Compression of a bit string x of length n involves creating a program shorter than n bits that returns

x. The Kolmogorov complexity of a string K(x) is the length of shortest program that returns x (i.e.

the length of a maximally compressed version of x).

(a) Explain why "the smallest positive integer not definable in under 100 characters" is paradoxical.

(b) Prove that for any length n, there must be at least one bit string that cannot be compressed to

fewer than n bits.

(c) Imagine you had the program K, which outputs the Kolmogorov complexity of string. Design

a program P that when given integer n outputs the bit string of length n with the highest

Kolmogorov complexity. If there are multiple strings with the highest complexity, output the

lexicographically first (i.e. the one that would come first in a dictionary).

(d) Suppose the program P you just wrote can be written in m bits. Show that P and by extension,

K, cannot exist, for a sufficiently large input n.

Find an equation of the ellipse with foci at (−4,3) and (−4,−9) and whose major axis has length 30. Express your answer in the form P(x,y)=0, where P(x,y) is a polynomial in x and y such that the coefficient of x^2 is 225.

A group of 268 overweight people signed up for a weight loss program. Participants were asked to choose whether they wanted to go on a vegetarian diet or follow a traditional low-calorie diet that included some meat. 153 people chose the vegetarian diet, and the rest chose to be in the control group and continue to eat meat. There was a greater weight loss in the vegetarian group.

1. What is the sample size of each group? What percentage of people was in the vegetarian group? What percentage was in the meat group? (Round to the nearest percent.)
2. What are the variables in this study? Remember that variables are used to record the characteristics of individual people or things. The description above does not necessarily name all the variables - you have to make an educated guess about what characteristics were measured and recorded in order to conduct this study. For each variable, indicate whether it is a categorical variable or a numerical variable, and explain why.
3. What confounding variables might account for the increased weight loss in the vegetarian group? Explain.
4. an observational or a controlled study?
5. should I design the study that is more likely to remove the effect of confounding variables

There are three forms of payment to buy a car. The first is to buy the car in cash at a price of $ 120,000. The second is to pay 60 equal monthly payments of $ 3,164.47 each month, the first one a month after purchase. The third way to acquire the car is by paying 48 equal monthly payments of $ 1,955.00 each, starting to pay a month after making the purchase, and also paying four equal annuities of $ 21, 877.83 at the end of the months 12, 24, 36 and 48. With an annual interest of 24% capitalized monthly, it determines the best form of payment from the economic point of view. a) Perform the corresponding calculations of the three options that allow you to compare total cost. b) Discuss which of the three options is best for the buyer and why.

Please explain each step very carefully and make sure your handwriting is easy to read. Thank you

Question: Suppose p(x) is a polynomial of degree n with coefficients in R and suppose p(x) has exactly n real roots. Show that p'(x) has exactly n-1 real roots.

Compute the following:

(a) 13^{^2018} (mod 12)

(b) 8^{^11111} (mod 9)

(c) 7^{^256} (mod 11)

(d) 3^¹⁶⁰ (mod 23)

This is the Cryptography question . I wanted to know correct period with explanation

Let the LFSR be xn+5 = xn + xn+3, where the initial values are x0=0, x1=1, x2=0, x3=0, x4=0.

(a) Compute first 20 bits of the above LFSR.

(b) What is the period?

5. To get a real feeling for how adjusted winner procedure works, it is a good idea to try it yourself. Make up a list of 10 items that you and a friend will divide using the adjusted winner procedure. Can you devise a strategy that will give you an advantage?

Let {λ_n} be a sequence of scalars that converges to zero, lim_n→∞ λ_n = 0. Show that the operator A : ℓ² → ℓ² , A(x₁, x₂, ..., x_n, ...) = (λ₁x₁, λ₂x₂, ..., λ_nx_n, ...) is compact. What is the spectrum of this operator?

Kim Walrath has a nutritional deficiency and is told to take at least 2400 mg of​ iron, 2100 mg of vitamin​ B-1, and 1800 mg of vitamin​ B-2. One Maxivite pill contains 40 mg of​ iron, 10 mg of vitamin​ B-1, and 8 mg of vitamin​ B-2 and costs ​$0.07. One Healthovite pill provides 10 mg of​ iron, 15 mg of vitamin​ B-1, and 18 mg of vitamin​ B-2 and costs ​$0.08. Complete parts​ (a) and​ (b) below.

(a) What combination of Maxivite and Healthovite pills will meet​ Kim's requirement at lowest​ cost? What is the lowest​ cost?

In your solution for part​ (a), does Kim receive more than the minimum amount she needs of any​ vitamin? If​so, which vitamin is​ it?

Students at Upscale University are required to take at least 4 humanities and 4 science courses. The maximum allowable number of science courses is 12. Each humanities course carries 4 credits and each science course 3 credits. The total number of credits in science and humanities cannot exceed 60. Quality points for each course are assigned in the usual​ way: the number of credit hours times 4 for an A​ grade, times 3 for a B​ grade, and times 2 for a C grade. Susan Katz expects to get​ B's in all her science courses. She expects to get​ C's in one dash fourth of her humanities​ courses, B's in​ one-fourth of​ them, and​ A's in the rest. Under these​ assumptions, how many courses of each kind should she take in order to earn the maximum possible number of quality​ points?

Hello.

This is an exercise is from Hoffman, Linear Algebra, chapter 7.4; but it has no solution, Can you help me to understand how to solve it? I have just a very general idea of how to solve it and I am afraid that, if the degree of the polynomial is changed, I may fail the solution.

4. Construct a linear operator T with minimal polynomial x^2 (x - 1)^2 and characteristic
polynomial x^3(x-1)^4. Describe the primary decomposition of the vector
space under T and find the projections on the primary components. Find a basis
in which the matrix of T is in Jordan form. Also find an explicit direct sum decomposition
of the space into T-cyclic subspaces as in Theorem 3 and give the invariant
factors.

Best Regards.

1. Find the half-range expansions of the given function. To illustrate the convergence of the cosine and sine series, plot several partial sums of each and comment on the graph (using words).
2. f(x) = 1 if 0 < x < 1.
3. f(x) = π-x if 0 ≤ x ≤ π.
4. f(x) = x² if 0 < x <1.

8)

Use the Laplace transform to solve the given initial-value problem.

y' − y = 2 cos(4t),  y(0) = 0

y(t)=???

9)

Use the Laplace transform to solve the given initial-value problem.

y'' − 5y' = 8e^4t − 4e^−t,    y(0) = 1, y'(0) = −1

y(t)=?

10)

Use the Laplace transform to solve the given initial-value problem.

y''' + 2y'' − y' − 2y = sin(4t),  y(0) = 0,  y'(0) = 0,  y''(0) = 1

y(t)=?