Use the subset construction to build DFAs that correspond to the given NFAs. Write out your DFA tables by listing all of the NFA states that are part of the same DFA state, and whether something is final.

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Discrete Math

State a b c Final? 1 3 — 2 x 2 4 x 3 3 5 x 4 6 ✓ 5 5 6 ✓ 6 1 ✓

Problem 2
Write regular expressions that match the following string patterns. Feel free to ask on Piazza whether example strings match the pattern or not, if you’re not sure. (a) All strings made of of as, bs, and cs, and ds that have only two different kinds of letters in them . The empty string counts. (b) All strings made up of as and bs that have an even number of letters .

Problem 3
(30 points) Draw NFAs that match the following regular expressions: (i) (10 points) ((ab)1(ba))* (ii) (10 points) (Pih)(ii0)(tIp) (iii) (10 points) e(Y1x)*Y”

Problem 4
Use the subset construction to build DFAs that correspond to the given NFAs. Write out your DFA tables by listing all of the NFA states that are part of the same DFA state, and whether something is final. For example:
(a) NFA 1:
State a b c Final?
1,2 2,4 — 1,4 x
y

 

Shows what the next state would be if you’re in state i and see symbol j. Consider the DFA below for the regular expression (yI(xx*))y.

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Discrete Math

Problem 1

You can build a transition table for any deterministic finite automaton. This table has as many rows as states, and as many columns as symbols in the alphabet that the strings are made out of . Entry T[i][x]

Shows what the next state would be if you’re in state i and see symbol j. Consider the DFA below for the regular expression (yI(xx*))y:

The transition table for this is:

Y

—00.

State x y Final? 1 2 3 x 2 2 4 x 3 — 4 x 4 ✓

Note that we don’t indicate a transition to the error state. Instead, we just mark the transition as —. We saw an example of this kind of transition table when building the subset construction. (a) (10 points) Draw the transition table for the following DFA: b —OP. d c b —No-(b) (10 points) Draw the DFA corresponding to the following transition table :

1

 

How many elements are in the conjugacy class of (123) in S 4? In A 4? How many elements are in the conjugacy class of (123) in S 5? In A 5?

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Algebra

Galois theory and its application

  • What is the degree of Q(\sqrt{2},\sqrt{3},\sqrt{6})?
  • State the field-theoretic condition that determines whether the point (x,y) can be constructed using compass and straight-edge.
  • Which subsets of S_3 can occur as kernels of maps of groups S_3 –> G?
  • Suppose that f: G –> H is a group homomorphism, briefly explain why the ker(f) must be a normal subgroup.
  • Consider the map from S_4 –> S_3 coming from the action of S_4 on partitions of {1,2,3,4} into two 2-element subsets. Exhibit a non-trivial element in the kernel of this map.
  • Give an example of a non-trivial proper subgroup of the dihedral group D {2*4} which is normal and another which is not normal.
  • Consider the action of S_n on ordered pairs of elements of {1, …, n}, what is the stabilizer of (1,2)?
  • Consider the action of S_n on unordered pairs of elements of {1, …, n}, what is the stabilizer of {1,2}?
  • How many elements are in the conjugacy class of (123) in S 4? In A 4?
  • How many elements are in the conjugacy class of (123) in S 5? In A 5?
  • The group Z/3Z acts on the set V of vertices of a cube via rotations around the long diagonal. Describe the orbits of this action. Find subgroups H i such that V is isomorphic to the disjoint union of G/H i as G-sets.
  • Suppose K is a field containing the pth roots of unity, and suppose L:K is a Galois extension with Galois group G, and a is an element of L which is not in K but whose pth power is in K. Describe a nontrivial map G –> Z/pZ.
  • Suppose f: S_n –> Z/pZ is a non-trivial group homomorphism, briefly explain why it follows that p = 2.
  • Suppose f: A n –> Z/pZ is a non-trivial group homomorphism, briefly explain why p = 3.
  • Give an explanation of no more than three sentences explaining the important difference between 4 and 5 that explains why there’s a quartic formula but no quintic formula.
  • Let p be a prime, and z be a primitive pth root of unity, write down all the elemeents of the Galois group \Gamma(Q(z):Q) as automorphisms and describe how they compose.
  • Give an example of a field extension L:K where the Galois correspondence does hold. Your answer should include a subgroup or an intermediate subfield where the correspondence breaks down.

 

Elaborate on why it works and what you found particularly interesting about it. Can you think of other tricks that aren’t listed? Lastly, do you think these ‘shortcuts’ would be worth teaching to lower grade school levels, Why or why not?

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Vedic mathematics

  1. Written Assignment: Follow the links provided below and read up some more on Vedic mathematics. There are plenty of more tricks than the ones we explored in class!

Afterward, write a 1-2-page response  explaining the methods behind your chosen techniques. Elaborate on why it works and what you found particularly interesting about it. Can you think of other tricks that aren’t listed? Lastly, do you think these ‘shortcuts’ would be worth teaching to lower grade school levels, Why or why not?

  • Link 1: https://youtu.be/grkWGeqW99c
  • Link 2: https://blog.pcmbtoday.com/10-vedic-maths-tricks-rapid-calculations/
  • Link 3: https://www.vedantu.com/blog/vedic-maths-tricks
  • Link 4: http://mathlearners.com/

Exercises

Calculate the products using the Vedic method of multiplication used in class/video. Show all your steps.

992 x 996

74 x 89

9835 x 10,003

 

Discuss the relationship between the p value and the significance level, including a comparison between the two, and decide to reject or fail to reject the null hypothesis.

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Hypothesis Testing for Regional Real Estate Company

Scenario

You have been hired by the Regional Real Estate Company to help them analyze real estate data. One of the company’s Pacific region salespeople just returned to the office with a newly designed advertisement. The average cost per square foot of home sales based on this advertisement is $280. The salesperson claims that the average cost per square foot in the Pacific region is less than $280. In other words, he claims that the newly designed advertisement would result in higher average cost per square foot in the Pacific Region. He wants you to make sure he can make that statement before approving the use of the advertisement. In order to test his claim, you will generate a random sample of size 750 using data for the Pacific region and use this data to perform a hypothesis test.

Prompt

Generate a sample of size 750 using data for the Pacific region. Then, design a hypothesis test and interpret the results using significance level α = .05. You will work with this sample in this assignment. Briefly describe how you generated your random sample.

Use the House Listing Price by Region Spreadsheet document and the National Summary Statistics and Graphs House Listing Price by Region PDF documents to help support your work on this assignment. You may also use the Descriptive Statistics in Excel and Creating Histograms in Excel tutorials for support.

Specifically, you must address the following rubric criteria, using the Module Five Assignment Template Word Document.

  • Hypothesis Test Setup: Define your population parameter, including hypothesis statements, and specify the appropriate test.
    • Define your population parameter.
    • Write the null and alternative hypotheses.
    • Specify the name of the test you will use.
      • Identify whether it is a left-tailed, right-tailed, or two-tailed test.
    • Identify your significance level.
  • Data Analysis Preparations: Describe sample summary statistics, provide a histogram and summary, check assumptions, and find the test statistic and significance level.
    • Provide the descriptive statistics (sample size, mean, median, and standard deviation).
    • Provide a histogram of your sample.
    • Describe your sample by writing a sentence describing the shape, center, and spread of your sample.
    • Determine whether the conditions to perform your identified test have been met.
  • Calculations: Calculate the p value, describe the p value and test statistic in regard to the normal curve graph, discuss how the p value relates to the significance level, and compare the p value to the significance level to reject or fail to reject the null hypothesis.
    • Calculate the sample mean and standard error.
    • Determine the appropriate test statistic, then calculate the test statistic.
      Note: This calculation is (mean – target)/standard error. In this case, the mean is your regional mean (Pacific), and the target is 280.
    • Calculate the p value.
      Note: For right-tailed, use the T.DIST.RT function in Excel, left-tailed is the T.DIST function, and two-tailed is the T.DIST.2T function. The degree of freedom is calculated by subtracting 1 from your sample size.
      Choose your test from the following:
      =T.DIST.RT([test statistic], [degree of freedom])
      =T.DIST([test statistic], [degree of freedom], 1)
      =T.DIST.2T([test statistic], [degree of freedom])
    • Using the normal curve graph as a reference, describe where the p value and test statistic would be placed.
  • Test Decision: Discuss the relationship between the p value and the significance level, including a comparison between the two, and decide to reject or fail to reject the null hypothesis.
    • Discuss how the p value relates to the significance level.
    • Compare the p value and significance level, and make a decision to reject or fail to reject the null hypothesis.
  • Conclusion: Discuss how your test relates to the hypothesis and discuss the statistical significance.
    • Explain in one paragraph how your test decision relates to your hypothesis

When Mendel conducted his famous genetics experiments with peas, he found that in sample of 600 peas, 168 were yellow peas. Find a 99% confidence interval for p, the true proportion of yellow peas in the sample.

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TEST #4 11M012022

#1 The mean cholesterol level of a random sample of 100 adults aged 45 – 54 was 216 milligrams with a standard deviation of 19.5 milligrams. Construct a 98% confidence interval for p, the true mean cholesterol level of adults aged 45 54.

#2 When Mendel conducted his famous genetics experiments with peas, he found that in sample of 600 peas, 168 were yellow peas. Find a 99% confidence interval for p, the true proportion of yellow peas in the sample.

#3 A special fat reducing diet is tested on 16 women. The results of the test indicated that the mean weight loss was 12 pounds. If the standard deviation was 1.8 pounds, construct a 90% confidence interval for p, the true mean weight Poss.

Evaluate the following integral, ffzdS, where S is the part of the sphere x2 + y2 + z2 = 81 that lies above the cone z = Ag■,/x2 + y2 .

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Problem #3:

Evaluate the following integral, ffzdS, where S is the part of the sphere x2 + y2 + z2 = 81 that lies above the cone z = Ag■,/x2 + y2 .

Enter your answer symbolically, as in these examples

What is the fully encrypted message that Alice receives?

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Message, “JAW”

The message, “JAW,” which he plans to encrypt using Alice’s RSA cypher with public key (pq, e) = (55, 3). To encrypt the message, Bob uses the method described in Example 8.4.9. He encodes one letter at a time using Next, he applies the encrypting formula where M is a plaintext letter and C is a block of ciphertext. Because 55 is a two-digit integer, each block of ciphertext is a two-digit integer with represented as

(a) When the first letter of the message is encoded, the result is . Bob then applies the encrypting formula to find that the first two-digit block of the encrypted message is

(b) The second two-digit block of Bob’s encrypted message is

(c) What is the fully encrypted message that Alice receives?

A = 01, B = 02, C = 03, , Z = 26.

C = M mod pq,e

0, 1, 2 , 9 01, 02, 03, , 09.

,

mod 55 = .3

mod 55 = .3

,

hat is the regression equation? What is the best predicted weight of a bear with a chest size of 39 inches?

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Math80 assignment

This is the week13 Math80 assignment Name_

Watch Week13 YouTube video-Ch10.1a – Create a Scatterplot in StatCrunch

Watch Week13 YouTube video-Ch10.1b – Create a Scatterplot in StatCrunch

Watch Week13 YouTube video-Ch10.1c – Determine Strongest Linear Correlation

Watch Week13 YouTube video-Ch10.1d – Predict the Linear correlation coefficient

Watch Week13 YouTube video-Ch10.2a – Find the Linear Regression Line – Ti84

Watch Week13 YouTube video-Ch10.2b – Find Linear Regression line – StatCrunch

Watch Week13 YouTube video-Ch10.2c – Find Linear Regression line – StatCrunch

Part(a) What is the regression equation?

Part(b) What is the best predicted weight of a bear with a chest size of 39 inches?

Verify the statement of the first isomorphism theorem from the lecture for the groups G = Z2 se Z2 se Z2, H = {0} ED Z2 ED Z2 and K = Z2 ED Z2 ED {0}.

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ALGEBRA

Verify the statement of the first isomorphism theorem from the lecture for the groups G = Z2 se Z2 se Z2, H = {0} ED Z2 ED Z2 and K = Z2 ED Z2 ED {0}. (5 Points)