In: Advanced Math
Suppose (A,∗) be an associative, unital, binary operation with inverses. Show that if|A|≤3,then in fact, (A,∗) isalsocommutative, even though we didn’t assume it at the beginning.
In: Advanced Math
A photoconductor film is manufactured at a nominal thickness of 25 mils. The product engineer wishes to increase the mean speed of the film, and believes that this can be achieved by reducing the thickness of the film to 20 mils. Eight samples of each film thickness are manufactured in a pilot production process, and the film speed (in microjoules per square inch) is measured. For the 25-mil film, the sample data result is x-bar1=1.13 and s1=0.11, while for the 20-mil film, the data yield x-bar2=1.08 and s2=0.09. Note that an increase in film speed would lower the value of the observation in microjoules per square inch.
(a) Do the data support the claim that reducing the film thickness increases the mean speed of the film? Use a=0.10 and assume that the two population variances are equal and the underlying population of film speed is normally distributed. What is the P-value for this test? Round your answer to three decimal places (e.g. 98.765).
The data ___support/do not support___ the claim that reducing the film thickness increases the mean speed of the film. The P-value is _______.
(b) Find a 95% confidence interval on the difference in the two means that can be used to test the claim in part (a). Round your answers to four decimal places (e.g. 98.7654).
_______ <= mu1 - mu2 <= _______
In: Advanced Math
Which one of the improper integrals below converges or
diverges?
[int _a ^b] means integral from a to b, we use
inf to indicate infinity.
a) [int _0 ^1] 1/x dx
b) [int _0 ^1] 1/x^(1/2) dx
c) [int _0 ^1] 1/x^2 dx
d) [int _1 ^inf] 1/x dx
e) [int _1 ^inf] 1/x^(1/2) dx
f) [int _1 ^inf] 1/x^2 dx
g) [int _1 ^inf] lnx / x^2 dx
h) [int _1 ^inf] lnx / x dx
i) [int _(-inf) ^inf] 1/(1+x^2) dx
j) [int _1 ^inf] exp(-x^2) dx
k) [int _1 ^inf] (sinx / x )^2 dx
l) [int _2 ^inf] 1 / sqrt(x^2 - 1) dx
m) [int _0 ^Pi/2] tanx dx
In: Advanced Math
Complete the proof for the claim that any open ball B(x0,r) in Euclidean space Rn is homeomorphic to Rn.
proof is given below the theorem. Show that suggested map g is in fact homeomorphism.
Theorem: Let X0, X1, and X2 be topological spaces and let f: X0 -> X1 and g : X1 -> X2 be continuous functions. Then g∘f : X0 -> X2 is continuous.
proof : Suppose that V is open in X2. Since g is continuous, g-1(V) is open in X1. Since f is continuous, f-1(g-1(V)) = (g∘f)-1(V) is open in X0. It follows that g∘f is continuous.
In: Advanced Math
In: Advanced Math
Prove: There are infinitely many primes congruent to 3 modulo 8. Hint: Consider N = (p1p2···pr)2 + 2.
In: Advanced Math
A small business owner contributes $2,000 at the end of each quarter to a retirement account that earns 10% compounded quarterly. (a) How long will it be until the account is worth at least $150,000? (Round your answer UP to the nearest quarter.) 43 quarters (b) Suppose when the account reaches $150,000, the business owner increases the contributions to $4,000 at the end of each quarter. What will the total value of the account be after 15 more years? (Round your answer to the nearest dollar.) $
In: Advanced Math
Use Gauss’s Lemma to find the Legendre symbol values (8/11), (5/19), and (6/31).
In: Advanced Math
Before we begin graphing systems of equations, a good starting point is to review our knowledge of 2-D graphs. These graphs are known as 2-D because they have two axes. Find an online image of a graph to use as the foundation of your discussion. (This is easily accomplished by searching within Google Images.) Using your graph as the example:
1.) Select any two points on the graph and apply the slope formula, interpreting the result as a rate of change (units of measurement required).
2.) Use rate of change (slope) to explain why your graph is linear (constant slope) or not linear (changing slopes).
Embed the graph into the post by copying and pasting into the discussion. You must cite the source of the image. Also be sure to show the computations used to determine slope.
In: Advanced Math
prove or disprove using logical equivalences
(a) p ∧ (q → r) ⇐⇒ (p → q) → r
(b) x ∧ (¬y ↔ z) ⇐⇒ ((x → y) ∨ ¬z) → (x ∧ ¬(y → z))
(c) (x ∨ y ∨ ¬z) ∧ (¬x ∨ y ∨ z) ⇐⇒ ¬y → (x ↔ z)
In: Advanced Math
Find Taylor series expansion of log(1+z) and show radius of convergence
In: Advanced Math
a) Let S ⊂ R, assuming that f : S → R is a continuous function,
if the
image set {f(x); x ∈ S} is unbounded prove that S is unbounded.
b) Let f : [0, 100] → R be a continuous function such that f(0) =
f(2),
f(98) = f(100) and the function g(x) := f(x+ 1)−f(x) is equal to
zero in at most
two points of the interval [0, 100].
Prove that (f(50) − f(49))(f(25) − f(24)) > 0.
In: Advanced Math
Consider a Math class with 15 female students and 14 male students.
a) How many different 5 people committees with exactly 3 females and 2 males are possible? Justify your answer
b) How many different 5 people committees with representation of both genders are there? Justify your answer
c) Suppose that two of the students refuse to work together. How many different 5 people committees are possible? Justify your answer
d) How many different ways to arrange them in a row with no two males together? Justify your answer
e) Show that there are at least 3 students with the same gender whose were born on the same day of the week.
In: Advanced Math
Find a characterization of primes p such that 3 is a square mod p and then prove it.
In: Advanced Math