A constructive study of the module structure of rings of partial differential operators

The purpose of this paper is to develop constructive versions of Stafford's theorems on the module structure of Weyl algebras A n (k) (i.e., the rings of partial differential operators with polynomial coefficients) over a base field k of characteristic zero. More generally, based on results of Stafford and Coutinho-Holland, we develop constructive versions of Stafford's theorems for very simple domains D. The algorithmization is based on the fact that certain inhomogeneous quadratic equations admit solutions in a very simple domain. We show how to explicitly compute a unimodular element of a finitely generated left D-module of rank at least two. This result is used to constructively decompose any finitely generated left D-module into a direct sum of a free left D-module and a left D-module of rank at most one. If the latter is torsion-free, then we explicitly show that it is isomorphic to a left ideal of D which can be generated by two elements. Then, we give an algorithm which reduces the number of generators of a finitely presented left D-module with module of relations of rank at least two. In particular, any finitely generated torsion left D-module can be generated by two elements and is the homomorphic image of a projective ideal whose construction is explicitly given. Moreover, a non-torsion but non-free left D-module of rank r can be generated by r+1 elements but no fewer. These results are implemented in the Stafford package for D=A n (k) and their system-theoretical interpretations are given within a D-module approach. Finally, we prove that the above results also hold for the ring of ordinary differential operators with either formal power series or locally convergent power series coefficients and, using a result of Caro-Levcovitz, also for the ring of partial differential operators with coefficients in the field of fractions of the ring of formal power series or of the ring of locally convergent power series. © 2014 Springer Science+Business Media

Differential and holomorphic differential operators on noncommutative algebras

Abstract This paper deals with sheaves of differential operators on noncommutative algebras, in a manner related to the classical theory of D-modules. The sheaves are defined by quotienting the tensor algebra of vector fields (suitably deformed by a covariant derivative). As an example we can obtain enveloping algebra like relations for Hopf algebras with differential structures which are not bicovariant. Symbols of differential operators are defined, but not studied. These sheaves are shown to be in the center of a category of bimodules with flat bimodule covariant derivatives. Also holomorphic differential operators are considered

Nodes of Ranvier and Paranodes in Chronic Acquired Neuropathies

Chronic acquired neuropathies of unknown origin are classified as chronic inflammatory demyelinating polyneuropathies (CIDP) and chronic idiopathic axonal polyneuropathies (CIAP). The diagnosis can be very difficult, although it has important therapeutic implications since CIDP can be improved by immunomodulating treatment. The aim of this study was to examine the possible abnormalities of nodal and paranodal regions in these two types of neuropathies. Longitudinal sections of superficial peroneal nerves were obtained from biopsy material from 12 patients with CIDP and 10 patients with CIAP and studied by immunofluorescence and in some cases electron microscopy. Electron microscopy revealed multiple alterations in the nodal and paranodal regions which predominated in Schwann cells in CIDP and in axons in CIAP. In CIDP paranodin/Caspr immunofluorescence was more widespread than in control nerves, extending along the axon in internodes where it appeared intense. Nodal channels Nav and KCNQ2 were less altered but were also detected in the internodes. In CIAP paranodes, paranodin labeling was irregular and/or decreased. To test the consequences of acquired primary Schwann cells alteration on axonal proteins, we used a mouse model based on induced deletion of the transcription factor Krox-20 gene. In the demyelinated sciatic nerves of these mice we observed alterations similar to those found in CIDP by immunofluorescence, and immunoblotting demonstrated increased levels of paranodin. Finally we examined whether the alterations in paranodin immunoreactivity could have a diagnosis value. In a sample of 16 biopsies, the study of paranodin immunofluorescence by blind evaluators led to correct diagnosis in 70±4% of the cases. This study characterizes for the first time the abnormalities of nodes of Ranvier in CIAP and CIDP, and the altered expression and distribution of nodal and paranodal proteins. Marked differences were observed between CIDP and CIAP and the alterations in paranodin immunofluorescence may be an interesting tool for their differential diagnosis

Histopathological differences of myotonic dystrophy type 1 (DM1) and PROMM/DM2

Muscle biopsy findings in DM2 have been reported to be similar to those in DM1. The authors used myosin heavy chain immunohistochemistry and enzyme histochemistry for fiber type differentiation on muscle biopsies. Their results show that DM2 patients display a subpopulation of type 2 nuclear clump and other very small fibers and, hence, preferential type 2 fiber atrophy in contrast to type 1 fiber atrophy in DM1 patients

Cerebellar ataxia, neuropathy, vestibular areflexia syndrome due to RFC1 repeat expansion

Ataxia, causing imbalance, dizziness and falls, is a leading cause of neurological disability. We have recently identified a biallelic intronic AAGGG repeat expansion in replication factor complex subunit 1 (RFC1) as the cause of cerebellar ataxia, neuropathy, vestibular areflexia syndrome (CANVAS) and a major cause of late onset ataxia. Here we describe the full spectrum of the disease phenotype in our first 100 genetically confirmed carriers of biallelic repeat expansions in RFC1 and identify the sensory neuropathy as a common feature in all cases to date. All patients were Caucasian and half were sporadic. Patients typically reported progressive unsteadiness starting in the sixth decade. A dry spasmodic cough was also frequently associated and often preceded by decades the onset of walking difficulty. Sensory symptoms, oscillopsia, dysautonomia and dysarthria were also variably associated. The disease seems to follow a pattern of spatial progression from the early involvement of sensory neurons, to the later appearance of vestibular and cerebellar dysfunction. Half of the patients needed walking aids after 10 years of disease duration and a quarter were wheelchair dependent after 15 years. Overall, two-thirds of cases had full CANVAS. Sensory neuropathy was the only manifestation in 15 patients. Sixteen patients additionally showed cerebellar involvement, and six showed vestibular involvement. The disease is very likely to be underdiagnosed. Repeat expansion in RFC1 should be considered in all cases of sensory ataxic neuropathy, particularly, but not only, if cerebellar dysfunction, vestibular involvement and cough coexist

Complete event-by-event α/γ(β) separation in a full-size TeO2 CUORE bolometer by simultaneous heat and light detection

The CUORE project began recently a search for neutrinoless double-beta decay ($0\nu\beta\beta$) of $^{130}$Te with a $\mathcal{O}$(1 ton) TeO$_2$ bolometer array. In this experiment, the background suppression relies essentially on passive shielding, material radiopurity and anti-coincidences. The lack of particle identification in CUORE makes $\alpha$ decays at the detector surface the dominant background, at the level of $\sim$0.01 counts/(keV kg y) in the region of interest ($Q$-value of $0\nu\beta\beta$ of the order of 2.5 MeV). In the present work we demonstrate, for the first time with a CUORE-size (5$\times$5$\times$5 cm) TeO$_2$ bolometer and using the same technology as CUORE for the readout of the bolometric signals, an efficient $\alpha$ particle discrimination (99.9\%) with a high acceptance of the $0\nu\beta\beta$ signal (about 96\%). This unprecedented result was possible thanks to the superior performance (10 eV RMS baseline noise) of a Neganov-Luke-assisted germanium bolometer used to detect a tiny (70 eV) light signal dominated by $\gamma$($\beta$)-induced Cherenkov radiation in the TeO$_2$ detector. The obtained results represent a major breakthrough towards the TeO$_2$-based version of CUPID, a ton-scale cryogenic $0\nu\beta\beta$ experiment proposed as a follow-up to CUORE with particle identification

Biallelic mutations in SORD cause a common and potentially treatable hereditary neuropathy with implications for diabetes

Here we report biallelic mutations in the sorbitol dehydrogenase gene (SORD) as the most frequent recessive form of hereditary neuropathy. We identified 45 individuals from 38 families across multiple ancestries carrying the nonsense c.757delG (p.Ala253GlnfsTer27) variant in SORD, in either a homozygous or compound heterozygous state. SORD is an enzyme that converts sorbitol into fructose in the two-step polyol pathway previously implicated in diabetic neuropathy. In patient-derived fibroblasts, we found a complete loss of SORD protein and increased intracellular sorbitol. Furthermore, the serum fasting sorbitol levels in patients were dramatically increased. In Drosophila, loss of SORD orthologs caused synaptic degeneration and progressive motor impairment. Reducing the polyol influx by treatment with aldose reductase inhibitors normalized intracellular sorbitol levels in patient-derived fibroblasts and in Drosophila, and also dramatically ameliorated motor and eye phenotypes. Together, these findings establish a novel and potentially treatable cause of neuropathy and may contribute to a better understanding of the pathophysiology of diabetes

Muscular dystrophy with arrhythmia caused by loss-of-function mutations in BVES

Objective To study the genetic and phenotypic spectrum of patients harboring recessive mutations in BVES. Methods We performed whole-exome sequencing in a multicenter cohort of 1929 patients with a suspected hereditary myopathy, showing unexplained limb-girdle muscular weakness and/or elevated creatine kinase levels. Immunohistochemistry and mRNA experiments on patients' skeletal muscle tissue were performed to study the pathogenicity of identified loss-of-function (LOF) variants in BVES. Results We identified 4 individuals from 3 families harboring homozygous LOF variants in BVES, the gene that encodes for Popeye domain containing protein 1 (POPDC1). Patients showed skeletal muscle involvement and cardiac conduction abnormalities of varying nature and severity, but all exhibited at least subclinical signs of both skeletal muscle and cardiac disease. All identified mutations lead to a partial or complete loss of function of BVES through nonsense-mediated decay or through functional changes to the POPDC1 protein. Conclusions We report the identification of homozygous LOF mutations in BVES, causal in a young adult-onset myopathy with concomitant cardiac conduction disorders in the absence of structural heart disease. These findings underline the role of POPDC1, and by extension, other members of this protein family, in striated muscle physiology and disease. This disorder appears to have a low prevalence, although it is probably underdiagnosed because of its striking phenotypic variability and often subtle yet clinically relevant manifestations, particularly concerning the cardiac conduction abnormalities