Hopf algebras and the logarithm of the S-transform in free probability

Let k be a positive integer and let G_k denote the set of non-commutative k-variable distributions \mu such that \mu (X_1) = ... = \mu (X_k) = 1. G_k is a group under the operation of free multiplicative convolution. We identify G_k as the group of characters of a certain Hopf algebra Y_k. Then, by using the log map from characters to infinitesimal characters of Y_k, we introduce a transform LS_{\mu} for distributions \mu in G_k. The main property of the LS-transform is that it linearizes commuting products in G_k. For \mu in G_k, the transform LS_{\mu} is a power series in k non-commuting indeterminates; its coefficients can be computed from the coefficients of the R-transform of \mu by using summations over chains in the lattices NC(n) of non-crossing partitions. In the particular case k=1 one has that Y_1 is naturally isomorphic to the Hopf algebra Sym of symmetric functions, and that the LS-transform is very closely related to the logarithm of the S-transform of Voiculescu, by the formula LS(z) = - z log S(z). In this case the group G_1 can be identified as the group of characters of Sym, in such a way that the S-transform, its reciprocal 1/S and its logarithm log S relate in a natural sense to the sequences of complete, elementary and respectively power sum symmetric functions.Comment: Version with minor revisions, to appear in Transactions of the American Mathematical Society. 39 pages, no figure

Additive Manufactured Scaffolds for Bone Tissue Engineering: Physical Characterization of Thermoplastic Composites with Functional Fillers

Thermoplastic polymer–filler composites are excellent materials for bone tissue engineering (TE) scaffolds, combining the functionality of fillers with suitable load-bearing ability, biodegradability, and additive manufacturing (AM) compatibility of the polymer. Two key determinants of their utility are their rheological behavior in the molten state, determining AM processability and their mechanical load-bearing properties. We report here the characterization of both these physical properties for four bone TE relevant composite formulations with poly(ethylene oxide terephthalate)/poly(butylene terephthalate (PEOT/PBT) as a base polymer, which is often used to fabricate TE scaffolds. The fillers used were reduced graphene oxide (rGO), hydroxyapatite (HA), gentamicin intercalated in zirconium phosphate (ZrP-GTM) and ciprofloxacin intercalated in MgAl layered double hydroxide (MgAl-CFX). The rheological assessment showed that generally the viscous behavior dominated the elastic behavior (G″ > G′) for the studied composites, at empirically determined extrusion temperatures. Coupled rheological–thermal characterization of ZrP-GTM and HA composites showed that the fillers increased the solidification temperatures of the polymer melts during cooling. Both these findings have implications for the required extrusion temperatures and bonding between layers. Mechanical tests showed that the fillers generally not only made the polymer stiffer but more brittle in proportion to the filler fractions. Furthermore, the elastic moduli of scaffolds did not directly correlate with the corresponding bulk material properties, implying composite-specific AM processing effects on the mechanical properties. Finally, we show computational models to predict multimaterial scaffold elastic moduli using measured single material scaffold and bulk moduli. The reported characterizations are essential for assessing the AM processability and ultimately the suitability of the manufactured scaffolds for the envisioned bone regeneration application.The work was supported by a Horizon 2020 Research and Innovation Programme grant from the European Union, called the FAST project (grant no. 685825, project website: http:// project-fast.eu). The authors acknowledge the support of the FAST project consortium for the various aspects of this wor

REM-Sleep Behavior Disorder in Patients With Essential Tremor: What Is Its Clinical Significance?

Objective: REM sleep behavior disorder (RBD) is an important risk factor for the dementia development and for the deterioration of autonomic functions in patients with Parkinson's Disease. RBD has also been reported in patients with Essential Tremor (ET). However, its clinical significance in ET remains still unknown. We aimed to investigate clinical, neuropsychological and cardiac autonomic scintigraphic differences between ET patients with and without RBD.Methods: To assess RBD symptoms, RBD Single-Question has been administered in a cohort of 55 patients with a clinical diagnosis of ET. Patients with clinical RBD underwent polysomnography (PSG) confirmation. All patients completed a battery of neuropsychological assessment of memory, executive function, attention, language, and visuospatial function. Cardiac MIBG scintigraphy was performed in order to measure the cardiac autonomic innervation.Results: Ten ET patients (18%) had a PSG-confirmed RBD (ETRBD+). Compared to ET patients without RBD (ETRBD−), significantly reduced scores on memory domain tests such as Rey auditory verbal learning test immediate recall (p = 0.015) and Rey auditory verbal learning test delayed recall (p = 0.004) and phonemic fluency test (p = 0.028) were present in ETRBD+. By contrast, no other significant clinical difference has emerged from the comparison between two ET groups. Similarly, ETRBD+ patients have cardiac MIBG tracer uptake in the normal value range as occurred in those with ETRBD−.Conclusions: This study improves the knowledge on clinical significance of RBD symptoms in ET patients. Our preliminary findings demonstrate that presence of RBD in ET is associated with neurocognitive impairment, but not with cardiac autonomic dysfunction. Further longitudinal studies are needed to investigate whether ET patients with RBD will develop a frank dementia over the time

The impact of cultural diversity on the innovation performance of MNC subsidiaries in strategic alliances

The strategy literature has widely acknowledged the negative impact of cultural diversity between the partners of an alliance on their innovation performance. We argue that innovation is even more challenging in alliances involving subsidiaries of multinational companies (MNCs), as they embody a dual background that encompasses the cultures of their host country and their home country. We also propose that the effect of cultural diversity is contingent on the specific content of the alliance, and that it is positive in explorative alliances and negative in exploitative alliances. Our findings, which are obtained from an analysis of 161 strategic alliances established by 31 MNC subsidiaries in the biotech industry during the period 1987 to 2010, confirm that subsidiaries are generally less innovative in alliances involving partners from other cultures. However, the impact of such cultural diversity becomes positive when those alliances focus on exploration activities, as the challenges of cultural diversity are offset by the benefits of exposure to novel cognitive schemes

I Principali indicatori dell'innovazione a Milano

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