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Let be a Noetherian local ring and let M be a finitely generated R-module of dimension d. Let be a system of parameters of M and let be a d-tuple of positive integers. In this paper we study the length of generalized fractions M(1/(x1, … , xd, 1)), which was introduced by Sharp and Hamieh. First, we study the growth of the function
Then we give an explicit calculation for the function in the case in which M admits a certain Macaulay extension. Most previous results on this topic are improved in a clearly understandable way.
The analysis of age-related changes in sleep requires the accumulation of data on a healthy population over 65 years of age. However, increased age raises the risk of cardiovascular, metabolic, cognitive, psychiatric, musculoskeletal, renal, hepatic, and hematological conditions. Structural decline in the elderly, including reduced brain mass and numbers of neurons, might lead to functional brain loss. Certain primary sleep disorders affecting sleep in the elderly includes insomnia, sleep-disordered breathing (SDB), periodic limb movement disorder and restless legs syndrome, circadian rhythm sleep disorders, and sleep in dementia. This chapter provides the clinical presentation, diagnosis and treatment for each disorder. As sleep and its restorative function become more widely recognized for their impact on both medical conditions and quality of life in the elderly, the healthcare community has ever-increasing responsibility for translating the growing knowledge of sleep medicine into clinical practice in geriatric medicine.
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