Calculate the nth divided difference of 1/x, on the nodal points X0 , X1 , ......., Xn .
For the function $f(x)=\dfrac{1}{x}$ on distinct nodes $X_0,X_1,\dots,X_n$, the $n$-th divided difference has the closed form $$ f[X_0,X_1,\dots,X_n]=\frac{(-1)^n}{X_0X_1\cdots X_n}. $$ ______ ______ ______ ________ ____ ____ ________ _______.
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Which of the following statements are true and which are false? Justify your answer with a short proof or a counter example.
(a) Every subgroup of S3 is normal.
(b) Every abelian group is cyclic.
(c) In a ring with unity, the sum of any two units is a unit.
(d) If a field has characteristic p, p a prime, the field is finite.
(e) If every element in group has finite order, the group is finite.
Show that, if G is a finite group and a ∈ G, o(a) | o(G). Further, show that a°(G) = e for all a ∈ G. Deduce the Euler-Fermat theorem a(n) = 1 (mod n) for all a, n ∈ N, n ≥ 2, (a, n) = 1.
From the following data, calculate the population in the year 1985:
| Year | Population (in , 000) |
| 1971 | 12 |
| 1981 | 15 |
| 1991 | 20 |
| 2001 | 27 |
| 2011 | 49 |
Find one root of the equation:
x3 - 2x 5 = 0
in the interval [3,2] using Birge-Vieta method. Perform only one iteration.
If H and K are normal abelian subgroups of a group, and if H∩ K = {e}, show that HK is abelian. Will the result be still true if we remove the condition that H and K are normal? Justify your answer.
Define a semigroup. Give an example of an infinite semigroup.
Using Runge-Kutta fourth order method with h = 0.1, find an approximation value of y(0.1) for the initial value problem:
b) Evaluate the integral:
using composite trapezoidal rule with h = ,2.0 compare with the exact value.