2. Give an example of a linear transformation L : lR^2 arrow lR^2 which has a repeated real eigenvalue, but in which lR^2 does posses a basis of eigenvectors.

Suppose f : R → R is measurable and g : R → R is monotone. Prove that g ◦ f is measurable

Developing filters using convolution theorem and Fourier Transform.

You have been hired as an Engineering Mathematician at a consulting firm located in Saint Louis. On your first job, you have been asked to mathematically design a frequency filter that removes from a standard beacon signal a periodic interference generated by a rotatory machine located in the basement of the company. Please see below for more details: Let s(t) be the standard beacon signal that is being communicated. Below you can find its Fourier Series representation ?(?)=2.5+2 sin(?)+3cos(t)+0.5cos(2?)+ 0.3 sin (2?). The periodic interference is given by ?(?)=0.5 cos (120 ?) and the measurement signal with noise is given as follows: ?(?)=?(?)+?(?) Let g( t) be the filter function and let ? Z(?) be the function that results from applying ? (?) to ?(?). Using the Fourier transform of the convolution theorem, propose the design of a filtering function g(t) which removes from ?(?) the effect of the periodic noise ?(?) assuming that we only know that its fundamental period is equal to 2pi/120. Make sure to write the analytical expression of ?(?) and its Fourier transform. Also, please write the mathematical expression that relates ? Z(?) as a function of ?(?)and ?(?)

How many eight-digit positive integers have the sum of digits being even?

Let P(n) := ” If n^3 is odd then n is also odd.” I.e., if ∃k ∈ Z, n3 = 2k + 1, ∃b ∈ Z, n = 2b + 1

a) Prove P(n) by contraposition

b) Prove P(n) contradiction

c) Prove P(n) using induction

(1 point) A bacteria culture starts with 560 bacteria and grows at a rate proportional to its size. After 3 hours there will be 1680 bacteria.

(a) Express the population after t hours as a function of t
population:  (function of t)

(b) What will be the population after 2 hours?


(c) How long will it take for the population to reach 1250 ?

1. (14) List the elements for each of the following sets:

(1) P({a, b, c})                                                         (Note: P refers to power set)

(2)   P{a, b}) - P({a, c})

(3)   P(Æ)

(4) {x Î ℕ: (x £ 7 Ù x ³ 7}                        (Note: ℕ is the set of nonnegative integers)

(5)   {x Î ℕ: $y Î ℕ (y < 10 Ù (y + 2 = x))}           

(6)    {x Î ℕ: $y Î ℕ ($z Î ℕ ((x = y + z) Ù (y < 5) Ù (z < 4)))}

(7)   {a, b, c} x {c, d}                                                 (Note: x refers to Cartesian product)

2. (12) True or False.

Let R = {(1, 2), (2, 3), (1, 1), (2, 2), (3, 3), (1, 3)}.

(1) R is reflexive.

(2) R is transitive.

(3) R is symmetric.

(4) R is antisymmetric.

3. (16) True or False.

(1) Subset-of is a partial order defined on the set of all sets.

(2) Subset-of is a total order defined on the set of all sets.

(3) Proper-subset-of is a partial order defined on the set of all sets.

(4) Proper-subset-of is a total order defined on the set of all sets.

(5) Less than or equal (<=) is a partial order defined on the set of real numbers.

(6) Less than or equal (<=) is a total order defined on the set of real numbers.

(7) Less than (<) is a partial order defined on the set of real numbers.

(8) Less than (<) is a total order defined on the set of real numbers.

4. (12) True or False.

(1) f (x) = 2x is onto where f: R -> R.           (Note: R is the set of real numbers)

(2) f (x) = 2x is one-to-one where f: R -> R.

(3) f(x) = x² is onto where f: R -> R.

(4) f(x) = x² is one-to-one where f: R -> R.

(5) f(x) = x² is onto where f: R -> [0, ∞).

(6) f(x) = x² is one-to-one where f: R -> [0, ∞).

5. (6) Let ℕ be the set of nonnegative integers. For each of the following sentences in first-order logic, state whether the sentence is valid, is satisfiable (but not valid), or is unsatisfiable.

(1) "x Î ℕ ($y Î ℕ (y < x)).

(2) "x Î ℕ ($y Î ℕ (y > x)).

(3) "x Î ℕ ($y Î ℕ f(x) = y).

6. (20) Are the following sets closed under the given operations? Answer yes or no. If the answer is no, please specify what the closure is.

(1) The negative integers under subtraction.

(2) The odd integers under the operation of mod 3.

(3) The positive integers under exponentiation.

(4) The finite sets under Cartesian product.

(5) The rational numbers under addition.

7. (20) True or False. If the answer is true, provide an example (Hint: use subsets of integers and real numbers) as a proof.

(1) The intersection of two countably infinite sets can be finite.

(2) The intersection of two countably infinite sets can be countably infinite.

(3) The intersection of two uncountable sets can be finite.

(4) The intersection of two uncountable sets can be countably infinite.

(5) The intersection of two uncountable sets can be uncountable.

Suppose we want to use Twitter activity to predict box office receipts on the opening weekend for movies. Assuming a linear relationship, the Excel output for this regression model is given below.

Excel output:

(a) State the regression equation for this problem.

(b) Interpret the meaning of b₀ and b₁ in this problem.

(c) Predict the box office receipts on the opening weekend for a movie that has a Twitter activity of 110,000.

(d) At the 0.05 level of significance, is there evidence of a linear relationship between the Twitter activity and the box office receipts on the opening weekend for a movie?

(e) Construct a 95% confidence interval estimate of the population slope β₁. Interpret the confidence interval estimate.

(f) How useful do you think this regression model is for predicting the box office receipts on the opening weekend for a movie?

A particular mattress company has three factories (1,2,3), each of which produces three types of mattresses: (1) spring, (2) foam, and (3) hybrid. In the matrix

? = [ 80 50 25

45 110 60]

??? represents the number of mattresses of type ? produced at factory ? in one day. Find the production levels if production increases by 15%.

Here are mass spectrometric signals for methane in H2:

CH2 (Vol %) 0 0.062 0.122 0.245 0.486 0.971 1.921

Signal (mV) 9.1 47.5 95.6 193.8 387.5 812.5 1671.9

a) Subtract blank value from all other values. Then use the method of least squares to find the slope and intercept and their uncertainties.

b)Replicate measurements of an unknown gave 152.1, 154.9, 153.9, and 155.1 mV and a blank gave 8.2, 9.4, 10.6, and 7.8 mV. Subtract average blank from average unknown to find average corrected signal for the unknown.

c) Find concentration of the unknown and uncertainty.

3) consider computing e^x-1/x

a) Explain when and why a loss of significance can happen.

b) show how you can remedy the problem. Hint: use a Taylor series.

Put a metric ρ on all the words in a dictionary by defining the distance between two distinct words to be 2^−n if the words agree for the first n letters and are different at the (n+1)st letter. A space is distinct from a letter. E.g., ρ(car,cart)=2^−3 and ρ(car,call)=2^−2.

a) Verify that this is a metric.
b) Suppose that words w1, w2 and w3 are listed in alphabetical order.
Find a formula for ρ(w1,w3) in terms of ρ(w1,w2) and ρ(w2,w3).

using / for integral

Evaluate the double integral //R cos( (y-x)/(y+x) )dA where R is the trapezoidal region with vertices (1,0), (2,0), (0,2), and (0,1)

A manufacturer of animal feed makes two grades of food. Each bag of high grade feed contains 10kg of wheat brand and 5kg of maize, while each bag of low-grade feed contains 12kg of wheat brand and 3kg of maize. There are 1920kg of wheat brand and 780kg of maize currently available. The manufacture can make a profit of ¢12,000 on each bag of high grade and ¢10000 on each bag of the low-grade feed. Determine the number of bags of each grade to produce to maximize profit.