A Note on Riemann-Liouville Fractional Sobolev Spaces

Taking inspiration from a recent paper by Bergounioux et al., we study the Riemann-Liouville fractional Sobolev space W-RL,a+(s,p), for I = (a, b) for some a, b is an element of R, a < b, s is an element of (0,1) and p is an element of [1, infinity]; that is, the space of functions u is an element of L-P(I) such that the left Riemann-Liouville (1 - s)-fractional integral I-a+(1-s)[u] belongs to W-1,W-p (I). We prove that the space of functions of bounded variation BV(I) and the fractional Sobolev space W-s,W-1 (I) continuously embed into W-s,(1)(RL,a+) (I). In addition, we define the space of functions with left Riemann-Liouville s-fractional bounded variation, BVRL,a+s (I), as the set of functions u is an element of L-1(I) such that I-a+(1-s)[u] is an element of BV (I), and we analyze some fine properties of these functions. Finally, we prove some fractional Sobolev-type embedding results and we analyze the case of higher order Riemann-Liouville fractional derivatives

The prescribed mean curvature equation in weakly regular domains

We show that the characterization of existence and uniqueness up to vertical translations of solutions to the prescribed mean curvature equation, originally proved by Giusti in the smooth case, holds true for domains satisfying very mild regularity assumptions. Our results apply in particular to the non-parametric solutions of the capillary problem for perfectly wetting fluids in zero gravity. Among the essential tools used in the proofs, we mention a \textit{generalized Gauss-Green theorem} based on the construction of the weak normal trace of a vector field with bounded divergence, in the spirit of classical results due to Anzellotti, and a \textit{weak Young's law} for $(\Lambda,r_{0})$-minimizers of the perimeter.Comment: 23 pages, 1 figure --- The results on the weak normal trace of vector fields have been now extended and moved in a self-contained paper available at: arXiv:1708.0139

Peripheral blood biomarkers in multiple sclerosis.

Multiple sclerosis is the most common autoimmune disorder affecting the central nervous system. The heteroge-neity of pathophysiological processes in MS contributes to the highly variable course of the disease and unpre-dictable response to therapies. The major focus of the research on MS is the identification of biomarkers inbiologicalfluids, such as cerebrospinalfluid or blood, to guide patient management reliably. Because of the diffi-culties in obtaining spinalfluid samples and the necessity for lumbar puncture to make a diagnosis has reduced,the research of blood-based biomarkers may provide increasingly important tools for clinical practice. However,currently there are no clearly established MS blood-based biomarkers. The availability of reliable biomarkerscould radically alter the management of MS at critical phases of the disease spectrum, allowing for interventionstrategies that may prevent evolution to long-term neurological disability. This article provides an overview ofthis researchfield and focuses on recent advances in blood-based biomarker researc

Autism as a disorder of neural information processing: directions for research and targets for therapy

The broad variation in phenotypes and severities within autism spectrum disorders suggests the involvement of multiple predisposing factors, interacting in complex ways with normal developmental courses and gradients. Identification of these factors, and the common developmental path into which theyfeed, is hampered bythe large degrees of convergence from causal factors to altered brain development, and divergence from abnormal brain development into altered cognition and behaviour. Genetic, neurochemical, neuroimaging and behavioural findings on autism, as well as studies of normal development and of genetic syndromes that share symptoms with autism, offer hypotheses as to the nature of causal factors and their possible effects on the structure and dynamics of neural systems. Such alterations in neural properties may in turn perturb activity-dependent development, giving rise to a complex behavioural syndrome many steps removed from the root causes. Animal models based on genetic, neurochemical, neurophysiological, and behavioural manipulations offer the possibility of exploring these developmental processes in detail, as do human studies addressing endophenotypes beyond the diagnosis itself

Clinical correlates of grey matter pathology in multiple sclerosis

Traditionally, multiple sclerosis has been viewed as a disease predominantly affecting white matter. However, this view has lately been subject to numerous changes, as new evidence of anatomical and histological changes as well as of molecular targets within the grey matter has arisen. This advance was driven mainly by novel imaging techniques, however, these have not yet been implemented in routine clinical practice. The changes in the grey matter are related to physical and cognitive disability seen in individuals with multiple sclerosis. Furthermore, damage to several grey matter structures can be associated with impairment of specific functions. Therefore, we conclude that grey matter damage - global and regional - has the potential to become a marker of disease activity, complementary to the currently used magnetic resonance markers (global brain atrophy and T2 hyperintense lesions). Furthermore, it may improve the prediction of the future disease course and response to therapy in individual patients and may also become a reliable additional surrogate marker of treatment effect

Pathogenic Huntingtin Repeat Expansions in Patients with Frontotemporal Dementia and Amyotrophic Lateral Sclerosis.

We examined the role of repeat expansions in the pathogenesis of frontotemporal dementia (FTD) and amyotrophic lateral sclerosis (ALS) by analyzing whole-genome sequence data from 2,442 FTD/ALS patients, 2,599 Lewy body dementia (LBD) patients, and 3,158 neurologically healthy subjects. Pathogenic expansions (range, 40-64 CAG repeats) in the huntingtin (HTT) gene were found in three (0.12%) patients diagnosed with pure FTD/ALS syndromes but were not present in the LBD or healthy cohorts. We replicated our findings in an independent collection of 3,674 FTD/ALS patients. Postmortem evaluations of two patients revealed the classical TDP-43 pathology of FTD/ALS, as well as huntingtin-positive, ubiquitin-positive aggregates in the frontal cortex. The neostriatal atrophy that pathologically defines Huntington's disease was absent in both cases. Our findings reveal an etiological relationship between HTT repeat expansions and FTD/ALS syndromes and indicate that genetic screening of FTD/ALS patients for HTT repeat expansions should be considered

The underpinning biology relating to multiple sclerosis disease modifying treatments during the COVID-19 pandemic

Since January 2020 Elsevier has created a COVID-19 resource centre with free information in English and Mandarin on the novel coronavirus COVID-19. The COVID-19 resource centre is hosted on Elsevier Connect, the company's public news and information website. Elsevier hereby grants permission to make all its COVID-19-related research that is available on the COVID-19 resource centre - including this research content - immediately available in PubMed Central and other publicly funded repositories, such as the WHO COVID database with rights for unrestricted research re-use and analyses in any form or by any means with acknowledgement of the original source. These permissions are granted for free by Elsevier for as long as the COVID-19 resource centre remains active

Behavioral and Cognitive Improvement Induced by Novel Imidazoline I2 Receptor Ligands in Female SAMP8 Mice

As populations increase their life expectancy, age-related neurodegenerative disorders such as Alzheimer's disease have become more common. I2-Imidazoline receptors (I2-IR) are widely distributed in the central nervous system, and dysregulation of I2-IR in patients with neurodegenerative diseases has been reported, suggesting their implication in cognitive impairment. This evidence indicates that high-affinity selective I2-IR ligands potentially contribute to the delay of neurodegeneration. In vivo studies in the female senescence accelerated mouse-prone 8 mice have shown that treatment with I2-IR ligands, MCR5 and MCR9, produce beneficial effects in behavior and cognition. Changes in molecular pathways implicated in oxidative stress, inflammation, synaptic plasticity, and apoptotic cell death were also studied. Furthermore, treatments with these I2-IR ligands diminished the amyloid precursor protein processing pathway and increased Aβ degrading enzymes in the hippocampus of SAMP8 mice. These results collectively demonstrate the neuroprotective role of these new I2-IR ligands in a mouse model of brain aging through specific pathways and suggest their potential as therapeutic agents in brain disorders and age-related neurodegenerative diseases. Keywords Imidazoline I2 receptors (2-imidazolin-4-yl)phosphonates Behavior Cognition Neurodegeneration Neuroprotection Agin

On BV functions and essentially bounded divergence-measure fields in metric spaces

By employing the differential structure recently developed by N. Gigli, we first give a notion of functions of bounded variation (BV) in terms of suitable vector fields on a complete and separable metric measure space (X, d, mu) equipped with a non-negative Radon measure mu finite on bounded sets. Then, we extend the concept of divergence-measure vector fields D M-p (X) for any p is an element of [1, infinity] and, by simply requiring in addition that the metric space is locally compact, we determine an appropriate class of domains for which it is possible to obtain a Gauss-Green formula in terms of the normal trace of a D M-infinity (X) vector field. This differential machinery is also the natural framework to specialize our analysis for RCD(K, infinity) spaces, where we exploit the underlying geometry to determine the Leibniz rules for D M-infinity (X) and ultimately to extend our discussion on the Gauss-Green formulas