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In doing so, it closely parallels our recursive multiplication algorithm (Figure 1.1). For instance, that algorithm would compute the product x 25 by an analogous decomposition to the one we just saw: x 25 = x 16 + x 8 + x 1. And whereas for multiplication the terms x 2 i i come from repeated doubling, for exponentiation the corresponding terms x 2 are generated by repeated squaring. Let n be the size in bits of x, y, and N (whichever is largest of the three). As with multiplication, the algorithm will halt after at most n recursive calls, and during each call it multiplies n-bit numbers (doing computation modulo N saves us here), for a total running time of O(n 3 ).

been described by Lugaresi and colleagues This disorder, with onset in middle age and a clinical course of 7 to 36 months, is characterized by a virtual incapacity to sleep and to generate EEG sleep patterns The cerebral changes consist mainly of profound neuronal loss in the anterior or anteroventral and mediodorsal thalamic nuclei These cases are thought to represent a familial form of prion disease similar to diseases that cause subacute spongiform encephalopathy and Gerstmann-Straussler-Scheinker disease (see Chap 33) Interestingly, the alcoholic form of the Korsakoff amnesic state, associated with less severe lesions in the same thalamic nuclei, is also characterized by a sleep disturbance, taking the form of an increased frequency of intermittent periods of wakefulness (Martin et al) Major head injury is an important cause of sleep disturbance.

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The abnormalities, which may persist for months or years, consist mainly of a decrease in stages 1 and 2 of NREM sleep and less than the expected amounts of REM sleep and dreaming Some patients in the persistent vegetative state show a cycle of changes in the EEG, progressing from a picture of abortive spindles and K complexes with cyclic alterations in respiration and pupil size to the acquisition of a more normally structured sleep activity This sequence usually presages the change from a state of coma to one of wakefulness Organized sleep activity is absent in virtually all types of coma that are the result of anatomic damage to the brain A possible exception occurs in the unusual condition known as spindle coma, in which persistent coma and the electrographic features of sleep coexist.

This particular combination of events has been described after head trauma and rarely in association with profound metabolic encephalopathies Despite what appears to be a genuine comatose state (not simply hypersomnolence) from a lesion of the reticular activating system, the EEG displays frequent spindle activity and vertex waves, attesting to the integrity of thalamocortical pathways for sleep activity (see Nogueira de Melo et al) Disturbed sleep patterns have been described in patients with Alzheimer disease, Huntington chorea, olivopontocerebellar degeneration, and progressive supranuclear palsy (Parkes) Dreaming is absent in some of these conditions The peculiarities of sleep in Parkinson disease have been extensively studied; many patients in early stages of the disease complain of fragmented and unrestful sleep, particularly in the early morning hours The loss of natural body movements and the alerting effects of L-dopa contribute to the insomnia.

Also, the directly acting dopaminergic agonist drugs used for the treatment of Parkinson disease may have the side effect of a pronounced and often rapid daytime sleepiness; however, a similar problem arises in some patients with advancing disease alone In striatonigral degeneration (multiple system atrophy), Lewy body disease, and other parkinsonian syndromes, REM sleep is particularly affected, although some of the effects may be the result of abnormal respiratory patterns and obstructive sleep apnea Migraine, cluster headaches, and paroxysmal hemicrania all have been linked to certain sleep stages Additionally, patients with epilepsy and myasthenia gravis and motor neuron disease may have sleep-related complaints; in the latter two the cause can be traced to mild respiratory failure or to pharyngeal weakness leading to apneic periods or aspiration A variety of sleep disturbances may accompany brain tumors or follow surgical resection of an intracranial tumor.

These include excessive daytime sleepiness, sleep apnea, and, rarely, nocturnal epilepsy The location of the lesion, rather than the tumor type, is predictive of such a disturbance; thus tumors affecting the hypo-.

It fails when you divide When you divide an integer by another integer, the order is important. If you divide 20 by 4, you get 5. But if you divide 4 by 20, you don t get 5. You don t even get an integer. A more dramatic example is the division of 0 by any other integer. If you divide 0 by 3, you get 0. But if you divide 3 by 0, you get an undefined quantity! Cross-multiplication Here s a fact that you will find useful in algebra. It s called the rule of cross-multiplication. Suppose you have two ratios of integers, a /b and c /d, and you re told that they re equal. You re also assured that neither b nor d is equal to 0. You write

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