Let (G,+) be an abelian group and U a subgroup of G. Prove that G is the direct product of U and V (where V a subgroup of G) if only if there is a homomorphism f : G → U with fU = IdU
In: Advanced Math
Boise Lumber has decided to enter the lucrative prefabricated housing business. Initially, it plans to offer three models: standard, deluxe, and luxury. Each house is prefabricated and partially assembled in the factory, and the final assembly is completed on site. The dollar amount of building material required, the amount of labor required in the factory for prefabrication and partial assembly, the amount of onsite labor required, and the profit per unit are as follows.
Standard Model  Deluxe Model  Luxury Model  

Material  $6,000  $8,000  $10,000 
Factory Labor (hr)  240  220  200 
OnSite Labor (hr)  180  210  300 
Profit  $3,400  $4,000  $5,000 
For the first year's production, a sum of $8,200,000 is budgeted for the building material; the number of laborhours available for work in the factory (for prefabrication and partial assembly) is not to exceed 215,000 hr; and the amount of labor for onsite work is to be less than or equal to 234,000 laborhours. Determine how many houses of each type Boise should produce to maximize its profit from this new venture. (Market research has confirmed that there should be no problems with sales.)
standard model  houses 
deluxe model  houses 
luxury model  houses 
In: Advanced Math
Let R be a UFD and let F be a field of fractions for R. If f(α) = 0, where f ∈ R [x] is monic and α ∈ F, show that α ∈ R
NOTE: A corollary is the fact that m ∈ Z and m is not an n^{th} power in Z, then ^{n}√m is irrational.
In: Advanced Math
The inputoutput matrix for a simplified economy with just three sectors (agriculture, manufacturing, and households) is given below.
Agriculture Manufacturing Households


A. How many units from each sector does the agriculture sector require to produce 1 unit?
The agriculture sector requires _____units from agriculture, ____units from manufacturing, and ___units from households.
B. What production levels are needed to meet a demand of 32 units of agriculture, 34 units of manufacturing, and 34 units of households?
Production levels of ____units of agriculture,___units of manufacturing, and _____units of households are needed. (Round to the nearest whole number as needed.)
C. How many units of manufacturing are used up in the economy's production process?
The economy's production process uses up_____ units of manufacturing.
(Round to the nearest whole number as needed.)
In: Advanced Math
Previously, we listed all 29 topologies on the set X={a,b,c}. However, some of the resulting topological spaces are homeomorphic. Which are homeomorphic? Divide the set of 29 topological spaces into homeomorphism classes, and be sure to justify your choices. There are 9 homeomorphism classes in total. (To justify your choices, explain why the spaces within each class are homeomorphic to each other. Your explanations can be somewhat loose).
In: Advanced Math
Suppose that a decisionmaker’s preferences over the set A={a, b, c} are represented by the payoff function u for which u(a) = 0, u(b) = 1, and u(c) = 4.
(a) Are they also represented by the function v for which v(a) =−1, v(b) = 0, and v(c) = 2?
(b) How about the function w for which w(a) =w(b) = 0 and w(c) = 8?
(c) Give another example of a function f:A→R that represents the decisionmaker’s preferences.
(d) Is there a function that represents the decisionmaker’s preferences and assigns negative numbers to all elements of A?
In: Advanced Math
In: Advanced Math
In: Advanced Math
Let u and v be orthogonal vectors in R3 and let w = 3u + 6v. Suppose that u = 5 and v = 4. Find the cosine of the angle between w and v.
In: Advanced Math
In: Advanced Math
A topological space X is zero − dimensional if it has a basis B consisting of open sets which are simultaneously closed. (a) Prove that the set C = {0, 1}^{N} with the product topology is zerodimensional. (b) Prove that if (X, d) is a metric space for which X < R, that is the cardinality of X is less than that of R, then X is zerodimensional.
In: Advanced Math
Suppose when searching for a large prime, our first step is to sieve by eliminating the first n primes. How large should n be to speed up our search by a factor of 10? This requires careful thought and inventiveness before doing a computation.
How to approach this: Eliminating all odd numbers speeds up our program by a factor of 2, since we've eliminated half of the choices that are composite. If we also immediately disqualify multiples of 3, then we speed up our program by a factor of 3, since we have eliminated 2/3 of all composites. Check that that argument is correct. Now see what happens when you eliminate all of 2, 3 and 5. Again for 2,3,5,7. Now try to figure out the mathematical result and use it to solve for n.
In: Advanced Math
Are the following statements true or false?
1. Let P(n) be the statement "If any string of length n over {a, b} has more a's than b's, then it has two a's in a row". We can prove this statement is true for all n with n ≥ 2 by proving P(2), P(3), and "for all n: P(n) → P(n+2)".
2. Let P(x) be a predicate with one free variable x of type natural. If I prove P(0), "for all x: P(x) → P(x+2)", and "for all x: P(x) → P(x+3)", I may conclude "for all x: P(x)".
In: Advanced Math
2. An industrial sewing machine uses ball bearings that are targeted to have a diameter of 0.75 inches. The lower and upper specification limits under which the ball bearings can operate properly are 0.74 inches and 0.76 inches, respectively.
Past experience has indicated that the actual diameter of the ball bearings is approximately normally distributed with a mean of 0.753 inches and a standard deviation of 0.004 inches.
What is the probability that a ball bearing is:
a. between the target and the actual mean?
b. between the lower specification limit and the target?
c. above the upper specification limit?
d. below the lower specification limit?
e. Of all the ball bearings, 93% of the diameters are greater than what value?
In: Advanced Math
We can write 13 as a sum of distinct powers of 2: 13 = 8+4+1 = 2 3 +2 2 +2 0 .
• Using strong induction, show every integer can be written as the sum of distinct powers of two.
• Show that every integer has a unique representation as the sum of distinct powers of two. It follows that every integer has a unique binary representation. For instance, 13 is uniquely represented by 1101 in binary
In: Advanced Math