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Hysteresis in material behaviour includes both signum nonlinearities as well as high dimensionality. Available models for component-level hysteretic behaviour are empirical. Here, we derive a loworder model for rate-independent hysteresis from a high-dimensional massless frictional system. The original system, being given in terms of signs of velocities, is first solved incrementally using a linear complementarity problem formulation. From this numerical solution, to develop a reduced-order model, basis vectors are chosen using the singular value decomposition. The slip direction in generalized coordinates is identified as the minimizer of a dissipation-related function. That function includes terms for frictional dissipation through signum nonlinearities at many friction sites. Luckily, it allows a convenient analytical approximation. Upon solution of the approximated minimization problem, the slip direction is found. A final evolution equation for a few states is then obtained that gives a good match with the full solution. The model obtained here may lead to new insights into hysteresis as well as better empirical modelling thereof.

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Background:Although magnesium is used through intravenous and inhalation route in the management of asthma, actual prevalence of hypomagnesemia in asthma is not known. We conducted this study: 1) to detect the prevalence of hypomagnesemia in stable asthma and 2) to assess the significance of hypomagnesemia in these patients.Design:Prospective clinical study.Setting:Department of Respiratory Medicine, Calcutta National Medical College, Kolkata.Period of Study:Four months from January, 2007, to April, 2007.Materials and Methods:Fifty patients attending outpatients department of respiratory medicine with stable asthma were randomly selected. They were assessed clinically and their serum magnesium levels were measured. This was compared with the serum magnesium values of 45 nonasthmatic healthy controls.Results:Out of 50 patients, 14 had hypomagnesemia. Possible relationship of hypomagnesemia with tachycardia, tachypnoea, severity of asthma, medication use, and previous and future exacerbations were analyzed.Conclusion:There was statistically significant association of hypomagnesemia with tachypnoea, severe asthma, use of long-acting β-agonist, inhaled corticosteroids, theophylline, use of ≥ 3 medications, previous and future exacerbations but not with tachycardia or use of short-acting β2 -agonist or montelukast.

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