Factorizations of Elements in Noncommutative Rings: A Survey

We survey results on factorizations of non zero-divisors into atoms (irreducible elements) in noncommutative rings. The point of view in this survey is motivated by the commutative theory of non-unique factorizations. Topics covered include unique factorization up to order and similarity, 2-firs, and modular LCM domains, as well as UFRs and UFDs in the sense of Chatters and Jordan and generalizations thereof. We recall arithmetical invariants for the study of non-unique factorizations, and give transfer results for arithmetical invariants in matrix rings, rings of triangular matrices, and classical maximal orders as well as classical hereditary orders in central simple algebras over global fields.Comment: 50 pages, comments welcom

Noncommutative generalizations of theorems of Cohen and Kaplansky

This paper investigates situations where a property of a ring can be tested on a set of "prime right ideals." Generalizing theorems of Cohen and Kaplansky, we show that every right ideal of a ring is finitely generated (resp. principal) iff every "prime right ideal" is finitely generated (resp. principal), where the phrase "prime right ideal" can be interpreted in one of many different ways. We also use our methods to show that other properties can be tested on special sets of right ideals, such as the right artinian property and various homological properties. Applying these methods, we prove the following noncommutative generalization of a result of Kaplansky: a (left and right) noetherian ring is a principal right ideal ring iff all of its maximal right ideals are principal. A counterexample shows that the left noetherian hypothesis cannot be dropped. Finally, we compare our results to earlier generalizations of Cohen's and Kaplansky's theorems in the literature.Comment: 41 pages. To appear in Algebras and Representation Theory. Minor changes were made to the numbering system, in order to remain consistent with the published versio

Counting and effective rigidity in algebra and geometry

The purpose of this article is to produce effective versions of some rigidity results in algebra and geometry. On the geometric side, we focus on the spectrum of primitive geodesic lengths (resp., complex lengths) for arithmetic hyperbolic 2-manifolds (resp., 3-manifolds). By work of Reid, this spectrum determines the commensurability class of the 2-manifold (resp., 3-manifold). We establish effective versions of these rigidity results by ensuring that, for two incommensurable arithmetic manifolds of bounded volume, the length sets (resp., the complex length sets) must disagree for a length that can be explicitly bounded as a function of volume. We also prove an effective version of a similar rigidity result established by the second author with Reid on a surface analog of the length spectrum for hyperbolic 3-manifolds. These effective results have corresponding algebraic analogs involving maximal subfields and quaternion subalgebras of quaternion algebras. To prove these effective rigidity results, we establish results on the asymptotic behavior of certain algebraic and geometric counting functions which are of independent interest.Comment: v.2, 39 pages. To appear in Invent. Mat

Mindful Aging: The Effects of Regular Brief Mindfulness Practice on Electrophysiological Markers of Cognitive and Affective Processing in Older Adults

There is growing interest in the potential benefits of mindfulness meditation practices in terms of counteracting some of the cognitive effects associated with aging. Pursuing this question, the aim of the present study was to investigate the influence of mindfulness training on executive control and emotion regulation in older adults, by means of studying behavioral and electrophysiological changes. Participants, 55 to 75 years of age, were randomly allocated to an 8-week mindful breath awareness training group or an active control group engaging in brain training exercises. Before and after the training period, participants completed an emotional-counting Stroop task, designed to measure attentional control and emotion regulation processes. Concurrently, their brain activity was measured by means of 64-channel electroencephalography. The results show that engaging in just over 10 min of mindfulness practice five times per week resulted in significant improvements in behavioral (response latency) and electrophysiological (N2 event-related potential) measures related to general task performance. Analyses of the underlying cortical sources (Variable Resolution Electromagnetic Tomography, VARETA) indicate that this N2-related effect is primarily associated with changes in the right angular gyrus and other areas of the dorsal attention network. However, the study did not find the expected specific improvements in executive control and emotion regulation, which may be due to the training instructions or the relative brevity of the intervention. Overall, the results indicate that engaging in mindfulness meditation training improves the maintenance of goal-directed visuospatial attention and may be a useful strategy for counteracting cognitive decline associated with aging

Giant breast tumors: Surgical management of phyllodes tumors, potential for reconstructive surgery and a review of literature

<p>Abstract</p> <p>Background</p> <p>Phyllodes tumors are biphasic fibroepithelial neoplasms of the breast. While the surgical management of these relatively uncommon tumors has been addressed in the literature, few reports have commented on the surgical approach to tumors greater than ten centimeters in diameter – the giant phyllodes tumor.</p> <p>Case presentation</p> <p>We report two cases of giant breast tumors and discuss the techniques utilized for pre-operative diagnosis, tumor removal, and breast reconstruction. A review of the literature on the surgical management of phyllodes tumors was performed.</p> <p>Conclusion</p> <p>Management of the giant phyllodes tumor presents the surgeon with unique challenges. The majority of these tumors can be managed by simple mastectomy. Axillary lymph node metastasis is rare, and dissection should be limited to patients with pathologic evidence of tumor in the lymph nodes.</p

Overexpression of endothelial nitric oxide synthase suppresses features of allergic asthma in mice

BACKGROUND: Asthma is associated with airway hyperresponsiveness and enhanced T-cell number/activity on one hand and increased levels of exhaled nitric oxide (NO) with expression of inducible NO synthase (iNOS) on the other hand. These findings are in paradox, as NO also relaxes airway smooth muscle and has immunosuppressive properties. The exact role of the endothelial NOS (eNOS) isoform in asthma is still unknown. We hypothezised that a delicate regulation in the production of NO and its bioactive forms by eNOS might be the key to the pathogenesis of asthma. METHODS: The contribution of eNOS on the development of asthmatic features was examined. We used transgenic mice that overexpress eNOS and measured characteristic features of allergic asthma after sensitisation and challenge of these mice with the allergen ovalbumin. RESULTS: eNOS overexpression resulted in both increased eNOS activity and NO production in the lungs. Isolated thoracic lymph nodes cells from eNOS overexpressing mice that have been sensitized and challenged with ovalbumin produced significantly less of the cytokines IFN-γ, IL-5 and IL-10. No difference in serum IgE levels could be found. Further, there was a 50% reduction in the number of lymphocytes and eosinophils in the lung lavage fluid of these animals. Finally, airway hyperresponsiveness to methacholine was abolished in eNOS overexpressing mice. CONCLUSION: These findings demonstrate that eNOS overexpression attenuates both airway inflammation and airway hyperresponsiveness in a model of allergic asthma. We suggest that a delicate balance in the production of bioactive forms of NO derived from eNOS might be essential in the pathophysiology of asthma

2022 World Hypertension League, Resolve To Save Lives and International Society of Hypertension dietary sodium (salt) global call to action

This fact sheet and global call to action is aimed at nutrition, hypertension, cardiovascular and other health care clinicians and scientists, and health advocates, as well as the organizations to which they belong. The ‘call’ is to align these audiences with the facts on: the burden of disease and key evidence supporting reductions in dietary sodium, the consistent recommendations for reducing dietary sodium from unbiased and comprehensive health and scientific reviews, the current levels of sodium intake, the cost savings expected from reducing high dietary sodium, the sources of controversial opinions, the current recommended approaches to reduce dietary sodium, and how to stay up to date with evidence on how to reduce dietary sodium and the evolving research on the adverse health effects of a high sodium intake. Health, nutrition, hypertension and cardiovascular organizations, and their members, need to become more engaged and advocate for reductions in dietary sodium, and for a greater priority to be given to high quality research on dietary sodium. The World Hypertension League, Resolve to Save Lives and International Society of Hypertension are committed to support reductions in dietary sodium as a high priority

Highly symmetric POVMs and their informational power

We discuss the dependence of the Shannon entropy of normalized finite rank-1 POVMs on the choice of the input state, looking for the states that minimize this quantity. To distinguish the class of measurements where the problem can be solved analytically, we introduce the notion of highly symmetric POVMs and classify them in dimension two (for qubits). In this case we prove that the entropy is minimal, and hence the relative entropy (informational power) is maximal, if and only if the input state is orthogonal to one of the states constituting a POVM. The method used in the proof, employing the Michel theory of critical points for group action, the Hermite interpolation and the structure of invariant polynomials for unitary-antiunitary groups, can also be applied in higher dimensions and for other entropy-like functions. The links between entropy minimization and entropic uncertainty relations, the Wehrl entropy and the quantum dynamical entropy are described.Comment: 40 pages, 3 figure