4. An element a in a ring R is called nilpotent if there exists a non-negative integer n such that a" = OR (a) Let a and m > O be integers such that if any prime integer p divides m then pſa. Prove that a is nilpotent in Zm. (b) Let N be the collection of all nilpotent elements of a ring R. Prove that N is an ideal of R. (c) Prove that the only nilpotent element in...
37. Show that if D is an integral domain, then 0 is the only nilpotent element in D. 38. Let a be a nilpotent element in a commutative ring R with unity. Show that (a) a = 0 or a is a zero divisor.. (b) ax is nilpotent for all x ER. (c) 1 + a is a unit in R. (d) If u is a unit in R, then u + a is also a unit in R.
where 7 is the region defined by >0, y >0, >0, r+y+z<3.
IF Let x(t) Show that e 20" σ>0, and let (o) be the Fourier transform of x(t) .
Prove that if ? is integrable on [?, ?] and ?(?) ≥ 0 for all ?
in [?, ?], then
[ f(x)dx > 0 7. Prove that if f is integrable on [a, b] and f(x) > 0 for all x in [a, b], then sof(x)dx > 0.
HW: Show that the series __, an n=0 converges whenever ſal < 1, and diverges whenever al > 0.
Solve the IVP 1 (31= [ -> ] (3) [6**) (O)= [-] +
7. Use the method of reduction of order to find a second solution of the differential equation xy" - y + 4x³y = 0, x > 0; y1(x) = sin x².
Please show detailed steps and in a clear writing. Thanks
Reduce the order of the following differential equation and solve 23 %>" + 22" – 32' = 3,3 > 0.
7 7. Let Xi, . . . , xn be iid based on f(x:0) = 2x e-x2/0 where x > 0, Show that θ =「X 2 is 2-1 efficient.