Atomistic subsemirings of the lattice of subspaces of an algebra

Let A be an associative algebra with identity over a field k. An atomistic subsemiring R of the lattice of subspaces of A, endowed with the natural product, is a subsemiring which is a closed atomistic sublattice. When R has no zero divisors, the set of atoms of R is endowed with a multivalued product. We introduce an equivalence relation on the set of atoms such that the quotient set with the induced product is a monoid, called the condensation monoid. Under suitable hypotheses on R, we show that this monoid is a group and the class of k1_A is the set of atoms of a subalgebra of A called the focal subalgebra. This construction can be iterated to obtain higher condensation groups and focal subalgebras. We apply these results to G-algebras for G a group; in particular, we use them to define new invariants for finite-dimensional irreducible projective representations.Comment: 14 page

Alien Registration- Sage, Wilfrid D. (Rumford, Oxford County)

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Evaluation of the Effects of Various Exercise Interventions on Parkinson’s Disease

The overall purpose of the current thesis was to evaluate the influence of various exercise strategies on Parkinson’s disease (PD). While countless exercise interventions have been investigated by PD, results have been weak and inconclusive at best. As such, there are currently no scientifically-validated recommendations for an optimal exercise intervention. The four studies comprising this thesis have attempted to address the shortcomings of previous literature, namely, inconsistent use of outcome measures, lack of PD symptomatic measures, varying lengths of exercise interventions, absence of a non-exercise control group, continued assessment of participants after exercise has ended, and verifying replicabilty of findings. The first study was focused on identifying objective outcome measures that are predictive or reflective of the classic symptoms associated with PD. Symptomatic assessment was conducted using the Unified Parkinson’s disease Rating Scale (UPDRS), the current gold standard for assessment of PD symptom severity. Objective outcome measures included the timed-up-and-go (TUG), grooved pegboard (GP), and spatiotemporal aspects of self-paced gait (velocity, step length). Backward elimination regression analysis demonstrated that the place phase of the GP was the most predictive of UPDRS score. Interestingly, no objective outcome measures were strongly correlated with change on the symptomatic subsets that they were believed to be theoretically evaluating. The results point to the continued need to identify objective outcome measures reflective of symptomatic assessment. Further, exercise rehabilitation trials should combine outcome measures with symptomatic assessment to ensure that improvements are reflective of symptomatic improvement. The second study compared the influence of four exercise interventions (in contrast to a non-exercising control group) on the symptoms of PD. The exercise interventions included aquatic exercise, aerobic training, strength training and sensory focused exercise (PD SAFEx). Each participant exercised three times per week for a twelve week period and the same trained evaluator (blinded to group assignment) performed symptomatic assessment of all participants before exercise began (pre-test), after exercise ended (post-test) and following a minimum six week non-exercise period (washout). Results displayed that the strength training and PD SAFEx interventions had the greatest symptomatic benefit from pre-test to post-test. The aerobic intervention had no apparent change to symptom severity across all three testing periods. Overall, the current study suggests that PD SAFEx and strength training are more beneficial for individuals with PD than aerobic or aquatic exercise. The third study attempted to determine the influence of increased focus on sensory feedback by comparing two identical exercise interventions that differed only in the presence (PD SAFEx) or absence (non-SAFE) of increased attention on sensory (specifically proprioceptive) feedback. Symptomatic assessment was combined with objective outcome measures that assessed upper limb motor control, functional gait and spatiotemporal aspects of self-paced gait. Findings suggested that both exercise interventions resulted in similar benefits on the objective outcome measures, including upper limb motor control (assessed using the grooved pegboard), functional gait (assessed using the timed-up-and-go) and velocity and step length of self-paced gait. Interestingly, only the PD SAFEx intervention resulted in improved PD symptoms assessed using the UPDRS and symptomatic improvement was maintained after a six week non-exercise period. Thus, the increased focus on sensory feedback present in the PD SAFEx intervention appears to have an important additional influence on the symptoms of PD. The final study assessed whether improved PD symptoms following a sensory attention focused exercise (PD SAFEx) intervention could be replicated across multiple administrations and secondly, whether the effect could be replicated when administered by minimally trained individuals in the community. The PD SAFEx intervention was administered to four separate groups at the Movement Disorders Research and Rehabilitation Center (MDRC) and twice at an exercise facility in the community (YMCA). Over the six administrations of the PD SAFEx intervention, similar symptomatic improvements were realized by participants. Interestingly, the community based intervention appeared to gain a greater symptomatic benefit than the PD SAFEx intervention administered by leaders knowledgeable in movement disorders and the underlying neurological deficits focused on in the PD SAFEx intervention suggests that the feasibility of global distribution of the PD SAFEx intervention would be a logical direction for future research. The methodological improvements employed in the current thesis allowed for detailed and thorough comparisons to be drawn between various exercise interventions. It has been shown that strength training and PD SAFEx interventions have the greatest symptomatic benefit for individuals with PD. Further, the beneficial effect of increased focus on sensory feedback and the simplicity of application of the PD SAFEx intervention suggest that the PD SAFEx intervention should be further explroed for its ability to be globally implemented

A construction of representations of affine Weyl groups

Let G be a complex, semisimple, simply connected algebraic group with Lie algebra g. We extend scalars to the power series field in one variable C((π)), and consider the space of Iwahori subalgebras containing a fixed nil-elliptic element of g ⊗ C((π)), i.e. fixed point varieties on the full affine flag manifold. We define representations of the affine Weyl group in the homology of these varieties, generalizing Kazhdan and Lusztig\u27s topological construction of Springer\u27s representations to the affine context

The Economic and Philosophic Manuscripts of Data

Society is filled with words and images that elucidate the positive force radiating from technology entities. I push back against this imprecise and inaccurate narrative by breaking down the illusions created by surveillance capitalism. I argue that there exists a unique relationship between an individual and their environment in creating value, especially in the form of data. This relationship tears down the smokescreens prompted up by the surveillance state because it demonstrates the costs of technology and surveillance capitalism. I found that how data is created and made monetarily valuable has significant, adverse repercussions on the capability to flourish as a human being. The world is increasingly shaped by the digital economy incentivizing the collection of data, and consequently, beliefs about people as no different than commodities proliferate, damage on people’s epistemic capacities continue, and deeply intimate costs are incurred in a person’s personal life. I conclude by imagining a relevant alternative scenario to a surveillance state: blockchain technology. While the surveillance state is a totalizing and powerful system that operates discreetly, blockchain has the potential to be a solution to the extant problems of the surveillance state because blockchain technology can establish and facilitate trust in the digital in a way that is decentralized. As a result, people can fundamentally trust each other in a manner that is not dependent on the centralized data storages. In this thesis, I not only evaluate the history of the surveillance state, but I also look to the future by imagining how a different system of valuation has the potential to respect the digital identity of a person through a combined economic and philosophic lens