#18 on the textbook b MATLAB: An Introduction with A XThomas W. Hungerford - Abstrac ×+O File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(2014).pdf-- A' Read aloud V DrawF HighlightO Erase117of 621(b) Give an example in Z[x] to show that part (a) may be false if the leadingcoefficient of g(x) is not a unit. [Hint: Exercise 5(b) with Z in place of Q.]Ceursu15. Let R be a commutative ring with identity and a e R.(a) If a = Or, show that 12 + ax is a unit in R[x]. [Hint: Consider 1 - ax +a*x*](b) If a' = Og, show that 1g + ax is a unit in R[x].16. Let R be a commutative ring with identity and a e R. If 12 + ax is a unit inR[x], show that d = Og for some integer n> 0. [Hint: Suppose that the inverseof 1g + ax is bo + bjx + bx + ...+ b. Since their product is 1g, bo = 1r(Why?) and the other coefficients are all Og.]17. Let R be an integral domain. Assume that the Division Algorithm alwaysholds in R[x]. Prove that R is a field.18. Let o:R[x] → R be the function that maps each polynomial in R[x] onto itsconstant term (an element of R). Show that o is a surjective homomorphismof rings.19. Let ç:Z[x] →Z,[x] be the function that maps the polynomial a, + ajx + . ..+ax in Z[x] onto the polynomial [ao] + [q]x + • . . + [a]x*, where [a] denotesthe class of the integer a in Z,. Show that o is a surjective homomorphism ofrings.20. Let D:R[x]→ R[x] be the derivative map defined byD(a + ax + ax++ax) = a¡ + 2azx + 3azx +...+ na,-.Is Da homomorphism of rings? An isomorphism?C.21. Let h:R- S be a homomorphism of rings and define a function h:R[x] -> S[x]by the ruleh(ao + ajx + ...+ ax) = h(a) + h(aj)x + h(a)x² +...+ h(a,)xª.Prove that Snt(a) h is a homomorphism of rings.(b) k is injective if and only if h is injective.(c) h is surjective if and only if h is surjective.(d) If R = S, then R[x] = S[x].22. Let R be a commutative ring and let k(x) be a fixed polynomial in R[x). Provethat there exists a unique homomorphism o:R[x] → R[x] such thatP(r) = r for allre Randp(x) = k(x).4.2Divisibility in F[x]All the results of Section 1.2 on divisibility and greatest common divisors in Z nowcarry over, with only minor modifications, to the ring of polynomials over a field.11:12 AMe Type here to searchEPICAiEPIC99+10/30/2020

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17. Let R be an integral domain. Assume that the Division Algorithm always holds in R[x). Prove that Ris a field. 18. Let e:R- R be the function that maps each polynomial in Rx] onto its constant term (an element of R). Show that e is a surjective homomorphism of rings 19. Let eZx] -ZJx] be the function that maps the polynomial a + ax +...+

17. Let R be an integral domain. Assume that the Division Algorithm always holds in R[x). Prove that Ris a field. 18. Let e:R- R be the function that maps each polynomial in Rx] onto its constant term (an element of R). Show that e is a surjective homomorphism of rings 19. Let eZx] -ZJx] be the function that maps the polynomial a + ax +...+

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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In the construction process for solid-state flash memory drives (thumb drives), nearly all drives areconstructed with a large potential amount of storage space, and are sold base on the actual amountof memory that forms correctly during the process (For example, drives that are designed to store 64gb but contain a large number of bad sectors will instead be formated and sold as 32 gb). Consider amanufacturing process where every chip has a 21% of having enough bad sectors to be downgraded,and these errors are random and independent. (a) If we wish to compute the probability that x number of chips out of a production run of size n are downgraded, do we have a probability distribution that will work for that? Are the assumptionsfor that met? Explain. (b) Compute the probability that between 3 and 5 (inclusive)or between 2 and 8 (inclusive) are downgraded. (c) Graph the probability mass function for x number of chips out of a production run of 10 aredowngraded.
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From time to time, the UTM Human Resource (UTMHR) department observes various employees fortheir work productivity. Recently UTMHR wanted to check whether the four employees at theDepartment XYZ counters, serve on average the same number of customers per hour. The HR managerobserved each of the four employees for a certain number of hours. The following Table 5 gives thenumber of customers served by the four employees during each of the observed hours. employee a employee b employee c employee d 19 14 11 24 21 16 14 19 26 14 21 21 24 13 13 26 18 17 16 20 employee a employee b employee c employee d mean 21.6 14.8 15.0 22.0 s 3.4 ? 3.8 ? c) Find variance between sample,d) Find variance within sample, e) Calculate the test statistic, F.f) Determine the numerator and denominator.g) Find the critical value of F at the 5% significance level, and test the claim that the mean numberof customers served per hour by each of these four employees is…
From time to time, the UTM Human Resource (UTMHR) department observes various employees fortheir work productivity. Recently UTMHR wanted to check whether the four employees at theDepartment XYZ counters, serve on average the same number of customers per hour. The HR managerobserved each of the four employees for a certain number of hours. The following Table 5 gives thenumber of customers served by the four employees during each of the observed hours. employee a employee b employee c employee d 19 14 11 24 21 16 14 19 26 14 21 21 24 13 13 26 18 17 16 20 a) Define the hypothesis null, H0 and hypothesis alternative, H1.b) Given below are means and standard deviation for some of the employees. Calculate thestandard deviation for Employee B and D.
A large operator of timeshare complexes requires anyoneinterested in making a purchase to first visit thesite of interest. Historical data indicates that 20% of allpotential purchasers select a day visit, 50% choose aone-night visit, and 30% opt for a two-night visit. Inaddition, 10% of day visitors ultimately make a purchase,30% of one-night visitors buy a unit, and 20%of those visiting for two nights decide to buy. Supposea visitor is randomly selected and is found to havemade a purchase. How likely is it that this person madea day visit? A one-night visit? A two-night visit?
Suppose that in manufacturing a very sensitive electronic component, a company and its customers have tolerated a 2% defective rate. Recently, however, several customers have been complaining that there seem to be more defectives than in the past. Given that the company has made recent modifications to its manufacturing process, it is wondering if in fact the defective rate has increased from 2%. For quality assurance purposes, you decide to randomly select 1,000 of these electronic components before they are shipped to customers. Of the 1,000 components, you find 25 that are defective. Assume that the company produces a very large number of these components on any given day. Set up an appropriate hypothesis to test whether or not the defect rate has increased. Before proceeding to test your hypothesis, check that all assumptions and conditions are satisfied for such a test. Conduct the test using a .05 level of significance (alpha) and state your decision about…
Suppose that in manufacturing a very sensitive electronic component, a company and its customers have tolerated a 2% defective rate. Recently, however, several customers have been complaining that there seem to be more defectives than in the past. Given that the company has made recent modifications to its manufacturing process, it is wondering if in fact the defective rate has increased from 2%. For quality assurance purposes, you decide to randomly select 1,000 of these electronic components before they are shipped to customers. Of the 1,000 components, you find 25 that are defective. Assume that the company produces a very large number of these components on any given day. Conduct the test using a .05 level of significance (alpha) and state your decision about whether or not you believe that the defect rate has increased. What would be the minimum number of defectives in a random sample of 1,000 would you need to find in order to statistically decide that the defect…
Data on the 4000 largest mutual funds shows which funds provided a high 5-year return and a high 10-year return. Of the 4000 mutualfunds surveyed, 3000 funds had a high 5-year return, 2000 had a high 10-yearreturn, and 1500 had both a high 5-year return and a high 10-year return Given that a mutual fund had a high 5-year return, what is theprobability of a mutual fund having a high 10-year return?
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Three components are connected to form a system asshown in the accompanying diagram. Because the componentsin the 2–3 subsystem are connected in parallel, thatsubsystem will function if at least one of the two individualcomponents functions. For the entire system to function,component 1 must function and so must the 2–3 subsystem. The experiment consists of determining the condition ofeach component [S (success) for a functioning componentand F (failure) for a nonfunctioning component].a. Which outcomes are contained in the event A thatexactly two out of the three components function?b. Which outcomes are contained in the event B that atleast two of the components function?c. Which outcomes are contained in the event C that thesystem functions?d. List outcomes in C9, A ø C, A ù C, B ø C, andB ù C.