6.1.5. Problem. Let J be the open unit interval (0, 1). For each a let Ua...

6.1.5. Problem. Let J be the open unit interval (0, 1). For each a let Ua = ?a, a + 1 ?, and let

U = {Ua : 0 ≤ a ≤ 34 }. Then certainly U covers J .

(a) Find a finite subfamily of U which covers J.

(b) Explain why a solution to (a) does not suffice to show that J is compact.

(c) Show that J is not compact.

Expert Solution

Colby Messinger answered 2 years ago

Problem 3. Consider the unit interval [0, 1], and let ξ be fixed real number with...

Problem 3. Consider the unit interval [0, 1], and let ξ be fixed real number with ξ ∈ (0, 1) (note that the case ξ = 1/3 corresponds to the regular Cantor set we learned in our lectures). In stage 1 of the construction, remove the centrally situated open interval in [0, 1] of length ξ. In stage 2 remove the centrally situated open intervals each of relative length ξ (i.e. if the interval has length a you remove an...

Problem 3. Consider the unit interval [0, 1], and let ξ be fixed real number with...

Problem 3. Consider the unit interval [0, 1], and let ξ be fixed real number with ξ ∈ (0, 1) (note that the case ξ = 1/3 corresponds to the regular Cantor set we learned in our lectures). In stage 1 of the construction, remove the centrally situated open interval in [0, 1] of length ξ. In stage 2 remove the centrally situated open intervals each of relative length ξ (i.e. if the interval has length a you remove an...

1) Let U1, U2, ... be independent random variables, each uniformly distributed over the interval (0,...

1) Let U1, U2, ... be independent random variables, each uniformly distributed over the interval (0, 1]. These random variables represent successive bigs on an asset that you are trying to sell, and that you must sell by time = t, when the asset becomes worthless. As a strategy, you adopt a secret number \Theta and you will accept the first offer that's greater than \Theta . The offers arrive according to a Poisson process with rate \lambda = 1....

Let L = {aibj | i ≠ j; i, j ≥ 0}. Design a CFG and...

Let L = {aibj | i ≠ j; i, j ≥ 0}. Design a CFG and a PDA for this language. Provide a direct design for both CFG and PDA (no conversions from one form to another allowed).

2. Let X be a uniform random variable over the interval (0, 1). Let Y =...

2. Let X be a uniform random variable over the interval (0, 1). Let Y = X(1-X). a. Derive the pdf for Y . b. Check the pdf you found in (a) is a pdf. c. Use the pdf you found in (a) to find the mean of Y . d. Compute the mean of Y by using the distribution for X. e. Use the pdf of Y to evaluate P(|x-1/2|<1/8). You cannot use the pdf for X. f. Use...

Let U be a continuous uniform variable over the interval [0, 1]. What is the probability...

Let U be a continuous uniform variable over the interval [0, 1]. What is the probability that U falls within kσ of its mean for k = 1, 2, 3?

3) Prove that the cardinality of the open unit interval, (0,1), is equal to the cardinality...

3) Prove that the cardinality of the open unit interval, (0,1), is equal to the cardinality of the open unit cube: {(x,y,z) E R^3|0<x<1, 0<y<1, 0<Z<1}. [Hint: Model your argument on Cantor's proof for the interval and the open square. Consider the decimal expansion of the fraction 12/999. It may prove handdy]

Let A = [ 0 2 0 1 0 2 0 1 0 ]  . (a)...

Let A = [ 0 2 0 1 0 2 0 1 0 ]  . (a) Find the eigenvalues of A and bases of the corresponding eigenspaces. (b) Which of the eigenspaces is a line through the origin? Write down two vectors parallel to this line. (c) Find a plane W ⊂ R 3 such that for any w ∈ W one has Aw ∈ W , or explain why such a plain does not exist. (d) Write down explicitly...

a) The J = 0 to J =1 rotational transition of the CO molecule occurs at...

a) The J = 0 to J =1 rotational transition of the CO molecule occurs at a frequency of 1.15×1011 Hz. Use this information to calculate the moment of inertia of the molecule about its center of mass. b) The CO molecule shows a strong absorption line at the frequency 6.42×1013 Hz due to transition from thev= 0 to the v = 1 vibrational levels. Calculate the effective force constant for this molecule. c) Why does CO molecule have pure...

. Let xj , j = 1, . . . n be n distinct values. Let...

. Let xj , j = 1, . . . n be n distinct values. Let yj be any n values. Let p(x) = c1 + c2x + c3x 2 + · · · + cn x ^n−1 be the unique polynomial that interpolates the data (xj , yj ), j = 1, . . . , n (Vandermonde approach). (a) Remember that (xj , yj ), j = 1, . . . , n are given. Derive the n...

Question

6.1.5. Problem. Let J be the open unit interval (0, 1). For each a let Ua...

Solutions

Expert Solution

Related Solutions

Problem 3. Consider the unit interval [0, 1], and let ξ be fixed real number with...

Problem 3. Consider the unit interval [0, 1], and let ξ be fixed real number with...

1) Let U1, U2, ... be independent random variables, each uniformly distributed over the interval (0,...

Let L = {aibj | i ≠ j; i, j ≥ 0}. Design a CFG and...

2. Let X be a uniform random variable over the interval (0, 1). Let Y =...

Let U be a continuous uniform variable over the interval [0, 1]. What is the probability...

3) Prove that the cardinality of the open unit interval, (0,1), is equal to the cardinality...

Let A = [ 0 2 0 1 0 2 0 1 0 ]  . (a)...

a) The J = 0 to J =1 rotational transition of the CO molecule occurs at...

. Let xj , j = 1, . . . n be n distinct values. Let...