Construction Of Real Numbers

Construction Of Real Numbers All mathematicians know (or think they know) all about the real numbers. However usually we just accept the real numbers as being there rather than considering precisely what they are. In this project I will attempts to answer that question. We shall begin with positive integers and then successively construct the rational and finally the real numbers. Also showing how real numbers satisfy the axiom of the upper bound, whilst rational numbers do not. This shows that all real numbers converge towards the Cauchys sequence. 1 Introduction What is real analysis; real analysis is a field in mathematics which is applied in many areas including number theory, probability theory. All mathematicians know (or think they know) all about the real numbers. However usually we just accept the real numbers as being there rather than considering precisely what they are. The aim of this study is to analyse number theory to show the difference between real numbers and rational numbers. Developments in calculus were mainly made in the seventeenth and eighteenth century. Examples from the literature can be given such as the proof that Ï€ cannot be rational by Lambert, 1971. During the development of calculus in the seventeenth century the entire set of real numbers were used without having them defined clearly. The first person to release a definition on real numbers was Georg Cantor in 1871. In 1874 Georg Cantor revealed that the set of all real numbers are uncountable infinite but the set of all algebraic numbers are countable infinite. As you can see, real analysis is a somewhat theoretical field that is closely related to mathematical concepts used in most branches of economics such as calculus and probability theory. The concept that I have talked about in my project are the real number system. 2 Definitions Natural numbers Natural numbers are the fundamental numbers which we use to count. We can add and multiply two natural numbers and the result would be another natural number, these operations obey various rules. (Stirling, p.2, 1997) Rational numbers Rational numbers consists of all numbers of the form a/b where a and b are integers and that b ≠  0, rational numbers are usually called fractions. The use of rational numbers permits us to solve equations. For example; a + b = c, ad = e, for a where b, c, d, e are all rational numbers and a ≠  0. Operations of subtraction and division (with non zero divisor) are possible with all rational numbers. (Stirling, p.2, 1997) Real numbers Real numbers can also be called irrational numbers as they are not rational numbers like pi, square root of 2, e (the base of natural log). Real numbers can be given by an infinite number of decimals; real numbers are used to measure continuous quantities. There are two basic properties that are involved with real numbers ordered fields and least upper bounds. Ordered fields say that real numbers comprises a field with addition, multiplication and division by non zero number. For the least upper bound if a non empty set of real numbers has an upper bound then it is called least upper bound. Sequences A Sequence is a set of numbers arranged in a particular order so that we know which number is first, second, third etc and that at any positive natural number at n; we know that the number will be in nth place. If a sequence has a function, a, then we can denote the nth term by an. A sequence is commonly denoted by a1, a2, a3, a4†¦ this entire sequences can be written as or (an). You can use any letter to denote the sequence like x, y, z etc. so giving (xn), (yn), (zn) as sequences We can also make subsequence from sequences, so if we say that (bn) is a subsequence of (an) if for each n∈ à ¢Ã¢â‚¬Å¾Ã¢â‚¬ ¢ we get; bn = ax for some x ∈ à ¢Ã¢â‚¬Å¾Ã¢â‚¬ ¢ and bn+1 = by for some y ∈ à ¢Ã¢â‚¬Å¾Ã¢â‚¬ ¢ and x > y. We can alternatively imagine a subsequence of a sequence being a sequence that has had terms missing from the original sequence for example we can say that a2, a4 is a subsequence if a1, a2, a3, a4. A sequence is increasing if an+1 ≠¥ an ∀ n ∈ à ¢Ã¢â‚¬Å¾Ã¢â‚¬ ¢. Correspondingly, a sequence is decreasing if an+1 ≠¤ an ∀ n ∈ à ¢Ã¢â‚¬Å¾Ã¢â‚¬ ¢. If the sequence is either increasing or decreasing it is called a monotone sequence. There are several different types of sequences such as Cauchy sequence, convergent sequence, monotonic sequence, Fibonacci sequence, look and see sequence. I will be talking about only 2 of the sequences Cauchy and Convergent sequences. Convergent sequences A sequence (an) of real number is called a convergent sequences if an tends to a finite limit as n→∞. If we say that (an) has a limit a∈ F if given any ÃŽ µ > 0, ÃŽ µ ∈ F, k∈ à ¢Ã¢â‚¬Å¾Ã¢â‚¬ ¢ | an a | < ÃŽ µ n ≠¥ k If an has a limit a, then we can write it as liman = a or (an) → a. Cauchy Sequence A Cauchy sequence is a sequence in which numbers become closer to each other as the sequence progresses. If we say that (an) is a Cauchy sequence if given any ÃŽ µ > 0, ÃŽ µ ∈ F, k∈ à ¢Ã¢â‚¬Å¾Ã¢â‚¬ ¢ | an am | < ÃŽ µ n,m ≠¥ k. Gary Sng Chee Hien, (2001). Bounded sets, Upper Bounds, Least Upper Bounds A set is called bounded if there is a certain sense of finite size. A set R of real numbers is called bounded of there is a real number Q such that Q ≠¥ r for all r in R. the number M is called the upper bound of R. A set is bounded if it has both upper and lower bounds. This is extendable to subsets of any partially ordered set. A subset Q of a partially ordered set R is called bounded above. If there is an element of Q ≠¥ r for all r in R, the element Q is called an upper bound of R 3 Real number system Natural Numbers Natural numbers (à ¢Ã¢â‚¬Å¾Ã¢â‚¬ ¢) can be denoted by 1,2,3†¦ we can define them by their properties in order of relation. So if we consider a set S, if the relation is less than or equal to on S For every x, y ∈ S x ≠¤ y and/or y ≠¤ x If x ≠¤ y and y ≠¤ x then x = y If x ≠¤ y and y ≠¤ z then x ≠¤ z If all 3 properties are met we can call S an ordered set. (Giles, p.1, 1972) Real numbers Axioms for real numbers can be spilt in to 3 groups; algebraic, order and completeness. Algebraic Axioms For all x, y ∈ à ¢Ã¢â‚¬Å¾Ã‚ , x + y ∈ à ¢Ã¢â‚¬Å¾Ã‚  and xy ∈ à ¢Ã¢â‚¬Å¾Ã‚ . For all x, y, z ∈ à ¢Ã¢â‚¬Å¾Ã‚ , (x + y) + z = x (y + z). For all x, y ∈ à ¢Ã¢â‚¬Å¾Ã‚ , x + y = y + x. There is a number 0 ∈ à ¢Ã¢â‚¬Å¾Ã‚  such that x + 0 = x = 0 + x for all x ∈ à ¢Ã¢â‚¬Å¾Ã‚ . For each x ∈ à ¢Ã¢â‚¬Å¾Ã‚ , there exists a corresponding number (-x) ∈ à ¢Ã¢â‚¬Å¾Ã‚  such that x + (-x) = 0 = (-x) + x For all x, y, z ∈ à ¢Ã¢â‚¬Å¾Ã‚ , (x y) z = x (y z). For all x, y ∈ à ¢Ã¢â‚¬Å¾Ã‚  x y = y x. There is number 1 ∈ à ¢Ã¢â‚¬Å¾Ã‚  such that x x 1 = x = 1 x x, for all x ∈ à ¢Ã¢â‚¬Å¾Ã‚  For each x ∈ à ¢Ã¢â‚¬Å¾Ã‚  such that x ≠  0, there is a corresponding number (x-1) ∈ à ¢Ã¢â‚¬Å¾Ã‚  such that x (x-1) = 1 = (x-1) x A10. For all x, y, z ∈ à ¢Ã¢â‚¬Å¾Ã‚ , x (y + z) = x y + x z (Hart, p.11, 2001) Order Axioms Any pair x, y of real numbers satisfies precisely one of the following relations: (a) x < y; (b) x = y; (c) y < x. If x < y and y < z then x < z. If x < y then x + z < y +z. If x < y and z > 0 then x z < y z (Hart, p.12, 2001) Completeness Axiom If a non-empty set A has an upper bound, it has a least upper bound The thing which distinguishes à ¢Ã¢â‚¬Å¾Ã‚  from is the Completeness Axiom. An upper bound of a non-empty subset A of R is an element b ∈R with b a for all a ∈A. An element M ∈ R is a least upper bound or supremum of A if M is an upper bound of A and if b is an upper bound of A then b M. That is, if M is a least upper bound of A then (b ∈ R)(x ∈ A)(b x) b M A lower bound of a non-empty subset A of R is an element d ∈ R with d a for all a ∈A. An element m ∈ R is a greatest lower bound or infimum of A if m is a lower bound of A and if d is an upper bound of A then m d. If all 3 axioms are satisfied it is called a complete ordered field. John oConnor (2002) axioms of real numbers Rational numbers Axioms for Rational numbers The axiom of rational numbers operate with +, x and the relation ≠¤, they can be defined on corresponding to what we know on N. For on +(add) has the following properties. For every x,y ∈ , there is a unique element x + y ∈ For every x,y ∈ , x + y = y + x For every x,y,z ∈ , (x + y) + z = x + (y + z) There exists a unique element 0 ∈ such that x + 0 = x for all x ∈ To every x ∈ there exists a unique element (-x) ∈ such that x + (-x) = 0 For on x(multiplication) has the following properties. To every x,y ∈ , there is a unique element x x y ∈ For every x,y ∈ , x x y = y x x For every x,y,z ∈ , (x x y) x z = x x (y x z) There exists a unique element 1 ∈ such that x x 1 = x for all x ∈ To every x ∈ , x ≠  0 there exists a unique element ∈ such that x x = 1 For both add and multiplication properties there is a closer, commutative, associative, identity and inverse on + and x, both properties can be related by. For every x,y,z ∈ , x x (y + z) = (x x y) + (x x z) For with an order relation of ≠¤, the relation property is a. we can claim that < b. if not then since < a and > b we would have > b a. John OConnor (2002) axioms of real numbers Theorem: The limit of a sequence, if it exists, is unique. Proof Let x and x†² be 2 different limits. We may assume without loss of generality, that x < x†². In particular, take ÃŽ µ = (x†² x)/2 > 0. Since xn→ x, k1 s.t | xn x | < n ≠¥ k1 Since xn→ x k2 s.t | xn x†²| < ÃŽ µ n ≠¥ k2 Take k = max{k1, k2}. Then n ≠¥ k, | xn x | < ÃŽ µ, | xn x†²| < ÃŽ µ | x†² x | = | x†² xn + xn x | ≠¤ | x†² xn | + | xn x | < ÃŽ µ + ÃŽ µ = x†² x, a contradiction! Hence, the limit must be unique. Also all rational number sequences have a limit in real numbers. Gary Sng Chee Hien, (2001). Theorem: Any convergent sequence is bounded. Proof Suppose the sequence (an) ®a. take = 1. Then choose N so that whatever n > N we have an within 1 of a. apart from the finite set {a1, a2, a3†¦aN} all the terms of the sequence will be bounded by a + 1 and a 1. Showing that an upper bound for the sequence is max{a1, a2, a3†¦aN, a +1}. Using the same method you could alternatively find the lower bound Theorem: Every Cauchy Sequence is bounded. Proof Let (xn) be a Cauchy sequence. Then for | xn xm | < 1 n, m ≠¥ k. Hence, for n ≠¥ k, we have | xn | = | xn xk + xk | ≠¤ | xn xk | + | xk | < 1 + | xk | Let M = max{ | x1 |, | x2 |, , | xk-1|, 1 + | xk | } and it is clear that | xn | ≠¤ M n, i.e. (xn) is bounded. Gary Sng Chee Hien, (2001). Theorem: If (xnx, then any subsequence of (xn) also converges to x. Proof Let (yn) be any subsequence of (xn). Given any > 0, s.t | xn x | < n ≠¥ N. But yn = xi for some so we may claim | yn x | < also. Hence, ( Gary Sng Chee Hien, (2001). Theorem: If (xn) is Cauchy, then any subsequence of (xn) is also Cauchy. Proof Let (yn) be any subsequence of (xn). Given any s.t | xn xm | . But yn = xi for so we may claim | yn ym | Hence (yn) x Gary Sng Chee Hien, (2001). Theorem Any convergent sequence is a Cauchy sequence. Proof If (an) a then given > 0 choose N so that if n > N we have |an- a| < . Then if m, n > N we have |am- an| = |(am- a) (am- a)| |am- a| + |am- a| < 2. We use completeness Axiom to prove Suppose X ∈ à ¢Ã¢â‚¬Å¾Ã‚ , X2 = 2. Let (an) be a sequence of rational numbers converging to an irrational 12 = 1 1.52 = 2.25 1.42 = 1.96 1.412 = 1.9881 1.41421356237302 = 1.999999999999731161391129 Since (an) is a convergent sequence in à ¢Ã¢â‚¬Å¾Ã‚  it is a Cauchy sequence in à ¢Ã¢â‚¬Å¾Ã‚  and hence also a Cauchy sequence in . But it has no limit in. An irrational number like 2 has a decimal expansion which does not repeat: 2 =1.4142135623730 John OConnor (2002) Cauchy Sequences. Theorem Prove that is irrational, prove that ≠¤ à ¢Ã¢â‚¬Å¾Ã‚  Proof We will get 2 as the least upper bound of the set A = {q Q | q2 < 2}. We know that a is bounded above and so its least upper bound b does not exists. Suppose x ∈ , x2 0 be given. Then k1, k2 s.t | xn xm | < ÃŽ µ/(2Y) n, m ≠¥ k1 | yn ym | < ÃŽ µ/(2X) n, m ≠¥ k2 Take k = max(k1, k2). Then | xn xm | < ÃŽ µ/(2Y) | yn ym | < ÃŽ µ/(2X) n, m ≠¥ k Hence, | xn yn xm ym | = | (xn yn xm yn) + (xm yn xm ym) | ≠¤ | xn yn xm yn | + | xm yn xm ym | = | yn | | xn xm | + | xm | | yn ym | ≠¤ Y | xn xm | + X | yn ym | < Y(ÃŽ µ/(2Y)) + X(ÃŽ µ/(2X)) n, m ≠¥ k = Hence, (xn yn) is also Cauchy. 5 Conclusion Real numbers are infinite number of decimals used to measure continuous quantities. On the other hand, rational numbers are defined to be fractions formed from real numbers. Axioms of each number system are examined to determine the difference between real numbers and rational numbers. Conclusion of the analysis of axioms resulted to be both real numbers and rational numbers contain the same properties. The properties being addition, multiplication and there exist a relationship of zero and one. The four fundamental results are obtained from this study. First concept is that the property of real number system being unique and following the complete ordered field. Second is that if any real number satisfies the axioms then it is upper bound, whilst rational numbers are not upper bound. The third being that all Cauchy sequences are converges towards the real numbers. Finally found out that all real numbers are equivalence classes of the Cauchy sequence. Appendices List of symbols à ¢Ã¢â‚¬Å¾Ã¢â‚¬ ¢ = Natural number à ¢Ã¢â‚¬Å¾Ã‚  = Real number = Rational number ∈ = is an element of = There exists = For all s.t. = Such that

Lyric Analysis

Kayla Keeney English 131 Ms. Jones 20 Feb. 2013 â€Å"Not Ready To Make Nice† Rhetorical Analysis â€Å"Not Ready To Make Nice,† a song released in 2006 by the Dixie Chicks is a controversial song written after Natalie Maines commented that she was â€Å"ashamed the president of the United States is from Texas† (Tyrangie), in between songs at a concert in Britain in 2003. The comment resulted in The Dixie Chicks being dropped from playlists at many radio stations across the south (Tyrangie). This song is about how Maines is not willing to forgive all of the negative remarks and actions made towards the band, some which include death threats.One rhetorical tool used by the Dixie Chicks is pathos. The first verse, â€Å"Forgive sounds good/ Forget I’m not sure I could/ They say time heals everything/but I’m still waiting,†opens the song showing forgiveness, anger and sadness (Dixie Chicks). They are letting it be known that they would like to f orgive all of the people that have hurt Natalie, and the rest of the band, due to her comment, but some of the actions taken towards them were so extreme that they are not sure that they could ever forget them.The next verse includes the lines â€Å"I’m through with doubt/There’s nothing left for me to figure out/ I’ve paid a price and I’ll keep paying† (Dixie Chicks). When this, and the lines â€Å"It turned my whole world around and I kind of like it† are sung, Maines is making it clear that she does not have any regret about her comment towards President Bush and she is done doubting herself because of it. She is letting it be known that she may have had to pay a price of the Dixie Chicks music no longer being on the radio, but she is no longer going to doubt herself or regret her comment because she is entitled to freedom of speech.This brings a bitter sweet emotion into play, because she realizes many other people probably feel the same way; she just voiced her opinion publicly and is not ashamed of it. Pathos is used in that verse when she is talking about the price they will keep paying. Music is their career and it got taken away by one band member’s comment. That verse brings heartbreak to not only the song, but the band members who are so passionate about their music and career choice, along with the listeners that loved the Dixie Chicks music. Joy and peace is brought into the song when â€Å"I made my bed and I sleep like a baby. With no regrets†¦ † is said.Natalie is saying she is content with what she said and all the trouble she got herself and her band into. She has no regrets. â€Å"It’s a sad, sad story when a mother will teach her daughter that she ought to hate a perfect stranger, and how in the world can the words that I said send somebody so over the edge that they’d write me a letter sayin’ that I better shut up and sing or my life will be over† (Dix ie Chicks), brings a confused emotion. In the song when this verse is sung, it sounds like Natalie is going on a rant about how sad our world is today, and an angry tone of voice is used to really show her anger about the situation.She sounds scared, sad, and mad all at the same time. She is confused at society. All of the pathos used already throughout the song, bittersweet, heart break, joy, peace, and confused, is revisited again in the chorus, which states â€Å"I’m not ready to make nice/ I’m not ready to back down/ I’m still mad as hell and I don’t have time to go round and round and round/ It’s too late to make it right/ I probably wouldn’t if I could/ ‘cause I’m mad as hell can’t bring myself to do what it is you think I should† (Dixie Chicks).Ethos is also a rhetorical tool the Dixie Chicks used throughout the song. The Dixie Chicks have credibility writing and singing this song, because it goes a long wit h a life event of theirs. If another singer or band would have published â€Å"Not Ready To Make Nice,† it would not have had as much meaning or emotion, and it would not have been expressed as it was by Maines, unless they had a similar experience. After Maines was asked if she was sorry about her London comments, she said no and responded with â€Å"Sorry about what? Sorry about what?Sorry about not wanting to go to war? And not wanting people to die? â€Å"(Schorn) Many people felt the same way as Natalie Maines did, but did not voice their opinion in such a way as she did. She has every right to do so though. It is common for people to not like the idea of going to war. Americans have the right of freedom of speech, in other words, â€Å"the right to express any opinions without censorship or restraint. † Natalie Maines should be able to voice her opinion as she pleases, due to the first amendment, which brings logos into play.Maines Logos in â€Å"Not Ready To Make Nice† includes the comment itself, that was made and freedom of speech. Since the first amendment exists the Dixie Chicks should not have been punished like they were. Knowledge about the comment, and some background about the Dixie Chicks lives afterwards, is needed to make sense of the song and the meaning behind it. After finding out about the death threats made towards the band, and the banning of their songs on the radio, there is more of an understanding as to why there is so much pathos throughout.It is logical to think that the Dixie Chicks wrote this song as feedback towards country music listeners and southerners, to voice how they feel about the actions done towards them; it is their way of expressing their feelings publically a couple of years later. Through pathos, ethos, and logos, the Dixie Chicks are able to speak out publically to let the world know how Maines’ comment has affected them. Though Maines states she does not regret what she said, she m akes it clear that her life has changed forever, along with the Emily and Martie’s, the other band members.This song is a way for Maines to stand behind her comment and to stand up to President Bush. Behind the words is passion. The Dixie Chicks are passionate about the meaning they are trying to get across to listeners. They want listeners to feel what this song means to them and they want us to feel their emotions that they have poured into writing this song. â€Å"Not Ready To Make Nice† is a way for them to show courage, by telling the world they are at peace with what was said back in 2003. Works CitedChicks, Dixie. â€Å"Not Ready To Make Nice. † Cowboy Lyrics. cowboylyrics. com. Web. 19 Mar 2013. . â€Å"freedom of speech. † Dictionary. com Unabridged. Random House, Inc. 19 Mar. 2013. . Schorn, Daniel . â€Å"Dixie Chicks: Not Ready to Make Nice. † http://www. cbsnews. com/8301-18560_162-1611424. html. CBS, 11 February 2009.

Alcuin and Charlemagne

Charlemagne was the king of the Franks from 768 to 814. He was known to be the most powerful Christian ruler and brought success to his country. Charlemagne was well educated and good looking. His strong voice allowed him to express what he had to say in a very eloquent manner. He was most famous for doubling the territory that his father had previously conquered. With his determination and persistence, Charlemagne became one of the most dignified rulers of the early middle ages. Charlemagnes determination allowed him to expand his empire.He undertook 54 military campaigns during his rule. He also lead his armies into Italy to conquer the Lombard State. His army also invaded the land of the Bavarians and took them under control. During his rule, Charlemagne insisted the Saxons convert to Christianity and soon took them over. In turn, adding more land to the growing Carolingian empire. (Speilvogel p. 138) An interesting characteristic of Charlemagne was his strong desire to learn. He studied foreign languages such as Latin and Greek. He learned from excellent scholars such as Peter of Pisa and Alcuin of York.Charlemagne established a palace school and encouraged other scholars from across Europe to come to the Carolingian court. He focused learning about liberal arts also took lessons in grammar. Charlemagnes love of learning inspired others to obtain educations and maintained the intellectual life of the Catholic church. (Speilvogel p140-141) (Einhard: Life of Charlemagne) Not only was Charlemagne good at conquering land but he also did very well at governing the land he conquered. He was a clever ruler and knew he had to keep the nobles in his service.To do this he granted part of the royal lands as lifetime holdings to nobles who assisted him. Charlemagne also knew that he could not let the counts gain more power then him. To hold more control over his kingdom, Charlemagne required counts to serve outside their own family lands. He also sent out â€Å"messen gers of the lord king† to check on the counts and make sure they were following the kings demands. One last thing that Charlemagne realized was the valuable assistance that the Catholic church could provide for him. He decided to create new bishoprics and archbishoprics while restoring old ones. Speilvogel p. 138-139) Charlemagne made many accomplishments during his rule. His first and biggest accomplishment was increasing the Frank Kingdom. His territory stretched over the majority of Europe. Charlemagne was able to subdue the barbarous tribes in Germany which was something other Kings could not accomplish. He also made good relationships with emperors of Constantinople. Being a devote Christian, Charlemagne built the beautiful church at Aix-la-Chapelle. Finally, Charlemagne took care of the poor in his country and sent money to the poor in other countries. (Speilvogel p. 139)As you can see, Charlemagne had many characteristics of a successful ruler. He was determine expand h is empire and he did. He also was good at governing his people. Charlemagne was very well educated and it showed through when whenever he spoke with eloquence. Also, Charlemagne was able to make constant accomplishments such as working on the Church reform, taking care of the poor, and building good relationships with other countries. Charlemagne was so respected that even the Pope called upon him to help during the Roman rebellion. All of these things are what made Charlemagne one of the greatest Kings of all time.

The fastest growing jobs and industries through 2023

The fastest growing jobs and industries through 2023 Even in a good economy, with significant job growth and high employment, there are shifts that favor some industries over others. According to a recent Careerbuilder study, this means good news and bad news for the U.S. job scene between now and 2023. First, the bad news: middle-wage jobs (like customer service representatives, maintenance workers, construction workers, or truck drivers, for some examples) are not expected to keep pace with high-wage jobs (like nurses, accountants, and IT specialists) and low-wage jobs (like home health aides, retail sales, and receptionists), which are both poised for serious growth. The study identified 121 jobs that will decline in growth between 2018 and 2023, and 75 of those jobs were considered middle-wage.But now the good news: these high- and low-wage fields are about to experience significant growth, meaning millions of job openings- approximately 8 million by 2023. Let’s look at some of the industries in the study, divided by high-, mid-, and low-wage job types.Fastest growing jobs   Fast-Growing Occupations By Wage Category Jobs Added, 2018-2023 % Change, 2018-2023 Median Hourly Pay High-Wage Registered Nurses 255,047 8.39% $33.55 Software Developers, Applications 143,466 15.57% $48.49 Postsecondary Teachers 110,955 7.25% $33.53 Accountants and Auditors 86,079 6.02% $32.33 Market Research Analysts and Marketing Specialists 83,187 12.60% $30.21 Computer User Support Specialists 54,044 7.48% $24.16 Plumbers, Pipefitters, and Steamfitters 43,625 8.58% $23.72 Middle-Wage Customer Service Representatives 120,673 4.21% $15.88 Medical Assistants 102,274 14.51% $15.62 Construction Laborers 92,182 6.56% $14.73 Maintenance and Repair Workers, General 83,931 5.41% $18.08 Licensed Practical and Licensed Vocational Nurses 55,345 7.34% $21.56 Light Truck or Delivery Service Drivers 48,837 5.12% $15.04 Billing and Posting Clerks 44,283 8.59% $17.85 Low-Wage Home Health Aides 207,732 22.42% $11.17 Waiters and Waitresses 146 ,281 5.49% $10.01 Retail Salespersons 108,229 2.37% $11.29 Cooks, Restaurant 100,664 7.46% $12.06 Nursing Assistants 96,384 6.33% $13.23 Security Guards 61,964 5.12% $12.97 Receptionists and Information Clerks 69,461 6.29% $13.70Trends in hiring over the next 5 yearsIf you’re in a field that’s expected to decline, this news can be dismaying for your career outlook. However, it’s also a great time to take stock of your career goals and your near future, and decide whether you’re able to adapt your skills to be more industry-flexible, or whether you’d like to change careers altogether to maximize your job potential.HealthcareHealthcare is one of the fields that is exploding now, and is likely to continue growing at a very fast pace for the foreseeable future.The healthcare field is popular because with a growing population (especially one that skews older and more in need of medical care), the need will continue to grow. But healthcare is also one of the most innovative fields, with digital equipment and recordkeeping requiring ever more tech-literate employees.TechnologyAs everything becomes more technology-focused, more and more companies will need dedicated tech teams and services to provide the digital infrastructure necessary to do business. People with IT expertise and skills will find themselves in demand in many different fields and companies. Having a flexible skill set that’s technologically advanced can help guarantee a spot in the digital jobs boom over the next several years.Data AnalysisEverything comes down to data these days, from marketing and customer service to accounting and financial data. This is also an area where a strong set of analytical and problem-solving skills can be applied across different industries, in different roles.Basically, the professional future is flexibility- if you’re willing to develop future-facing skills to go along with your education and experience base.

Prepositional Phrases Sentence Building Exercise

Prepositional Phrases Sentence Building Exercise In this exercise, you will continue to apply the basic strategies outlined in Introduction to Sentence Combining.  Combine the sentences in each set into a single clear sentence containing at least one prepositional phrase. Omit words that are needlessly repeated, but dont leave out any important details.   After you have completed the exercise, compare your new sentences with the original sentences on page two. Keep in mind that many combinations are possible, and in some cases, you may prefer your own sentences to the original versions. A mouse darted.It darted across the salad bar.This happened during the luncheon.We traveled this summer.We traveled by train.We traveled from Biloxi.We traveled to Dubuque.The convertible swerved, crashed, and caromed.It swerved off the road.It crashed through the guardrail.It caromed off a maple tree.Mick planted seeds.He planted them in his garden.He did this after the quarrel.The quarrel was with Mr. Jimmy.Grandpa dropped his teeth.His teeth were false.His teeth dropped into a glass.There was prune juice in the glass.Lucy played.She was behind the sofa.She was with her friend.Her friend was imaginary.They played for hours.There was a man.He wore a chicken costume.He dashed across the field.He did this before the ballgame.The ballgame was on Sunday afternoon.A man stood, looking down.He stood upon a railroad bridge.The bridge was in northern Alabama.He was looking down into the water.The water was twenty feet below.The water was swift.The gray-flannel fog closed off the Salinas Val ley.It was the fog of winter.The fog was high.The Salinas Valley was closed off from the sky.And the Salinas Valley was closed off from all the rest of the world. I climbed to my perch.I did this one night.The night was hot.The night was in the summer.The night was in 1949.It was my usual perch.My perch was in the press box.The press box was cramped.The press box was above the stands.The stands were wooden.These were the stands of the baseball park.The baseball park was in Lumberton, North Carolina. After you have completed the sentence building  exercise on  page one, compare your new sentences with the sample combinations below. Keep in mind that many combinations are possible, and in some  cases, you may prefer your own sentences to the original versions. Sample Combinations During the luncheon, a mouse darted across the salad bar.This summer we traveled by train from Biloxi to Dubuque.The convertible swerved off the road, crashed through the guardrail, and caromed off a maple tree.After his quarrel with Mr. Jimmy, Mick planted seeds in his garden.Grandpa dropped his false teeth into a glass of prune juice.Lucy played  behind  the couch for hours with her imaginary friend.Before the ballgame on Sunday afternoon, a man in a chicken costume dashed across the field.A man stood upon a railroad bridge in northern Alabama, looking down into the swift waters twenty feet below.​  (Ambrose Bierce, An Occurrence at Owl Creek Bridge)The high gray-flannel fog of winter closed off the Salinas Valley from the sky and from all the rest of the world. (John Steinbeck, The Chrysanthemums)One hot night in the summer of 1949, I climbed to my usual perch in the cramped press box above the wooden stands of the baseball park in Lumberton, North Carolina.  (Tom W icker, Baseball)

Excutions Essay Example | Topics and Well Written Essays - 2000 words

Excutions - Essay Example Keeping in view this social condition, the sociologists try to explore crime, crime control and the administration of criminal justice from the point of social constructionism. They view 'crime' and 'criminals' a product of social and political interests and where besides other, the most dominating factor are class, race and gender. They also take into consideration the historical and contemporary practices of criminal justice which is shaped and experienced by the racial and ethnic minorities and majorities, the rich and poor and by men and women, so as to help us understand the numerous social realities of justice in the United States; as this essay will try to examine the pattern of execution based on race, gender and class. The study of social inequalities has always been the central focus of sociologists. They are not only interested in issues related to race/ethnicity, gender and class but also the intersections of these dimensions by employing a wide variety of methods from the ethnographic fieldwork and in-depth interviews to multilevel social and networks methods and statistical models. According to The ABA Kennedy Commission Report (June 23, 2004), the United States puts more people behind bars than any other country in the world and needs to eradicate the disproportionate impact 'tough on crime' laws it has for minorities. This is not because of higher criminal behavior among blacks, but because as compared to non-whites, they are more likely to be imprisoned especially as drug users. Though white drug dealers outnumber the black ones but 86.8% of those imprisoned for drug charges are blacks. In the 1980s, the media talked about drug-addicted mothers, perpetuating the racial stereotypes of African American women who trading sex for drugs rather than a white middle class woman snorting the more expensive cocaine powder. While the poor black pregnant women became targets of the criminal justice system, the middle and the upper class women escaped scrutiny of the criminal justice agents into the private facilities of detoxification. Similarly, most studies on crime take a narrow approach to the subject by treating a crime as simply a violation of legalized social norm that carries a penal sanction. The figure among black males was 3,405. Much of the history of sentencing reform both in capital and non-capital punishment has been influenced by implicit concerns about racial disparities and discriminatory decision making in the criminal justice system. In a study carried out to find whether the four delinquency theories, strain, social learning, low self-control and control theories could better explain juvenile offending in comparison to gender, race and class impact on delinquency. The findings suggested that the quantitative analysis is an effective tool for detecting intersectional differences resulting from gender, race and class to support feminist assertions that general theories are less universal than claimed by their proponents. Based on the unfair racial disparities in federal sentencing, a report released in 1984 by the United States Sentencing Commission confirmed, that the average federal prison sentence for black offenders was about five months longer than for whit es. By 2001, the average sentence

Strategic Management Essay Example | Topics and Well Written Essays - 2500 words - 10

Strategic Management - Essay Example Positioning of the organisation through strategy, responding to real time issues through strategic management and managing the resistance offered by the competitors through systematic planning are some of the broader aims of strategic management (Ayanda, M., n.d.). There are a few key attributes that strategic management addresses. It helps the organisation to move towards its goal and achieve its objectives. It helps the stake holders to be a part of the decision making body. The need of incorporating short term and long term goals can be identified. It also helps in understanding the trade off between efficiency and effectiveness in order to achieve the goals of the organisation. Some theorist do believe that the traditional approach is the standard approach to strategic management however it can no longer cope up with the complexities of the new demands (Dess, et.al., n.d). Brinkerhoff had a very simple way of defining strategic management. He defined strategic management as ‘looking out’, ‘looking in’ and ‘looking ahead’. According to Brinkerhoff looking out means, evaluating the environment in order to set organisational goals and also recognise the potential stakeholders. By looking in he means to identify the strengths that the firm possesses meaning the resources like finance and the man power. Looking ahead points out at formulating strategies and allocating resources to set targets and evaluate performance. Strategic management mainly consists of the following five factors. They are setting goals, analysing, strategy formation, strategy implementation and evaluating the strategy. These factors need continuous interaction and feedback between them (Susan. n.d.). The Balance Scorecard is an effective tool by which organisations can evaluate its performance which in turn helps to accomplish the vision.