arm, Shoulder and Hand (DASH) Outcome Measure will be monitored as subjective parameters. Data on complications, bone union, cosmetic aspects and use of painkillers will be collected with follow-up questionnaires. The follow-up time will be two years. All patients will be monitored at regular intervals over the subsequent twelve months (two and six weeks, three months and one year). After two years an interview by telephone and a written survey will be performed to evaluate the two-year functional and mechanical outcomes. All data will be analysed on an intention-to-treat basis, using univariate and multivariate analyses. DISCUSSION: This trial will provide level-1 evidence for the comparison of consolidation and functional outcome between two standardised treatment options for dislocated midshaft clavicular fractures. The gathered data may support the development of a clinical guideline for treatment of clavicular fractures. TRIAL REGISTRATION: Netherlands National Trial Register NTR2399