Serre Duality, Abel's Theorem, and Jacobi Inversion for Supercurves Over a Thick Superpoint

The principal aim of this paper is to extend Abel's theorem to the setting of complex supermanifolds of dimension 1|q over a finite-dimensional local supercommutative C-algebra. The theorem is proved by establishing a compatibility of Serre duality for the supercurve with Poincare duality on the reduced curve. We include an elementary algebraic proof of the requisite form of Serre duality, closely based on the account of the reduced case given by Serre in Algebraic Groups and Class Fields, combined with an invariance result for the topology on the dual of the space of repartitions. Our Abel map, taking Cartier divisors of degree zero to the dual of the space of sections of the Berezinian sheaf, modulo periods, is defined via Penkov's characterization of the Berezinian sheaf as the cohomology of the de Rham complex of the sheaf D of differential operators, as a right module over itself. We discuss the Jacobi inversion problem for the Abel map and give an example demonstrating that if n is an integer sufficiently large that the generic divisor of degree n is linearly equivalent to an effective divisor, this need not be the case for all divisors of degree n.Comment: 14 page

Deeper Into the Knight: Exploring Deans Knight and Its Effects on the Canadian GAAR

This article discusses the most recent decision of the Supreme Court of Canada in Deans Knight Income Corporation v. Canada (2023) and explores its implications for the Canadian GAAR

Degeneration and impaired regeneration of gray matter oligodendrocytes in amyotrophic lateral sclerosis.

Oligodendrocytes associate with axons to establish myelin and provide metabolic support to neurons. In the spinal cord of amyotrophic lateral sclerosis (ALS) mice, oligodendrocytes downregulate transporters that transfer glycolytic substrates to neurons and oligodendrocyte progenitors (NG2(+) cells) exhibit enhanced proliferation and differentiation, although the cause of these changes in oligodendroglia is unknown. We found extensive degeneration of gray matter oligodendrocytes in the spinal cord of SOD1 (G93A) ALS mice prior to disease onset. Although new oligodendrocytes were formed, they failed to mature, resulting in progressive demyelination. Oligodendrocyte dysfunction was also prevalent in human ALS, as gray matter demyelination and reactive changes in NG2(+) cells were observed in motor cortex and spinal cord of ALS patients. Selective removal of mutant SOD1 from oligodendroglia substantially delayed disease onset and prolonged survival in ALS mice, suggesting that ALS-linked genes enhance the vulnerability of motor neurons and accelerate disease by directly impairing the function of oligodendrocytes

D-modules on 1|1 Supercurves

It is known that to every 1|1 dimensional supercurve X there is associated a dual supercurve \hat{X}, and a superdiagonal \Delta in their product. We establish that the categories of D-modules on X, \hat{X}, and \Delta are equivalent. This follows from a more general result about D-modules and purely odd submersions. The equivalences preserve tensor products, and take vector bundles to vector bundles. Line bundles with connection are studied, and examples are given where X is a superelliptic curve.Comment: 18 page

Arachidonic Acid Activates a Proton Current in the Rat Glutamate Transporter EAAT4

The excitatory amino acid transporter EAAT4 is expressed predominantly in Purkinje neurons in the rat cerebellum (1-3), and it participates in postsynaptic reuptake of glutamate released at the climbing fiber synapse (4). Transporter-mediated currents in Purkinje neurons are increased more than 3-fold by arachidonic acid, a second messenger that is liberated following depolarization-induced Ca2+ activation of phospholipase A2 (5). In this study we demonstrate that application of arachidonic acid to oocytes expressing rat EAAT4 increased glutamate-induced currents to a similar extent. However, arachidonic acid did not cause an increase in the rate of glutamate transport or in the chloride current associated with glutamate transport but rather activated a proton-selective conductance. These data reveal a novel action of arachidonate on a glutamate transporter and suggest a mechanism by which synaptic activity may decrease intracellular pH in neurons where this transporter is localized

Aberrant RNA Processing in a Neurodegenerative Disease: the Cause for Absent EAAT2, a Glutamate Transporter, in Amyotrophic Lateral Sclerosis

AbstractAmyotrophic lateral sclerosis (ALS) is a neurodegenerative disease that is characterized by selective upper and lower motor neuron degeneration, the pathogenesis of which is unknown. About 60%–70% of sporadic ALS patients have a 30%–95% loss of the astroglial glutamate transporter EAAT2 (excitatory amino acid transporter 2) protein in motor cortex and spinal cord. Loss of EAAT2 leads to increased extracellular glutamate and excitotoxic neuronal degeneration. Multiple abnormal EAAT2 mRNAs, including intron-retention and exon-skipping, have now been identified from the affected areas of ALS patients. The aberrant mRNAs were highly abundant and were found only in neuropathologically affected areas of ALS patients but not in other brain regions. They were found in 65% of sporadic ALS patients but were not found in nonneurologic disease or other disease controls. They were also detectable in the cerebrospinal fluid (CSF) of living ALS patients, early in the disease. In vitro expression studies suggest that proteins translated from these aberrant mRNAs may undergo rapid degradation and/or produce a dominant negative effect on normal EAAT2 resulting in loss of protein and activity. These findings suggest that the loss of EAAT2 in ALS is due to aberrant mRNA and that these aberrant mRNAs could result from RNA processing errors. Aberrant RNA processing could be important in the pathophysiology of neurodegenerative disease and in excitotoxicity. The presence of these mRNA species in ALS CSF may have diagnostic utility

Deciphering Amyotrophic Lateral Sclerosis: What Phenotype, Neuropathology and Genetics Are Telling Us about Pathogenesis

Amyotrophic lateral sclerosis (ALS) is characterized phenotypically by progressive weakness and neuropathologically by loss of motor neurons. Phenotypically, there is marked heterogeneity. Typical ALS has mixed upper motor neuron (UMN) and lower motor neuron (LMN) involvement. Primary lateral sclerosis has predominant UMN involvement. Progressive muscular atrophy has predominant LMN involvement. Bulbar and limb ALS have predominant regional involvement. Frontotemporal dementia has significant cognitive and behavioral involvement. These phenotypes can be so distinctive that they would seem to have differing biology. But they cannot be distinguished, at least neuropathologically or genetically. In sporadic ALS (SALS), they all are characterized by ubiquitinated cytoplasmic inclusions of TDP-43. In familial ALS (FALS), where phenotypes are indistinguishable from SALS and similarly heterogeneous, each mutated gene has its own genetic and molecular signature. Putting this together, since the same phenotypes can have multiple causes including different gene mutations, there must be multiple molecular mechanisms causing ALS and ALS is a syndrome. But since multiple phenotypes can be caused by one single gene mutation, a single molecular mechanism can cause heterogeneity. What the mechanisms are remain unknown, but active propagation of the pathology neuroanatomically seems to be a principle component. Leading candidate mechanisms include RNA processing, cell-cell interactions between neurons and non-neuronal neighbors, focal seeding from a misfolded protein that has prion-like propagation, and fatal errors introduced during neurodevelopment of the motor system. If fundamental mechanisms can be identified and understood, ALS therapy could rationally target progression and stop disease—a goal that seems increasingly achievable.Stem Cell and Regenerative Biolog

Counteridenticals

A counteridentical is a counterfactual with an identity statement in the antecedent. While counteridenticals generally seem non-trivial, most semantic theories for counterfactuals, when combined with the necessity of identity and distinctness, attribute vacuous truth conditions to such counterfactuals. In light of this, one could try to save the orthodox theories either by appealing to pragmatics or by denying that the antecedents of alleged counteridenticals really contain identity claims. Or one could reject the orthodox theory of counterfactuals in favor of a hyperintensional semantics that accommodates non-trivial counterpossibles. In this paper, I argue that none of these approaches can account for all the peculiar features of counteridenticals. Instead, I propose a modified version of Lewis’s counterpart theory, which rejects the necessity of identity, and show that it can explain all the peculiar features of counteridenticals in a satisfactory way. I conclude by defending the plausibility of contingent identity from objections

Design and Initial Results of a Multi-Phase Randomized Trial of Ceftriaxone in Amyotrophic Lateral Sclerosis

Objectives: Ceftriaxone increases expression of the astrocytic glutamate transporter, EAAT2, which might protect from glutamate-mediated excitotoxicity. A trial using a novel three stage nonstop design, incorporating Phases I-III, tested ceftriaxone in ALS. Stage 1 determined the cerebrospinal fluid pharmacokinetics of ceftriaxone in subjects with ALS. Stage 2 evaluated safety and tolerability for 20-weeks. Analysis of the pharmacokinetics, tolerability, and safety was used to determine the ceftriaxone dosage for Stage 3 efficacy testing. Methods: In Stage 1, 66 subjects at ten clinical sites were enrolled and randomized equally into three study groups receiving intravenous placebo, ceftriaxone 2 grams daily or ceftriaxone 4 grams daily divided BID. Participants provided serum and cerebrospinal fluid for pharmacokinetic analysis on study day 7. Participants continued their assigned treatment in Stage 2. The Data and Safety Monitoring Board (DSMB) reviewed the data after the last participants completed 20 weeks on study drug. Results: Stage 1 analysis revealed linear pharmacokinetics, and CSF trough levels for both dosage levels exceeding the pre-specified target trough level of 1 µM (0.55 µg/mL). Tolerability (Stages 1 and 2) results showed that ceftriaxone at dosages up to 4 grams/day was well tolerated at 20 weeks. Biliary adverse events were more common with ceftriaxone but not dose-dependent and improved with ursodeoxycholic (ursodiol) therapy. Conclusions: The goals of Stages 1 and 2 of the ceftriaxone trial were successfully achieved. Based on the pre-specified decision rules, the DSMB recommended the use of ceftriaxone 4 g/d (divided BID) for Stage 3, which recently closed. Trial Registration ClinicalTrials.gov NCT00349622