h Consider a solid T enclosed by the paraboloid z = x^2 +y^2 and the plane z = 4 (the solid above the paraboloid and below the plane). Let M the (closed) surface representing the boundary surface of T. The surface M consists of two surfaces: the paraboloid M1 and the lid M2. Orient M by an outward normal. Let F=(z,2y,-2)
Compute the integral using the Divergence theorem. Carry out the computation of the triple integral using the spherical coordinates.
In: Advanced Math
In: Advanced Math
A is allowed to select any date of the year other than December 31. B may then select any date later in the same month or the same day of any later month. For example, if A selects June 16, then B may choose any later date in June or the 16th of any month from July to December. Using the same rule with regard to the date that B has selected, A must select a new date, and so on. The winner is the player who arrives at December 31
(a) What date should A select to begin with in order to ensure the fastest possible win?
(b) What if A must start with a date in January? ([27], May 1971).
Full shown work and answer please.
In: Advanced Math
Given ( x + 2 )y" + xy' + y = 0 ; x0 = -1
In: Advanced Math
For the Fibonacci sequence, f0 = f1 = 1 and fn+1 = fn + fn−1 for all n > 1. Prove using induction: fn+1fn−1 − f2n = (−1)n.
In: Advanced Math
Let a1 ≥ a2, . . . , an be a sequence of positive integers whose
sum is 2n − 2. Prove that there exists a tree T on n vertices whose
vertices have degrees a1, a2, . . . , an.
Sketch of solution: Prove that there exist i and j such that ai = 1
and aj ≥ 2. Remove ai, subtract 1 from aj and induct on n.
In: Advanced Math
MAT 204 Discrete Structures – Assignment #10
Number theory is the branch of mathematics concerned with the integers. Traditionally, number theory was a pure branch of mathematics – known for its abstract nature rather than its applications. The great English mathematician, G.H. Hardy (1877 – 1947), used number theory as an example of a beautiful, but impractical, branch of mathematics. However, in the late 1900s, number theory became extremely useful in cryptosystems – systems used for secure communications.
Find the following for each pair of integers:
(a) The prime factorization;
(b) The greatest common divisor;
(c) The least common multiple;
(d) Verify that gcd (m, n) * lcm(m, n) = mn.
(i) 315, 825
(ii) 2091, 4807
In: Advanced Math
f(x) = (x^2 )0 < x < 1, (2−x), 1 < x < 2
A) Solve this integral, writing An as an expression in terms of
n. Write down the
values of A1,A2,A3,A4,A5 correct to 8 significant
figures.
b) Use MATLAB to find the coefficients of the first five
harmonics and compare the
results with those from part (e). Your solution should include a
copy of the m-file
fnc.m which you use to obtain the coefficients
c) Using MATLAB, plot the function and its approximating five-term Fourier series.
In: Advanced Math
Find the general solution of the linear system x ̇1 = x1, x ̇2 = ax2 Where a is a constant. Draw the phase planes for a = −1, 0, 1. Comment on the changes of the phase plane
In: Advanced Math
. Three Dice of a Kind
Consider the following game: You roll six 6-sided dice d1,…,d6 and you win if some number appears 3 or more times. For example, if you roll:
(3,3,5,4,6,6)
then you lose. If you roll
(4,1,3,6,4,4)
then you win.
In: Advanced Math
2. Drinking Warm Beer
We put 10 bottles of Molson Export and 3 bottles of Labatt 50 into the trunk of our black car on a hot summer day. We reach into the the cooler, pull out a random bottle b1 and drink it. Then we reach into the cooler, pull out a second bottle b2 and drink it.
In: Advanced Math
In: Advanced Math
Let F be a finite field. Prove that the multiplicative group F*,x) is cyclic.
In: Advanced Math
In: Advanced Math
In: Advanced Math