For all n > 2 except n = 6, show how to arrange the numbers 1,2,...,n2 in an n x n array so that each row and column sum to the same constant.
In: Advanced Math
Find the first four terms of the sequence (an)n≥1 with the given definition. Determine if they are potentially arithmetic or geometric.
(a) an is the number of n-bit strings which have more 1’s than 0’s. (Also, write down the strings for n ≤ 4.)
(b) an is the number of n-bit strings in which the number of 1’s is greater than or equal to the number of 0’s in every prefix. For example, 010111 would not qualify, since the prefix 010 has more 0’s than 1’s. (Also, write down the strings for n ≤ 4.)
(c) an is the number of lattice paths from (0, 0) to (n, n). (Refer to Section 1.2 if you need a refresher on lattice paths.)
In: Advanced Math
et A be a 157 x 157 upper-triangular matrix. Suppose that every diagonal entry of A is 1 and that there is at least one nonzero off-diagonal entry in A. Is A diagonalizable? Explain how you can answer this question mentally, with no non-trivial calculations.
In: Advanced Math
In: Advanced Math
In: Advanced Math
Let L1 be the line passing
through the
point P1=(−3, −1, 1) with direction
vector →d1=[1, −2, −1]T, and
let L2 be the line passing through the
point P2=(8, −3, 1) with direction
vector →d2=[−1, 0, 2]T.
Find the shortest distance d between these two lines, and
find a point Q1 on L1 and
a point Q2 on L2 so
that d(Q1,Q2) = d. Use the
square root symbol '√' where needed to give an exact value for your
answer.
d =
Q1 =
Q2 =
In: Advanced Math
Let A be a set with m elements and B a set of n elements, where m; n are positive integers. Find the number of one-to-one functions from A to B.
In: Advanced Math
Convert to standard maximum form and apply two iterations of simplex process using slack form.
Maximize
2x1 -6x3
Subject to
x1 + x2 – x3 <= 7
3x1 – x2 >= 8
-x1 + 2x2 + 2x3 >= -2
x2, x3 >=0
Please write the answer very clearly.
In: Advanced Math
In: Advanced Math
Find general solutions of the following systems using undetermined coefficients.
X′ =(2x2 matrix) ( 2 2; 3 1 ) X + (column matrix) ( e^−4t; 0 )
In: Advanced Math
In: Advanced Math
1. Find general solutions of the following systems using undetermined coefficients.
a.) x′ = −5x + 6y + 1
y′ = −7y + t .
b.) x′ = 6x − 5y + e^5t
y′ = x + 4y .
c.) x′ = −6x − 3y + te^2t
y′ = 4x + y .
In: Advanced Math
Consider the function, ? (?) = 20?² + 28? - 17
Determine both real roots of the quadratic equation as follows:
i. Graphically and determine the intervals for both roots to be used in iii & iv.
ii. Using the quadratic formula
iii. Using the bisection method until ?? is less than ?? = 0.5%
iv. Using the false-position method until ?? is less than ?? = 0.5%
In: Advanced Math
Let A be such that its only right ideals are {¯0} (neutral
element) and A. Show that A or is a ring with division or
A is a ring with a prime number of elements in which a · b = 0 for
any, b ∈ A.
In: Advanced Math
Approximately how many flops are needed to find the LU factorization of an n x n matrix using Doolittle’s method? If a computer requires 1 second to find an LU factorization of a 500 x 500 matrix, what would you estimate is the largest matrix that could be factored in less than 1 hour?
In: Advanced Math