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### Jeffrey A Brinker, M.D. • Professor of Medicine

https://www.hopkinsmedicine.org/profiles/results/directory/profile/0001297/jeffrey-brinker

Diseases

• Bolivian hemorrhagic fever
• Synovial cancer
• Spinocerebellar atrophy type 3
• Humero spinal dysostosis congenital heart disease
• Uniparental disomy of 14
• Dyscalculia
• Primary lateral sclerosis
• Myoclonus hereditary progressive distal muscular atrophy  Topics studied include formulas fr areas and volumes cheap zantac 300 mg fast delivery gastritis symptoms in tamil, continued factions order zantac cheap online gastritis espanol, sums of power series cheap zantac 150 mg online gastritis prevention, an approximation fr n, and tables of sines. Aryabhata also described a metod fr fnding greatest common divisors tat is the same as the metod described by Euclid. His frmulas fr the areas of triangles and circles are corect, but tose fr the volumes of spheres and pyramids are wrong. Aryabhata also produced an astronomy text, Sidhanta, which includes a number of remarkably accurate statements (as well as oter statements tat are not corect). For example, he states tat the orbits of the planets are ellipses, and he corectly describes the causes of solar and lunar eclipses. India named its frst satellite, launched in 1975 by the Russians, Arabhata, in recognition of his fundamental contributions to astronomy and matematics. We summarize tese steps in the fllowing table: j r· rj+l qj+l rj+2 J 0 252 198 1 54 1 198 54 3 36 2 54 36 1 18 3 36 18 2 0 the last nonzero remainder (fund in the next-to-last row in the last column) is the greatest common divisor of 252and 198. Later, we will see this when we estimate the maximum number of divisions used by the Euclidean algorithm to fnd the greatest common divisor of two positive integers. However, we frst show that, given any positive integer n, there are integers a and b such tat exactly n divisions are required to fnd (a, b) using the Euclidean algoritm. The reason that the Euclidean algorithm operates so slowly when it fnds the greatest common divisor of successive Fibonacci numbers is tat the quotient in all but the last step is 1, as illustated in the fllowing example. Observe that when the Euclidean algorithm is usedt find the greatest common divisor of /9=34 and /10=55, a total of eight divisions ae required. Then the Euclidean algorithm takes exactly n divisions to show tat C/n+l• fn+2}=1. Hence, the Euclidean algorithm takes exactly n divisions, t show that (/n+2, fn+1)= h=l. The number ofdivisions needed to find the greatest common divisor of two positve integers using the Euclidean algorithm does not exceed five tmes the number of decima digits in the smaler of the two integers. Proof When we apply the Euclidean algoritm to find the greatest common divisor of a =r0 and b=r1 with a > b, we obtain the fllowing sequence of equatons: 106 Pmes ad Greatest Common Dvisors r =r1q1+r2, 0 < r2 < r1, o 71 = r2q2 + T3, Q! Hence, b>a11-1• Now, becase log 01 a> 1/5, we see that log10 b> (n 1) log10 a> (n 1)/5. The geatest common divisor of two positve intgers a and b wit 3 a b can be fund using O((log2 a)) bit operatons. A civil and railway engine, he advanced the matematcal thery of elastcity ad invented cecoos. We illustate this by expressing (252, 198) =18 as a linear combination of252 and198. Refring to the steps of the Euclidean algoritm used to fnd (252, 198), by the next to the last step we see tat 18 =54 1. In general, to see how d =(a, b) may be expressed as a linea combination of a and b, refr to the series of equations that is generated by the Euclidean algoritm. By the penultimate equation, we have r =(a, b) =rn-2 rn-lqn-1· n this expresses (a, b) as a linear combination of rn 2 and rn-l· the second to the last equation can be used to express rn-1 as rn-3 rn-2qn-2· Using this last equation to eliminatern-l in the previous expression fr (a, b), we fnd tat so that (a, b) =rn-2 (rn-3 rn-2qn-2)qn-1 =(1+ qn-lqn 2)rn-2 qn-lrn-3• which expresses (a, b) as a linear combination of rn 2 and rn 3. We continue workng backwad through the steps of the Euclidean algorithm to express (a, b) as a linear combination of each preceding pair of remainders, until we have fund (a, b) as a linear combination ofr0 =a andr1 =b. This metod fr expressing (a, b) as a linear combinaton of a ad bis somewhat inconvenient fr calculation, because it is necessary to work out the steps of the Euclidean algoritm, save altese steps, and then proceed backwad through the steps t write (a, b) as a linea combination of each successive pair of remanders. There is another method fr finding (a, b) tat requires workng through the steps of the Euclidean c algoritm only once. The fllowing theorem gives this method, which is called the etended Euclidean algorith. Then, fom the kth step of the Euclidean algorithm, we have rk=rk-2-rk-lqk-1· Using the induction hypothesis, we fnd tat rk=(sk-2a+tk-2b)-(sk-la+tk-1b)qk-l =(sk-2-sk-lqk 1)a+(tk-2-tk-lqk 1)b =ska+t bk. We summarze the steps used by the extended Euclidean algorthm to express (252, 198) as a linear combination of 252 and 198 in the fllowing table. Note tat the greatest common divisor of two integers, not both 0, may be expressed as a linea combination of these integers in an infnite number of ways. In oter words, there are infnitely many pairs of Bezout coefcients fr every pair integers, not both zero. To see tis, let d=(a, b) and let d=sa+th be one way to write d as a linear combination of a and b, so thats and tare Bezout coeffcients fr a and b, guaranteed to exist by the previous discussion. Then fr all integers k, s+k(b/d) and t-k(a/d) are also Bezout coeffcients fr a and b because d=(s+k(b/d))a+(t-k(a/d))b. Wita=252adb=198, we have 18=(252, 198)= 4(+llk)252+ (-5 14k) 198 fr any integer k. For each pair of integers in Exercise 1,express the greatest common divisor of the integers as a linear combination of these integers. For each pair of integers in Exercise 2,express the greatest common divisor of the integers as a linear combination of these integers. Express the greatest common divisor of each set of numbers in Exercise 5 as a linear combinaton of the numbers in tat set. Express the greatest common divisor of each set of numbers in Exercise 6 as a linear combination of the numbers in tat set. The greatest common divisor of two positive integers can be fund by an algoritm tat uses only subtactions, paity checks,and shifs of binary expansions, witout using any divisions. Purchase discount zantac line. Who Needs an Upper Endoscopy?.

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