Team Analysis of Competing Values Framework Coursework

Team Analysis of Competing Values Framework - Coursework Example Every member of the team depicts effective both at personal and organizational levels. Every quadrant is represented by the values. Control is highly represented in the quadrant. Most of the team members embrace control, which involves doing things right. Create follows the control character, which involves doing things first. Besides doing things rightful, the team members embrace characters of innovation. Compete is the third dominant managerial characteristic of the team members. Competition implies undertaking managerial duties fast and at the desired pace. The least represented in the framework is collaborate. Collaboration demands togetherness when performing roles. Avoiding negative extremes in the team is a vital aspect of management. Ensuring people focus is a sure strategy to enhance effectiveness. People focus managers show compassion, are concerned and expressive. Authenticity is the second key factor to enhance effectiveness. An authentic manager is reflective, mindful, and principled. Practicality is the third important character for an effective manager. Practicality involves being factual, realistic and

Categorical imperative Essay Example for Free

Categorical imperative Essay Philosophers usually tend to think and respond in totally different and opposite ways. However, in Glaucon’s challenge, Aristotle, Spinoza, Hume, Kant and Mill agreed that it’s an uncontrollable system of desire. In Glaucon’s challenge he describes three important ideas. The first idea is â€Å"of the nature and origin of unjust, according to the common view of them† (488). The second idea is â€Å"I will show that all men who practice justice do so against their will, of necessity but not as a good† (488). The third idea is â€Å"I will argue that there is reason in this view, for the life of the unjust is after all better far than the life of the just† (488). Each one of these philosophers responded to this challenge similar and different manners. Aristotle believed that to be a good man is to have good morality. In his point of view he sees that morality gives a reason that leads to actions. Artistotle respond toward Glaucon’s challenge is reason guides passion. This makes him an injustice man because he just cares about how to please himself and follow his desire. Spinoza’s idea is emotions have nothing to do with human beings its reasons that guide us. He believed that passion and emotions leads us to be changeable due to how the world around us and not necessarily our knowledge. Reasons give us the opportunity to gain knowledge of life. Spinoza becomes unjust and gives reasons to Glaucon’s challenge. From Hume’s point of view he believed that benevolence makes us social being instead of self interest. Hume’s main concern is sympathy and empathy. Sympathy is the nature of human that makes them human beings. It’s the ability to see feel pity toward other. Empathy is the understanding of pain and happiness of one another. Hume believed that all human are selfish and that their passion guides their reasons. Passion and reasons is not having emotions or feeling pity. Hume’s view is that we should the ability to feel others emotions and understand them. Kant sees that the categorical imperative is the only moral way to act. Categorical imperatives are based on ideals of logic and acts on duty. He believed that if you feel satisfied about something you did, you shouldn’t consider it a reward but rather as a bonus for you. We only do moral things because it’s in fulfill our interest and benefit. Kant responds to Glaucon’s challenge that if the act is immoral, then it cannot be considered into categorical imperative. Mill’s view is that everyone should be satisfied and happy. He believes that everyone should think of the consequences of their actions before deciding to make them. Mill puts others as well as himself into consideration before any action. He doesn’t argue towards or against Glaucon’s challenge, he just cares about his pleasure and others. Each philosopher had his own argument toward Glaucon’s challenge. Their ideas were unjust happiness towards Glaucon’s challenge. Aristotle and Hume had opposite views. Aristotle‘s idea towards the challenge was reason guides passion, where Hume’s idea was passion guides reason. Mill and Kant had the same assumptions. Mill expresses about pleasure and Kant was certain about duty. Mill agrees with Aristotle as well, they believe that happiness is a man’s ambition and human existence. Spinoza is just unjust to the whole idea of the challenge. These philosophers’ ideas were very similar that a man would behave unjust if he had the ring.

Aristotles View of Slavery Essay -- Slave Slavery Aristotle Paper Ess

Born in the year of 384 B.C. Aristotle was seen as conventional for his time, for he regarded slavery as a natural course of nature and believed that certain people were born to be slaves due to the fact that their soul lacked the rational part that should rule in a human being; However in certain circumstances it is evident that Aristotle did not believe that all men who were slaves were meant to be slaves. In his book Politics, Aristotle begins with the Theory of The Household, and it is here that the majority of his views upon slavery are found. With the beginning of Chapter IV, Aristotle's idea of slavery is clearly defined. "The instruments of the household form its stock of property : they are animate and inanimate : the slave is an animate instrument, intended (like all the instruments of the household) for action, and not for productions." This distinction between action and production, is based upon the understanding that 'production' is a course in which a result is desired beyond the immediate act of doing. Where as, the simple act of completing a task is identified as 'action'. Aristotle, who believed that life was action and not production theorized that slaves were instruments of life and were therefore needed to form a complete household. In fact Aristotle went as far as to say that a slave was comparable to a tame animal, with their only divergence in the fact that a slave could apprehend reason. For he concluded that a slave and animals only use was to supply their owners with bodily help. At the end of the Theories of the Household, Aristotle explains how slaves are different from andy other types of people, in the sence that they are the only class who are born into their occupation and become property of their masters. In examining this relationship we find that he thought that while masters were the masters of the slaves, they still held a life other than that of being master; However, Aristotle believed that not only was the slave a slave to his master, but the slave had no other life or purpose than belonging. From this consideration we begin to understand Aristotle's views on the relationship between Master and Slave. At the beginning of Chapter V of the Theory of the Household, the distinct role of master and slave is defined. There is a principle of rule and subordin- ... ... Aristotle we find that he was a man of great curiosity, wisdom and ideas. Although his views on slavery seemed to hold true to the times, he had many variations on the conservative norms and beliefs. He had believed that slavery was a just system where both master and slave were beneficial from this relationship. And with this he thought that by nature, certain people were born to be slaves, yet with these beliefs we find many exceptions, where Aristotle allocates areas to describe those who by chance became slaves but in his opinion were born to be free. And in such incidence where men born free were not fit to be masters Aristotle explained how it would be easier for the master to obtain a steward who was more adept at giving instructions to run the household and leave the master of the house to more prudent issues. We can only guess as to what made Aristotle believe that by the human soul one could delineate whether or not a man was meant to be a slave or a freeman. And with his arguments we find that it was just as difficult for him to make that distinction as well. "Though it is not as easy to see the beauty of the soul as it is to see that of the body."

What Is Orientalism? Essay -- Definition Oriental Orientalism Essays

What Is Orientalism? Said made a clear analogy between orientalism and colonialism. They are both set with the same binary opposition. white/ non white occidental/ non occidental In a very detailed and structured study of the orient (behavior, habit, tradition ...) we document a large amount of fact and data. All compile in a general study they produce the illusion of a well understood and objectively constructed knowledge. These are, in fact, mere observations and purely subjective entities (seen only with the western eyes) which do not explain nor reflect the true nature of the object. We then generalize from theses singles observations, set up categories and labels. We are now able to answer questions very simply creating this illusion of knowledge. We witness the realization of cliches; single sided beliefs that fuels themselves upon their own ignorance. SAID then explains how this technique empowers his creator. The rigorous discourse, the elaboration of thought and ideas, subjectively authenticated itself. The content is no longer address and leaves the victim of the discourse reduced to plain and pure denial. The content has become a set of prejudicial belief (cliches). To remind you of the famous aphorism: "the pen is mightier than the sword" He writes: "Orientalism is fundamentally a political doctrine willed over the orient because the orient was weaker than the west." Though injustice has already been served and the...

Expert Hair Testimony Essay

Hello ladies and gentlemen of the jury my name is Brittany Marroquin. I am a hair analysis expert. I am here today to determine whether or not the suspects are responsible for the crime that has been committed. I have here the lab results depicting the crime scene. The trace evidence (small but measurable amounts of physical or biological material found at a crime scene) found at the crime scene was the killers hair, and the victim’s own hair to exclude any hair samples found similar to the victim. There are three suspect and one sample hair found at the scene of the crime. After analyzing the crime scene hair we found that the hair was light brown, transparent and have no medulla, (the central core of a hair fiber) with an imbricate (cuticles scales that are flattened and narrow) cuticle (the tough outer covering of a hair composed of overlapping scales). When analyzing the hair follicles (the actively growing root or base of a hair containing DNA and living cells), you can g o at it two ways macroscopically (in a such a way as to be large enough to be visible to the naked eye; in a way which is comprehensive or is concerned with large units) and microscopically (Too small to be seen by the unaided eye but large enough to be studied under a microscope). When analyzing the hair macroscopically you should look at the length, color, sheen, whether it’s wavy or straight, and the presence or absence of follicle tissue or not.

Wireless Routers essays

Wireless Routers essays If you walk through a school and ask any kid thats sitting at a computer they could probley tell you what a router does, may not be able to explain how it does this, but at very least they do know its function. So you wont be reading any explination of how a router works here, but rather an over view of a special type of router. One not confined by the laws that control almost all of todays electronics. Im not writing this here to give you vague hints of the topic, as the title gives that away, but rather to inform and teach. To borrow a line, It's depressing how often we see that those who don't remember history are doomed to repeat it. When cordless phones and the first analog cell phones hit the market, anybody with a scanner that operated at the right frequency could easily listen to calls not intended for them. History truly does repeat it self as wireless routers first hit the market any one with a scanner could listen in on whats going across your network, just like they could with the cordless phone. Security was a major issue for companys, even home set ups of wireless networks. Though most werent aware of the problem in the beginning it did become very clear to the public as rouge individuels worked there way into the various networks. You see the way a wireless router works is by sending signals through the air with radio waves, on a frequency much higher then what phones use making them a bit harder to detect. Cause of that reason companys orginaly claimed the networks were secure cause no one had the equipment needed to listen to the information. But if you truly think about it what was the wireless router other then a receaver for that frequency, proving right there that it wasnt nearly as hard as they planned to listen in. Security with the wirless networks has come a long way over the years, but they still dont have the same securi ...

How to Use the French Expression NImporte Quoi

How to Use the French Expression N'Importe Quoi The French expression nimporte quoi,  pronounced  neh(m) puhr t(eu) kwa,  means literally no matter what. But in use, the sense is anything, whatever or nonsense. Nimporte quoi has a few different uses. Most often it means anything, as in: Je ferais nimporte quoi pour gagner.   Id do anything to win. Informally, nimporte quoi  or cest du nimporte quoi are used to convey  nonsense. A less literal translation would be What the heck are you talking about?! or the exclamatory Rubbish! Though not a perfect equivalent, nimporte quoi is also probably the best translation  for whatever, when its used as an expression of dismissal. Examples Ce magasin vend tout et nimporte quoi.   This store sells anything and everything.Nà ©coute pas Philippe. Il dit nimporte quoi.   Dont listen to Philippe. Hes talking nonsense. /  Hell say anything!Il ferait nimporte quoi pour obtenir le rà ´le. Hed do anything.  /  Hed go to any lengths to get the part.Tu dis vraiment nimporte quoi  ! Youre talking absolute nonsense !  Cest un bon investissement. Thats a good investment.Nimporte quoi  !  (familiar) Dont talk rubbish / nonsense !Je ferais nimporte quoi pour elle.  Ã‚  Id do anything for her.  Ã‚  Comme qualità ©, cest nimporte quoi.  Ã‚  Ã‚  In terms of quality / As for quality, its rubbish. Almost Famous Theres a well-known saying in French popular culture that goes:  Cest en faisant nimporte quoi, quon devient nimporte qui  (or ...que lon devient...). This expression means literally, Its by doing nonsensical things that you become nonsensical, but its better expressed as Its by doing anything that you become anyone, and its the motto of French prankster and video maker Rà ©mi Gaillard, who calls himself Nimporte qui. The phrase is a play on the French proverb Cest en forgeant quon devient forgeron (the equivalent of Practice makes perfect, but literally Its by forging that one becomes a blacksmith). Part of the NImporte Family of Expressions Nimporte quoi  is a popular  combining  form of the French indefinite expression  nimporte, which literally means no matter. It can be followed by an  interrogative pronoun  like quoi, an  interrogative adjective,  or an interrogative adverb in order to designate an unspecified person, thing, or characteristic. NImporte With Interrogative Pronouns Interrogative pronouns imply the question who, what, and which one, or qui, quoi, and lequel / laquelle / lesquels / lesquelles. These phrases can function as subjects,  direct objects, or  indirect objects. 1) Nimporte qui   anyone, anybody   Nimporte qui peut le faire.   Anyone can do it.Tu peux inviter nimporte qui.   You can invite anyone.Ne viens pas avec nimporte qui.   Dont come with just anyone.   2)  Nimporte quoi   anything Nimporte quoi maiderait.   Anything would help me.Il lira nimporte quoi.   Hell read anything.​Jà ©cris sur nimporte quoi.   I write on anything. 3) Nimporte lequel,  laquelle   any (one)​ Quel livre veux-tu  ?   Which book do you want?Nimporte lequel.   Any one. / Any of them.Aimes-tu les films  ?   Do you like movies?Oui, jaime nimporte lesquels.   Yes, I like any at all. Nimporte With  Interrogative Adjectives In this case,  nimporte  is combined with the interrogative adjectives  quel or quelle, which pose  the question what. This combined form produces  nimporte quel / quelle, which translates to  any.  Nimporte quel  is used in front of a noun to indicate a nonspecific choice, as in: Nimporte quel, quelle any Jaimerais nimporte quel livre.   Id like any book.Nimporte quelle dà ©cision sera...   Any decision will be... Nimporte with Interrogative Adverbs Here nimporte is combined with  interrogative adverbs that pose the questions  how, when, and where. These indicate that the how, when, or where is unspecified and are translated as: (in) any way, anytime, and anywhere. 1)  Nimporte comment   (in) any way   Fais-le nimporte comment.   Do it any way / any old way. (Just do it!)Nimporte comment, il part ce soir.   Hes leaving tonight no matter what. 2) Nimporte quand   anytime Ecrivez-nous nimporte quand.   Write to us anytime. 3) Nimporte oà ¹Ã‚   wherever, anywhere Nous irons nimporte oà ¹.   Well go wherever / anywhere.

Psychology Coursework

Psychology Coursework Psychology Coursework Psychology Coursework. What Do The Students Expect? Students, who want to unite their lives with psychology in the future must get a masters degree on this subject, therefore they should obtain psychology courses and write a psychology coursework: After the graduation they can work as doctors, teachers and have other occupation which is connected with peoples psychology. Their career depends on the doing their psychology courseworkproperly, so it is an important step in their education and it should be treated corresponding. There are different types of the assignments which can be done by the student for the obtaining the psychology degree. Format For Psychological Coursework A psychology courseworkcan be written in the form of essay, dissertation, term paper and other study assignments. As it must cover a plenty of literature material, so the demonstration by the students of their talent of disposing facts and analyzing material should be on the high level If the practical part is present, the student should observe the actions and the results of the experiments and then to put down these clinical observations in the form of the report to their psychology coursework And the students should remember that any theoretical statements should be supported by the hands-on experiment If the student is insuch a level of education that there is a turn for the dissertation, so it should be done perfectly and have much efforts for the better understanding of the problem. The student should be deeply absorbed in the issued problem and the practical part is a must-be requirement here. But it is not the only thing which is the element of the dissertation or extended psychology coursework. The students are expected to show view of other scholars and make a conclusion, which study is more appropriate for the demonstration of the solution to the problem or to the hide question within the topic of the assignment. There is another one type of the psychology coursework, which can be called as case study, when a student should do a research basing on the profile of the patient of a group of them. There must be a lot of arguments and the psychological ability of patients should be observed in the different situations. Their behavior, their justifying of the done actions should be considered also. Then a student can offer a suggestion how the mental problem or life conditions could be improved and the danger of the mental illnesses can be avoided. Areas Of Psychology There some areas of psychology, which can be observed by students, they are: Abnormal psychology Applied psychology Behavioral psychology Child psychology Clinical psychology Developmental psychology Educational psychology Experimental psychology general psychology Gestalt psychology Social psychology. So if you are not sure in the efforts, so you can entrust the writing to our writing service. We should notice that no customer had any complaint about our service. Our experts can perform any order concerning any topic or form of the assignment. Good luck to you and be happy with us! Related posts: Informative Essay Writing How to Write a Good College Essay? Essay about Bosnia College Essay Online College Essay Help

Common Assessment Research Paper Example | Topics and Well Written Essays - 1000 words

Common Assessment - Research Paper Example The main aim and goals is to help obtain the necessary management skills to manage the Information Technology department. Goals and Objectives The chief role of this channel is to make sure that the organization accomplishes its goals through having a good knowledge on how to use the Information Technology systems that store, process, and transmit the required information of the organization. Secondly, the management can achieve this by ensuring that the management make good risk management decisions to minimize expenditures that are also part of IT budget, and thirdly is by helping the Information Technology systems management on the purpose of making use of the supporting software documentation resulting from the performance of the risk management. Target Audience The main audience used to identify these issues, and concerns leading to poor Technology management are the personnel who support the management process for their Information Technology systems. ... The system and application programmers IT quality assurance personnel Information system auditors IT consultants Strategies and techniques used to improve technology management Information Technology management is a technique used by many companies to take advantage of the available Information Technology resources to increase and improve the pace of production and create a competitive business operations. Many business or managers may fail to fully implement and make use of Information Technology in their due to the expenses, maintenance and upgrade requirements. However, this can be through developing good Information Technology management strategies in the company. The managers should also find a good way of using technology to improve and increase the company’s productivity and operations. Corporate strategy Managers to set the overall direction of a company use corporate strategy. The business shareholders and department managers determine the financial markets the corpor ation will function in, how to get the required materials and resources, and how to get its products into the marketplace (Tesar, 2003). The information Technology department should assist the company in achieving these goals. For example, the Information Technology systems can use the company’s website to advertise and sell the company goods and services online in markets. The technology department can also use the systems to order materials from dealers, and using commerce software to make forecasts for productions and sales of the company. Business strategy This strategy only focuses on one division, department, and product line in the company. Managers and the

Human Resource Management Employee Selection Processes Research Paper

Human Resource Management Employee Selection Processes - Research Paper Example Methods of selection on the other hand should have some reasonable quality as dictated by the concept of utility. Selected employees should add value to the organization. A method that selects employees who cannot add value to the organization is an ineffective method. Lastly, selection process must adhere to the legal regulations and provisions of employment such as discrimination and equal employment regulations (Ullah, 2010). Organizations selecting employees should ensure that there are no discriminations and equal employment opportunities. Equally suited candidates must be subjected to equal interviews or selection process. One of the easiest and most effective approaches in selecting one of the two equally suited candidates is the thorough analysis and evaluation of experiences and job specifications. There is no doubt that there must be one candidate who is slightly more experienced that the other or has an additional qualification. In the event that they have the same years of experience, academic qualifications, and admirable track or records, the two equally suited candidates should be subjected to various selection processes such as aptitude of panel interview to critically evaluate their uniqueness (Knouse,

Phase 1 Discussion Board 2 Posting Assignment Example | Topics and Well Written Essays - 1000 words

Phase 1 Discussion Board 2 Posting - Assignment Example However, it should be noted that it basically depends upon the situation and in some cases, it is quite important for experts to take up the role of project management. Project management is usually a wide area involving a wide array of things. In the process of developing a new technology or implementing a highly technical process in an organization, it is imperative that the whole process is conducted by a person who has technical knowledge in the workings of such technology. The present world is very dynamic and new technology comes by the day. In this case, it normally takes much time before people get acquainted with such technology. It is therefore important for the project manager to be an expert in the field so as to offer the necessary instruction or guidance on how the whole field should be handled. In a highly technical working environment like NASA, it is almost impossible to have a project manager who is not an expert in the project. This is because there is a high degree of technical excellence required and there is no room for any mistakes. As such, a project manager in such a case is required to have knowledge on the technology in use (Wysocki, 2010). It has always been seen that in situations where project manager do not have expert knowledge on the subject matter, challenges often arise. For instance, a layman in medical equipment might not really realize that such equipment require a lot of maintenance which might be very expensive. In that case, some of the advice directed to the project manager might be ignored since the manager does not clearly understand the weight of the whole issue. Having a project manager who has knowledge on the subject matter is very important as it greatly saves time and cost of executing roles. This is because the manager already understands much of the operations and does not really need to study the process much. On the contrary, a lay project

Discussion Research Paper Example | Topics and Well Written Essays - 250 words - 3

Discussion - Research Paper Example Both pieces are white and have columns and arches, and are decorated by sculptures. Similarly, they both are examples of representational art. However, the subject matters of the two pieces are different. The Temple of Athena’s purpose is to help the Greeks in war, while the Arch of Constantine celebrates the war that has already been won. The Temple of Athena had a religious function of worshiping the goddess of victory. The statue of Athena inside the Temple held a pomegranate in one hand, which was a symbol of fertility, and a helmet, a symbol of war, in the other hand. In addition, Athena had no wings, as usual. She was depicted without wings in order not to ever leave the city, as the people believed (Ancient-Greece.org, 2013). On contrast, the Arch of Constantine had a social and political function of celebrating, immortalizing and commemorating Constantine’s victory over his enemy. The use of materials from the monuments of Marcus Aurelius, Hadrian and Trajan, who were considered good emperors, also created positive associations between them and

International economics Assignment Example | Topics and Well Written Essays - 500 words

International economics - Assignment Example These fears that caused by the emergence of globalization mainly focused on jobs that might be lost due to the competition from foreign countries. The main threat was on wages and not on jobs as many analyst thought. Trade had little impact on the total number of employment in most of the developed countries. The other thing that trade had minimal effect on was the composition of the level of employment. Given that the labor markets are always flexible, the loss of jobs in one field should be to create other new jobs in another field. Therefore both the relative and the average wages stand a chance of being impacted by trade, this is due to the emergence of economies. For a very long time, real wage has been traced back to the average productive growth. This has however changed in modern times, the real pay of workers in many developed countries has slowly increased than labor productivity. For instance, the real wages that an employee gets in a week in the United States of America has fallen in the middle of income distribution by at least 4% in 2001 alone. During this period, the level of labor productivity has risen by 15%; this is based on real terms. In other economies such as that of Germany and Japan the income distribution is either flat or maybe falling. This has fueled an argument in favor of globalization that is aimed at making the workers to get better wages, it has also improved the security of jobs of the low-skilled workers hence the job loss is reduced. According to the ideas from David Ricardo, he argues that many economies tend to gain from trade through specializing in products that has the capability of comparative advantage. Looking at the economies of many developed countries, there is existence of many skilled workers but that is not the case in countries that are experiencing the emergence of economy since they have low skilled workers. The economist argues that the real wage in the United States of America

Discussion Research Paper Example | Topics and Well Written Essays - 250 words - 3

Discussion - Research Paper Example Both pieces are white and have columns and arches, and are decorated by sculptures. Similarly, they both are examples of representational art. However, the subject matters of the two pieces are different. The Temple of Athena’s purpose is to help the Greeks in war, while the Arch of Constantine celebrates the war that has already been won. The Temple of Athena had a religious function of worshiping the goddess of victory. The statue of Athena inside the Temple held a pomegranate in one hand, which was a symbol of fertility, and a helmet, a symbol of war, in the other hand. In addition, Athena had no wings, as usual. She was depicted without wings in order not to ever leave the city, as the people believed (Ancient-Greece.org, 2013). On contrast, the Arch of Constantine had a social and political function of celebrating, immortalizing and commemorating Constantine’s victory over his enemy. The use of materials from the monuments of Marcus Aurelius, Hadrian and Trajan, who were considered good emperors, also created positive associations between them and

Culture and Organisation Essay Example | Topics and Well Written Essays - 3000 words - 1

Culture and Organisation - Essay Example At the time, the company was generating revenue that amount to over 21.5 billion euros per year during which time it had the biggest market share in the UK (Wei & Zhou, 2007). Currently, IKEA operates around 301 stores in over 35 countries. Most of the stores are in Europe, Asia, North America, and Australia. This billion dollar company was founded in 1943 by Mr. Ingvar Kamprad (Edvardsson & Enquist, 2008). It is owned by Stichging INKGA group although the company is partly controlled by the founder’s family. INKGA group is a charitable foundation registered in the Netherlands. The company’s main competitors include Argos, Asda, Next, Debenhams, Tesco, and John Lewis (Moon, 2004). There exist many ways through which companies can gain a competitive advantage over other companies. With the current economic uncertainties in the market, it is critical that companies get to engage in activities that will ensure their continued success in the market. Since economies from different part of the world are failing, one thing to do to ensure that a business is not affected and can stand the test of turmoil is to spread risks. In other words, most companies are not putting their eggs in the same basket. This means they are promoting their brand through establishing stores in as many countries as possible. One of these companies, and one that this paper will be discussing in detail is IKEA. As put by Porter, sustainable competitive advantage is one of the most sought after things by companies today (Porter, 2008). It requires that a company ensures that its level of profits are maintained if at all the company aims to be sustainable. As stated by Christopher (2012), â€Å"the most profitable competitor in any industry sector tends to be the lowest-cost producer or the supplier providing a product with the greatest perceived differentiated values.† This is, however, affected by important factors such as culture as will be discussed below. The issue of communication is

Benefits of a C Corporation Essay Example for Free

Benefits of a C Corporation Essay Firstly, a C Corporation can be defined as a legal unit of business that has its own separate identity from its owners or the persons that form it. To use the general legal term, it is a â€Å"body corporate†, a â€Å"legal entity†, which simply means a corporation can be treated just like an individual in law, it can sue and can be sued. This singular characteristic does not only make it quite unique but also gives it an edge over other business entities such as sole proprietor, partnership, limited liability Company to mention but few. However, the two major types of corporation, which are the C Corporation and S Corporation. The C Corporation has quite numerous benefits; the most technical of these is its ability to guard its owners’ personal assets from judgment against the business. Owners can also split profit and loss with the business for a lower overall tax rate. Yet another benefit is that it has no limits in number of shareholders and owner. Again, Owners do not need to be U. S. citizens or residents; both foreign and local are equally allowed to share in its ownership. A C corporation also has the flexibility of being owned by yet another business entity, rather than an individual or several individuals. Finally, the C Corporation has a package of much greater tax advantages over any other business entity, for it was primarily designated for tax purposes, while the â€Å"S† corporation passes it tax to the shareholders, the C Corporation on the other hand acts as it own tax entity. Reference Bakan J. , 2004, The Corporation: The Pathological Pursuit of Profit and Power. USA

An Analysis Of Italian National Interests Politics Essay

An Analysis Of Italian National Interests Politics Essay National interest incorporates within it many things. Since the beginning of the nineteenth century, events have shown us that Italy and many of its leaders and politicians have always found it difficult to opt for a particular direction in both internal as well as external affairs. Whilst fighting against the Hapsburg Empire for independence in the nineteenth century, various Italian politicians proposed several options and alternatives that Italy could take. All of this brought conflict of interests and ideas in Italy and no-one was sure which road Italy should follow. In the quest for independence, the Italians knew what to remove but not what to implement instead. In the twentieth century, national interest changed frequently and much of this depended on the leaders of the time and on external events. The fact that Italian governments have changed so frequently in the past century is a clear example of how Italian governments and politicians never agreed on where their national i nterests lied. This paper will focus on how hard the Italians have found it to agree on their national interests and the reasons behind such a statement. This paper will then analyse to what extent such a statement is true and whether agreements between the Italians and Italian politicians regarding the matter have been reached. Special focus will be given on Italys relationship with Europe and the US and certain domestic policies as well as on the Italians view on the Arab World and the Mediterranean. Italys national interests Italy has had a history of changing alliances at the opportune moment in order to suit its interests. This shows that the Italians were never certain on what they really wanted. In the First World War, Italy joined the Entente when it had been an ally of both Germany and Austria-Hungary for so long. On the other hand, in the run-up towards the Second World War, Italy was attracted to Germany and its way of thinking and, when Germany rearmed and was pictured by many as the current strongest power, Italy did not hesitate to join Hitler in an alliance  [1]  . When Mussolini came into power in the 1920s, national interest lied in that of expansion and irredentismo. Mussolini wanted to promote the grandeur of Italy by conquering territories in Northern Africa and in some parts of Europe. Such an expansionist foreign policy agenda changed abruptly with the fall of the fascist regime in Italy and a new direction of national interest became important. Italian national interest then chang ed again with the end of the war as the Italian population and government sought the help and friendship of the United States  [2]  . The two main rival parties in the period were the Communists and the Christian Democrats and both had very diverse interests and foreign policy agendas. Immediately after the war, Italy promoted its interests in the international arena and this is seen in its quests for European Integration and in that of joining the North Atlantic Alliance yet such notions and ideas had to wait until the arrival of De Gasperi for them to become the new direction of Italys foreign policy: Italy under the leadership of Alcide De Gasperi became one of the founding countries of European integration, when the European Community and NATO were the two main pillars in postwar Italian foreign policy.  [3]   Putting these agendas aside, many argue that, during the First Italian Republic, national interest was not focused on the international context. This statement is quite correct especially when seeing how the culture of national interests did not reach its full potential due to the political elites interest in domestic rather than in foreign affairs. Due to certain issues occurring in Italy today it may seem as though the Italians are more concerned about their domestic affairs rather than their external ones  [4]  . The huge importance given to issues such as Silvio Berlusconis scandals and personal life is a clear example of this. Some newspapers are more focused on these particular affairs rather than what is happening abroad. The fact that domestic affairs was important then and is still so relevant now, brings to mind the thought that, in certain aspects, the Italians have managed to find a compromise between them. Others will say otherwise especially after analyzing Italian foreign policy in the second half of the twentieth century. Italy opted for an active foreign policy agenda as it tried to restore the honour it had lost as a result of The Second World War. Upon joining the European Union, Italys economy flourished considerably as can be witnessed in the 1960s. This was a result of increased industrial development and new economic reforms. When analyzing such issues, it could be said, therefore, that Italys national interests lay mostly in Europeanism as Italy became a strong and active member in European affairs and integration and it benefited a lot from its membership. Most political parties in Italy were and still are very pro-EU and Italy was always amongst the highest in Europe which had a strong public opinion with regards to the European integration process. Though the Communist party was quite anti-European in its policies, it changed attitude during the 1970s when it saw European integration as a means to promote its national interests. Hence, this led to support for integration becoming nearly unanimous in Italy  [5]  . Many believe that Italian national identity was weakened because of the relations between the state and the church in Italy especially during the nation-building process  [6]  . The church has always maintained a strong role in Italy and this is mostly due to the reason that the Vatican City is the centre of Catholicism. Hence, the church has been very influential in events happening both at the national level as well as on the international level and in fact, since unification and even in issues such as Italian-Arab relations, the church has always expressed its views and opinion on what Italys approach to such matters should be. Though Italys activity in European processes and activities was always present, it still wasnt recognized as a major power in the EU but was seen as a large member state among small ones, but incapable of taking its place among the big ones in the Union  [7]  . Italys inability to implement certain European legal norms, as can be witnessed in certain aspects such as that of freedom of expression, has reduced its credibility in the European context and even resulted in Italys less influential role when it comes to decision-making. Italy, however did try to promote its Europeanism especially in issues such as that of a European Monetary Union to which Italy gave much support. It was during Romano Prodis Italian Premiership when Italy became more credible as a result of its joining of the Euro-zone. This credibility was then again lessened with the arrival of Silvio Berlusconi who had a more Euro-skeptical attitude in his foreign policy. Many scholars and analysts believe that this attitude was a main reason of why national interest was seen as unstable and unclear. Until the 1990s, Italy was quite pro-European, however, when Berlusconi became Prime Minister, this changed. In fact, Berlusconi conflicted quite often with other heads of European states especially with the French President and German Chancellor. Further proof of this erosion of Europeanism can be witnessed in the resignation of Italys foreign affairs minister, Renato Ruggiero, who was a pro-European  [8]  . Berlusconi, apart from being Prime Minister, had been minister of foreign affairs and his motives and interests were clearly seen in his policies. Berlusconi preferred to maintain good relations with American President George W. Bush rather than focusing his strength on maintaining excellent EU relations. He wanted the American administration to see Italy as its closest European partner after Britain and that is why, Italy gave America its full support in many issues and crises. Italys conflicting interests between Europeanism and Atlanticism did not start with Berlusconi. Sergio Romano comments on how the Italians believed that the Second World War was not lost by the nation but rather by the fascist government and this explains why, after overthrowing Mussolinis regime, Italy was still willing to seek cooperation with other states. As from the late 1940s Italys cooperation with the US grew especially due to the Marshall Plan which, according to Romano, ebbe maggiore influenza sul profile internazionale del paese e sul sistema politico italiano negli anni seguenti  [9]  . In the meantime, Italy was very confident in the fact that it could play an important role in European as well as in Mediterranean affairs. The world order after World War II was very different than that prior to the war. Not only did countries like Italy lose certain territories such as Istria, Libya, Eritrea and Ethiopia but, now, there was the rise of communism and nation states suc h as Italy had to respond to such crises. The Marshall Plan wasnt only implemented for the restructuring of Europe, but also to combat the communist threat of the Soviet Union. The plan was also aimed to bring unity in Europe and this influenced the policies of Italian politicians and how they behaved at the international level  [10]  . National interest, in a few years, had changed radically from one being aimed to promote Italy and its grandeur to one which promoted Italys activeness and participation in an international community. After 1947, Italys policies started reflecting those of the United States of America. Hence, not only was it interested in economic development and restructuring, but was also intrigued by implementing a democratic government, rights and freedoms as well as having an anti-communist task force. Because Italy followed these concepts, it was also interested in having a united Europe and, as explained above, was among the first to promote cooperation am ong European states. Italys national interest was always divided between that of Europeanism and that of maintaining excellent cordial relations with the US and such a phenomenon may still be said to be going on today. Many countries were against Italy joining the Atlantic Alliance and the Italian government was particularly keen on changing the negative assumptions that these countries had about it. Such countries believed that: Se fosse stata ammessa avrebbe costretto lAlleanza a difendere non soltanto lAtlantico del Nord ma buona parte del mediterraneo, avrebbe dato un contributo insignificante alla difesa commune, e avrebbe probabilmente approfittato della sua partecipazione per avanzare una volta querule richieste coloniali.  [11]   De Gasperi and Sforza saw that these ideas had to change as Italy needed to develop economically as well as improve its security. Without the help of foreign powers it couldnt do much. Even if other countries had no faith in Italy, the latter knew that it could count on the US to provide for Italys needs. The USA was, and still is, considered as a super power and could provide for the security of others apart from itself. The transition that Italian national interest was taking clearly shows how Italian leaders realised that Italy should not be overconfident in its approaches and be as ambitious as it was in the time of Mussolini. Even nowadays, Italy is trying to maintain the best of relations with the US as the former knows that America can provide Italy with financial aid and security. Apart from this, by maintaining a relationship with the worlds superpower, Italy and Italian opinion could become more credible and attractive in the international arena. De Gasperi and Sforzas opposition and criticisms which were coming from within the mainland clearly proves how national interest differed from one party to another. Apart from being criticized by the national liberals, both statesmen found opposition in the Catholics of the left, the socialists and in the communists all of which had considerable influence in Italy  [12]  . Hence, national interest varied in Italy as some opted for expansion and the regaining of national prestige, others for the weakening of the Western front whilst others were willing to make Italy a determining power in international affairs. Nonetheless, both De Gasperi and Sforza tried to establish strong positions not only in the US, especially after joining NATO in 1949, but also in Europe and this is evident when analyzing the relationship Italy had with countries like France. Since the time of de Gaulle, and especially under Georges Bidault, France considered Italy as an important ally and friend. In this issue another phenomenon regarding national interest is evident. This is Italys interest in safeguarding the Mediterranean and Italy, since Mussolinis rule up until today, was one of the major players when it comes to security in the Mediterranean. France, who had interests in Northern Africa, agreed to help Italy in this task and, since EU membership, Italian statesmen, ministers and politicians have promoted the idea of Mediterranean security even more  [13]  . Italian Prime Minister Aldo Moro, as did many others, maintained good relations with Mediterranean countries including Malta so as to maintain a stable and secure Mediterranean region. Italys interest in the Mediterranean may be a reason why the country is so keen on maintaining stable relations with the Arab world. Since the era of Fanfani and Moro, and even before that, up till today, Italian relations with Arab countries have been quite positive. This can be seen in Italys involvement in the Suez Crisis and in the Six-Days War and also in the fact that Italy always expressed its opinion with regards to the Middle-East conflict and offered means to aid the people affected by the war  [14]  . Many say that Italy has boosted its market-share and maintained a high political profile in the Arab world and in the Persian-Gulf and this is evident in countries like Qatar, UAE, Oman, Kuwait and many others. Embassies have been set up in these countries, exports increased and relations with them have improved due to the issuing of new contracts. Also, with regards to some states, Italy is the only remaining European trading partner as can be seen in the case of Iran  [15]  . Many believe that, though Italy has had a strong pro-European perspective, European enthusiasm has fallen considerably in the past years. According to a survey last year, only a small percentage of the Italian population actually believed that Italy was benefiting from its relationship with the EU. Italy has indeed fought for the strengthening and widening of the European Union, however, with the acceptance of the Constitutional Treaty, the introduction of the Euro and with the incorporation of many new pacts and treaties allowing greater EU say in decision making, Italians are starting to question what role has the national sphere left to play. From the period 2001 till 2006 the European Union was rarely consulted and called upon when Italy discussed matters of policy and foreign affairs. Even though Prodi was a pro-European, the two years he spent as Italian Prime Minister following 2006 werent enough to change the European outlook towards Italy. It seemed that Italy was changing f rom being one of the most active and respected members in Europe into a country which acted as an obstacle to the European Union. Events such as Italys exclusion from key talks on important issues suggest that the above statement is true and that Europe and the European Union had lost faith in Italy  [16]  . In Italy there was always a diverging of interests between the centre-left and the centre-right parties. The issue of multilateralism in this context becomes important. Under Berlusconis centre-right government this process was threatened especially due to the fact that the Prime Minister had little faith in such a process. Berlusconi, always projected as being so Euro-skeptic, has always preferred to maintain bilateral relations with countries such as the US and with certain countries in Europe rather than having many relations on a multilateral level. This statement is clearly supported by the fact that international organizations have never regarded Italy under Berlusconi as an important asset especially since the centre-right participated mostly in multilateral agreements and in international organizations only if there was something to gain from them. Centre-left parties, on the other hand have always taken a different approach and have always preferred to maintain strong multil ateral bonds especially with the European Union and its institutions. On the other hand the centre-left was then skeptic on the relationship with the US and this is evident in the centre-left parties criticisms towards the US and some of its policies  [17]  . Conclusion The fact that so many different governments were set up after 1945 and the phenomenon that only nowadays is an Italian political party managing to stay in government a full term is a clear reason on the state of agreement on national interests is in Italy. The huge number of fragmented parties in Italy means that it is difficult for Italy to come to a consensus on both domestic affairs as well as on international affairs. All the parties have different national interests and this may be the cause of why many believe that the statement that there hasnt been agreement on where Italys national interests lied was and still is the reality of Italy. Even the events of 2008 when Romano Prodis government collapsed is a sign on how the Italians are still struggling to find an agreement on where their national interests lie. Never before in the post-World War II history of Italy, had a government been unsupported by its own majority because of disagreement (within that same majority) on how to run the countrys foreign policy.  [18]   Such an event, however, will not change the real aims of Italys foreign policy and will not affect the reasons of why Italy has acted the way it did in the international arena in the past years. Italian politics in the last decades has always been directed towards achieving a more peaceful world. What Italy has done in the Middle-East and in Europe was done to strengthen unity as well as to bring stability in societies. On this matter, national interest is quite concrete and everyone, starting from Italys political elite moving down to the lowest classes of society, has had the same ideology and realm of interests with regards to having peace in the world. Though this is a huge contrast to the Italy prior to the war, it shows how, though national interest in Italy does tend to change rapidly, the Italian population may find agreement on certain issues especially on matters which are considered important and vital to the international sphere. Hence, it can be concluded that the statement that the Italians have never agreed on where their national interests lie isnt entirely true. Though Italian national interest did change quite frequently along the years, there were times when it was stable and followed a clear path. The fact that Berlusconi was elected for the third time has resulted in national interest to become clearer. Before the end of the cold war, the diverging national interest between that of a pro-European and a pro-American agenda was more evident. Now, with Berlusconi as Prime Minister, it seems that the divide isnt as evident especially with Berlusconis close relationship with the US and his lack of faith in Europe. Agreement on national interest, however, was never entirely present in Italy and may still be seen like that nowadays. It is unclear if Italian politicians are more interested in domestic rather than in international affairs and. Apart from this, Italy still maintains good relations with Mediter ranean countries as well as with the Arab world and is always keen in intervening wherever there is an issue abroad and this further confuses scholars and analysts trying to come out with where Italian national interest really lies.

An Organic Dissection of a Tutoring Session :: Personal Narrative Writing

An Organic Dissection of a Tutoring Session When Delvin (that's what I will call him) walked into the Writing Resource Center I was a little worried because I knew him from another class and a few social events, besides talking with him on occasion around the campus. I had had a bad experience earlier in the year when tutoring a friend of mine who'd asked me to help him with punctuation. I ended up just giving him a punctuation sheet and tried to work on some deeper level problems with word choices and development. He didn't see the problems or seem to want help in those areas, so he was offended at my attitude towards the content of his paper. (He thought that it was a near final draft except for a little editing.) Right away, though, Delvin said that his paper was in an early draft stage and he need some help "making it clear" because he was dealing with a difficult topic for Philosophy 205, Aesthetics. Now, Aesthetics is a complex topic dealing with ideas, objects, feelings, and theories as to what is beautiful and what is ugly. And Philosophy has a mode of thinking that is complex. At the time of my session with Delvin I didn't make the connection between the two, but prior writing to this paper the magnets reached their poles: the connection between philosophical thinking and writing development and also their ability, when done properly, to deal with complex issues. The main connection between the two is their ability to start with a topic and break it down into simple ideas and then reconstruct it into a theory or paper that displays, argues, questions, or proves the topic to a reader or listener. The philosophers main mode of dissecting a topic is by questioning every factor that is relevant (and even some that are irrelevant.) This is basically the same thing a writer should do when he or she begins to organize his or her paper. They need to find out, "who did what, when, where, and why?" Many writers and philosophers can organize these questions and answers in their head so that it makes sense to them, but the problem lies when they have to communicate multitudes of simple ideas to someone other than themselves. The overall idea won't make much sense to a reader if the writer spits it out like a madman babbling about the apocalypse.

The impact of migration on families around the world

Migration has been an on going phenomena from past to present. Individuals, families or group of people may leave a country voluntarily because of events such as harsh environmental or economic conditions. Migration has always been an important part of human endeavour. Migration can have positive and negative impacts on the families and also the country. With pros such as freedom, job vacancies and benefits to host country, it's no surprise most people don't have second thoughts about the cons such as effects on child, decrease in economy and propagation of diseases. Firstly migrants frequently settle in places with lesser population. These places would be having a shortage of skilled labour due to the decreased population. The migrants bring with them enough skills that will help support the economy. The migrants may be skilled people like doctors or unskilled people like construction workers. Migrants may increase unemployment in certain areas, but this is compensated by their raising the overall level of demand like housing and household related goods, needs†¦This promotes both a higher level and a wider range of goods and services produced, stimulating the economy and job growth. Migrants help to reduce labor shortages and also attract international traders through their foreign knowledge and language skills. Secondly migrants will be usually willing to fill job vacancies that local people are unwilling to fill. These include jobs such as babysitting, cleaning and other activities. This is an advantage for the existing community. Migration benefits not only the migrants, but also the countries that receive them, and even the countries they have left. Migrants doing these jobs live a happy and free life in a country where they get equal opportunity and thereby secure a better future for themselves and their families. For many young people, the experience and skills they acquire in the jobs to which they have migrated can serve as a step to further migration for better paid jobs. In addition to this researchers concur unanimously in reporting that migratory workers use part of their savings from working to improve their homes, whether by decorating them, extending them or introducing basic services such as electricity, water and drainage. Similarly, it is common to find that localities with migrants benefit from their contributions by receiving monetary support that helps them to improve diverse services, whether urban infrastructure, health, religious, educational or entertainment services. In towns with migrants it is common to find acknowledgments from the residents to their fellow townsmen for having helped them build a school or health clinic, or to introduce drinking water or build or improve the church,†¦. However the importance of keeping in regular contact: e-mails and text messages can in no way substitute for the physical presence of a parent, but they can help a child feel connected with family members who are away. Finally, there is the importance of the family's relations and communication before the migration. Many parents presumably do not migrate unless they think their children can cope in the first place. A factor found to be extremely important is the child's understanding of, and support for, the family goal. In fact, an individual's migration can be highly valued within a family: it can give status, not only for the material objects the migration may bring, but in more symbolic form of being the family member who gives for others. Some of these observations might be extrapolated to the situation of children affected by HIV and AIDS. Moreover it is generally believed that those migrants who have had the courage to leave one country and move to another are often enterprising and entrepreneurial, even if poor. As such, in many countries, migrants often set up small businesses. They however, become easy targets when the general economic conditions in that host country worsen. In other cases, people become migrants because they have fleed worsening conditions or persecution. In that situation, although they may live in another country, it may initially be quite difficult to adapt and change practices and customs. In such situations migrants are clearly seen as different and in worsening economic times can be seen as sapping away resources that could otherwise have been used for local populations. Furthermore migrating parents may decide, whether by choice or due to untenable circumstances to leave their children in their country of origin, planning either to return to their household of origin or to reunite much later in their destination country. The decision of one or both parents to migrate and consequently, to leave children behind, may be the result of an individual altruistic decision to send remittances in order to make their family members' lives better, or the result of household utility maximization that may take into consideration also the risks and perils of travel. Children whose parents are working abroad have a similar profile to those living in mono-parental families resulting from the parents' separation or from the death of one parent. This shows that, although the work abroad is temporary, the impact on the children could be similar to that of the loss of a parent, through divorce or death. In conclusion, children whose parents are working abroad should be considered at risk. That would be a first recommendation to children protection authorities, who should find solutions to strengthen the relations between school and the social services system. For many migrants, migration represents an alternative for supporting their families. Nevertheless, while going to work abroad contributes significantly to household incomes, it also has many social costs. Migration also affects social relations. Migration also changes the roles within the family additional tasks must be assumed by those left behind.

Differences in the Myths of the Toad, the Snake, and the Medicine

The Limba people of Africa live in different villages, and each village puts its own spin on the myths that are passed down from generation to generation. One of these myths focuses on the god Kanu making medicine to immortalize the Limba, and the destruction of that medicine by the toad. There are three versions of the myth of the destruction of the medicine, but they vary in several ways. The first myth, â€Å"The Toad Did Not Love Us,† suggests that the toad dropped the medicine Kanu gave him on purpose.Although most Biblical tales paint the snake as a dangerous creature, in this myth the snake loved the people. This myth implies that Kanu finds it strange that the people kill the snake, but not the toad, considering that the snake loved them. This myth is also different because it mentions the â€Å"white people,† so the reader can be fairly sure that this myth was either thought up after European colonization, or it was modified to add them. The second myth,  "The Toad and the Snake,† tells that Kanu wanted to save both animals and people.Again, the toad insisted on carrying the human portion, and again he spilled it, but not out of ill will. The snake carried his portion, and arrived with it safely. While this myth is still about why the people die, it also points out that snakes live forever because of their medicine. Perhaps this idea came from seeing molted snake skins. The skin might come off, but the snake lived forever. The last myth, â€Å"The Toad and Death,† is a short version of the same myth, but it only concentrates on how the snake and toad feel about each other.They are enemies because they perpetually argue about who should have carried the medicine. This is not because one loved the people more than the other. It is simply a rivalry that goes on for eternity. This myth serves more as an explanation of why snakes and toads do not get along rather than why people die. These three short myths show how stories c hange as they are passed around and told by different people with different influences. It is intriguing to see how one tribe could have such varied views on the same tale.

Far From the Maddening Crowd Essay

Bathsheba Everdene is the main character in the book â€Å"Far From The Madding Crowd. † She is central to nearly every situation and is held in high regard by the village people. She has her ups and downs in the book, her highs and her lows, finding love and happiness, only for it to be torn away from her and handed back to her in the not too distant future! Bathsheba begins the book in a cold hearted way, first refusing to marry Gabriel Oak and then playing with the feelings of farmer Boldwood, this coming back to haunt her further in the book. She reluctantly sent Boldwood a Valentines card aided by her maid Liddy’s persuasion, declaring her love for him, and seemingly to her surprise, poor Boldwood takes it very seriously resulting in a cat and mouse game of love. In chapter 24 at the fir plantation, Bathsheba was heading home when she bumped into a dark stranger and became tangled up in his boots. She tried to wriggle free but to no avail. After a lot of fiddling by the stranger, he freed Bathsheba and rose face to face. Bathsheba asked him â€Å"Who are you then, who can so well afford to despise opinion? † The stranger replied â€Å"No stranger. Sergeant Troy†. Bathsheba was evidently spooked by Troy; little knowing he’d be her future husband. Bathsheba met up with Troy on several occasions and subsequently fell in love with him. They agreed to marry whilst on a visit to Bath and returned to Casterbrige to Boldwood’s and Oak’s despair as Mr. and Mrs. Troy. When Boldwood discovered the news, he was erratic in his actions. He offered to buy Bathsheba from Troy and eventually after torment from Troy, he murdered him in front of the whole of the village at a Christmas party. He then proceeded to kill himself with the same gun and in the same spot where Troy died. Bathsheba was deeply upset about the death of her husband despite the fact he declared he did not return her feelings after the death of his previous fianci , Fanny Robbin. Eventually our widow married Farmer Oak whom she clearly was in love with from the first time they met. Bathsheba’s toying with Oak, Troy and especially Boldwood could be seen as a reason for her not to find happiness in the book, but despite her unkind joke with Boldwood, Bathsheba didn’t set out to hurt anyone and was caught up in a battle for her love between these men. After the death of Fanny Robbin, it was evident that Troy was still in love with this woman, despite her failing to show for their wedding. He blamed all his misfortune on his wife, Bathsheba and he bullied her into feeling that she was to blame. The death of Troy was a blessing in disguise as she would have been sad for all her days and would not have ended up married to Oak. All in all, I feel that Bathsheba deserved to be happy, as she was an honest, hard working woman who was independent and capable. I think she was genuinely sorry for the Valentines card to Boldwood. She finally does find happiness in Oak and realises his companionship and true love is what she has wanted all her life. I think Gabriel Oak certainly was the right man for her as, like his name suggests, he was angelic, solid and dependable. This was evident when he stays by her, and by his actions on the night of the storm when he and Bathsheba worked together to cover the crops. It is evident they are in love when they cast eyes upon each other and fail to remove them for a few seconds, showing mutual unspoken understanding and purpose.

From Imitation to Innovation Zongshen Industrial Group Essay

From Imitation to Innovation Zongshen Industrial Group - Essay Example Zongshen acquired human sources from Japan, SOE leaders and Chinese government officials. It developed Masters in Combustion Engines and Mechanical Engineering to improve skills of local students. The company developed its staff’s skills on technology and engineering by organizing foreign visits to joint venture partners and other companies. The decision of reverse merger was too challenging for Zongshen and Zou because the idea for quick listening was attractive but, on the other hand, Zongshen was investing in a company which was about to be delisted. However, Zongshen took a bold step and the original company eventually transformed to Chongqing Zongshen Power Machinery Co. Ltd, the largest engine producing business. Another critical phase of Zongshen was its expansion to acquire Canadian based PEM technology. Zongshen took a smart decision to purchase PEM, it first merged with a Toronto listed capital asset pool company, Norstar Ventures. Then it formed Zongshen PEM Power Systems. This was later expanded to a whole Chinese subsidiary as Zongshen PEM (Canada) Chongqing Hydrogen Energy Ltd. In 2006, another subsidiary Zongshen Electric Vehicle Company Limited was developed. These two focused over the production of premium e-bikes, giving edge to Zongshen over its rivals. The initiative of ‘blue ocean’ strategy for Cyclone products was very attractive. But the way this strategy was implemented cannot be appreciated. The main flaw with this strategy was that the consumers couldn’t relate such a premium quality and price product to Zongshen. Following are the Yellow Tail wine case implications for Zongshen to implement their ‘blue ocean’ strategy. 2. Reduce consumer confusion. In Chinese market there was a need for quality and performance but it was not yet realized by the consumers at large. So they could not relate Cyclone products’ premium differentiation to their need. Therefore, the

Assessing Reading and Language Instruction Research Paper

Assessing Reading and Language Instruction - Research Paper Example The children will continue to read pictures and to improve their basic comprehension skills. First Grade Observation In my observation of the first grade classroom the students’ desks were setup in groups of six. The teachers’ desk was placed by the classroom door. The room was decorated with many posters and projects that the children had completed. The room had many bulletin boards that were used for classroom jobs and notes that the students could look at like the lunch menu and any important classroom events that they can pass on to their parents. Some bulletin boards were to show off their art work in the other areas of the curriculum. There was an independent reading area, a game area, a listening area for stories and music, an art area and a writing area. There was a section of the classroom that could be used to practice the skills with the other students and/or the teacher. The area for direct reading instruction was in the front of the room but to the left whe re there were many charts with the necessary skills that they are going to learn on the journey to learn reading. The teacher also led shared reading experiences here along with language experience where the students could share a story they â€Å"wrote† to the class. There was also a variety of books of all classifications that they can choose and read from. . Reading Instruction and the Curriculum Used In a first grade classroom Reading is considered a subject where students will learn to use word recognition skills and learn about consonants and vowels through Phonics and the Bottom- up theory to Reading. Reading is a subject that must build on what the children already know. These children will bring what they learned at home from watching their parents reading the newspaper, writing notes and lists, reading the mail they received like they did in preschool and Kindergarten. The teacher will need to assess these children to see what the level they will be at during the sc hool year. The children will need to start to make relationships between the pictures and words and to put these ideas together to understand a story that they are reading. The children will learn that the alphabet is the beginning of words. The first grader learns that letters and their arrangement matters in the words that they are learning and reading. First graders are still emergent readers and they will want to learn how to spell and then write the words that they are learning. The first grade classroom teacher must give plenty of opportunities to read and write words. They will learn that reading and writing is their way of communication. They will learn about the concepts of print and that the print represents the spoken words, they have boundaries, and are read from left to right. (Gunning, 2005) Reading can be taught in the classroom by using direct instruction and by using learning centers that were around the classroom. They will also use learning centers for Art, Music and even Computer education. The children can practice their reading and writing using their Dolch Vocabulary List and continue to practice even more sight words. Using these words the children can read and write notes and letters and drawing pictures. Reading is for exploration. â€Å"When the parents and/or the teacher read to the children they learn and develop vocabulary, expand their experiental background and makes them aware of the language of books and introduces them to basic concepts of print

The legalization of marijuanna Essay Example | Topics and Well Written Essays - 1750 words - 1

The legalization of marijuanna - Essay Example in regard to the crime of drug use is that it is a self abusive crime, the only victim of the commission of this crime being the person using the drug. The illegal nature of drugs creates all of the victimization associated with drug use. More specifically, marijuana has properties that have the ability to naturally support different types of illnesses. Therefore, it is ethically irresponsible for society to continue to support the criminalization of drugs, especially in the case of marijuana. Through the lens of virtue ethics, one can examine an issue in regard to moral good and moral evil (Darwall, 2003, p. 53). One of the ways in which society has collectively determined that drugs should be criminalized is through the concept that they represent a moral evil. The social point of view is that anything that threatens the perception of safety and normalcy should be criminalized in order to provide perceived protection. The problem with this point of view is that it is blind to the realities of the effects of criminalization which are creating a far more moral evil than the self abuse of drug use represents. The moral evils that are created through criminalization is that it empowers people who have constructed evil intent, most often through greed, to create collaborative efforts that are outside of the control of the state. Virtue ethics is defined by perception. An example is given by Darwall (2003), in regard to two men who can provide another with the same service or advantage, the first man doing the service because he feels it is his moral responsibility and does it through a sense of altruism, where the second man does it because it fulfills an aspect of his own greed. The discussion is centered upon which man will have the greatest amount of sentiment, in this case gratitude, from the person who receives the service. It is the perceived motivation of the person providing the service that creates the level of gratitude as a response (p. 53). The fact is,

Gospel Choir by Walter Hawkins Essay Example | Topics and Well Written Essays - 500 words

Gospel Choir by Walter Hawkins - Essay Example Somehow Walter Hawkins managed to combine being an attentive pastor, a successful musician and a loving father. He got two kids from his marriage with Tramaine Hawkins. Even though their marriage was quite short, they remained friends and link-minded people. The album series â€Å"Love Alive† were very popular and their success was growing with each next recorded hits. The audience loved that Gospel singer for his counter tenor and it can be said that his voice was so powerful that it made him sound almost like an opera singer. The role of the Choir in Walter’s performances cannot be neglected   because some velvety texture was added to his singing.   The worship of God, expressed in such talented manner, made its work and carried the necessary message to his audience . People got silent listening to his songs as they were full of his energy and love that was felt by people. It was not important what languages were understood by his listeners because language of lov e had always been understood by everyone. Walter Hawkins got many awards for his talent. Grammy Award (he was nominated nine times), Dove Awards and Stellar Awards are the most prestigious among all of them.Walter was an exceptional man because he was the best in accomplishing each task that he had started. His main contribution was his dedication to people, either to those ones who entered his life for long, or those ones who crossed it for a while. His singing career enriched the whole musical world with jazz improvisation connected.

Strategic HRD and HRM Essay Example | Topics and Well Written Essays - 4250 words

Strategic HRD and HRM - Essay Example Each cupboard is a complex ensemble of fume extraction units, plumbing, and electrical systems to be designed and fitted. Each cupboard is unique and requires flexibility and an attention to detail within the manufacturing process to suit client’s customization needs. The company delivers bespoke fume-cupboards to customers gate but does not get involved in its installation. The Company consists of 220 personnel, 175 of them work on manufacturing process-the core process. Balance of 45 personnel work in the sales department, design department and the support function. Support functions include payroll, accounts, HRM, and administration. HRD problem as identified in company structure appears to be the fact that HRM function is small and inadequately staffed within the company. Further evidence in the case reveals that this function is not able to achieve its objectives to a very large extent. Similarly, sales and design departments appear to be under Staffed.The Company has cor e operations in the manufacturing process in which 80% of employees are deployed. It comprises of several functional sections like the sheet metal cutting and folding for the cupboards, protective coating application, initial assembly, plumbing, and electrical sections. The first HRD/Organizational problem identified in core operations appears to be a loose span of supervisory control over laborers. Assuming 4 supervisors per functional section, we have about 20 supervisors in 5 sections and 155 laborers under such supervisors. Each supervisor on an average has about 8 laborers under him. The second problem relates to very high labor turnover on manufacturing shop floor. This disturbs the supervisor span of control further which is already high at 8 laborers and possibly obstructs the product customization process leading to defectives and quality deficiencies. Moreover, quality control function at the company is very weak too.

Drama Coursework Essay Example for Free

Drama Coursework Essay The main reason I chose this extract was that there were two female characters in the extract and two female actors in the group. I liked the way the relationship between the two characters developed during the extract. At the beginning of the play, there is a professional, quite friendly relationship but by the end, both characters hate each other. I also liked the way my character, Mrs Lyons, descends into madness. When I first started to rehearse this play, I found that it was harder than I expected. This is the first serious play I have been involved in, so I found it hard to say the lines convincingly and naturally. I suggested that I should talk with a more educated voice and that my partner, Becky, should talk with a more common voice to emphasise the social difference between the two characters. I also suggested some ideas for what we should wear. In my coursework I wanted to present a convincing portrayal of a wealthy, educated woman who, faced with a extremely distressing, seemingly insurmountable personal problem, sees what looks like a perfect solution, but which ends up driving her to madness. I wanted to initially gain the audiences sympathy for her predicament by showing how much she wanted children and to show her as a reasonable person. I considered the second scene the most important as this was when my character discovered that her employee was expecting twins and couldnt afford to keep them both; and when the plan for Mrs Lyons to pretend to be pregnant and to keep one of the babies was hatched. In this scene, she promises Mrs Johnstone that the baby will be better off with her, and that Mrs Johnstone will be able to see him every day as she comes to work. However, in the next scene, she breaks that promise by sacking her. I wanted to show that the sacking was motivated by Mrs Johnstones paranoia. In the final scene I wanted to demonstrate that my characters mental health had deteriorated. I tried to portray that she was wealthy and educated was by talking in an upper-class, educated accent, and by dressing in a smart suit. I could have improved my performance by making my accent more pronounced, doing my hair in a more sophisticated way, and wearing some tasteful make-up. In the first scene I tried to convey her longing for a child by delivering the words as if I was completely wrapped up in my problem and as if I was talking to myself rather than anyone in particular. To demonstrate this, when I said the lines about only buying such a big house in the hope of having children, I looked down at the table rather than looking at my partner because I wanted to make it seem as if Mrs Lyons felt vulnerable because she was being so open. I concentrated on what I was doing at the time, which was getting something out of my bag, to try and convey that my character was fighting back tears, and didnt want to look at Mrs Johnstone in case she showed her any sympathy or pity, which might have made her cry. I think I could have emphasised my characters reaction when she found out that Mrs Johnstone was pregnant in a similar way to Kara when she choked back her tea. In the second scene I wanted to show the first signs of my characters madness, when, on learning that Mrs Johnstone is expecting twins, she comes up with the extraordinary idea that she should fake her pregnancy and take one of the babies. I tried to express how she got more excited by speaking more quickly and by the tone of my voice. I also moved around a lot and started to talk more to myself than to my partner as my character got more carried away with her plan. In the third scene, where Mrs Lyons sacks Mrs Johnstone, I started off talking in a very authoritative tone and avoided eye contact with Becky because the supply teacher explained that when you have a problem with someone, you dont look at them. As the conversation develops, and Mrs Johnstone threatens to take the baby away or tell the police, I wanted to show that Mrs Lyons was manipulating Mrs Johnstone by playing on her superstition and lying to her that shell be locked up if she tells anyone what happens. I showed this by getting close up to Becky and talking quite slowly and with a deep voice, in a threatening way. Because I am quite a bit taller than Becky, I was also able to look down on her, which reflected the difference in class between the two characters. I think it was a mistake to avoid mannerisms as compared to some other groups, whose little individual touches made their plays stand out, our performance was quite boring.

Hopf Algebra Project

Hopf Algebra Project Petros Karayiannis Chapter 0 Introduction Hopf algebras have lot of applications. At first, they used it in topology in 1940s, but then they realized it has applications through combinatorics, category theory, Hopf-Galois theory, quantum theory, Lie algebras, Homological algebra and functional analysis. The purpose of this project is to see the definitions and properties of Hopf algebras.(Becca 2014) Preliminaries This chapter provides all the essential tools to understand the structure of Hopf algebras. Basic notations of Hopf algebra are: Groups Fields Vector spaces Homomorphism Commutative diagrams 1.Groups Group G is a finite or infinite set of elements with a binary operation. Groups have to obey some rules, so we can define it as a group. Those are: closure, associative, there exist an identity element and an inverse element. Let us define two elements U, V in G, closure is when then the product of UV is also in G. Associative when the multiplication (UV) W=U (VW) à ªÃ¢â‚¬Å" ¯ U, V, W in G. There exist an identity element such that IU=UI=U for every element U in G. The inverse is when for each element U of G, the set contains an element V=U-1 such that UU-1=U-1U=I. 2.Fields A field Ã’Å“ is a commutative ring and every element b à Ã‚ µ Ã’Å“ has an inverse. 3.Vector Space A vector space V is a set that is closed under finite vector addition and scalar multiplication. In order for V to be a vector space, the following conditions must hold à ªÃ¢â‚¬Å" ¯ X, Y à Ã‚ µ V and any scalar a, b à Ã‚ µ Ã’Å“: a(b X) = (a b) X (a + b) X=aX + bX a(X+Y)=aX + aY 1X=X A left ideal of K-algebra is a linear subspace that has the property that any element of the subspace multiplied on the left by any element of the algebra produces an element of the subspace. We say that a subset L of a K-algebra A is a left ideal if for every x and y in L, z in A and c in K, we have the following: X +y is in L cx is in L zà ¢Ã¢â‚¬ ¹Ã¢â‚¬ ¦ x is in L If we replace c) with xà ¢Ã¢â‚¬ ¹Ã¢â‚¬ ¦ z is in L, then this would define a right ideal. A two-sided ideal is a subset that is both a left and a right ideal. When the algebra is commutative, then all of those notions of ideal are equivalent. We denote the left ideal as à ¢Ã…  Ã‚ ³. 4.Homomorphism Given two groups, (G,*) and (H, °) is a function f: Gà ¢Ã¢â‚¬  Ã¢â‚¬â„¢H such that à ªÃ¢â‚¬Å" ¯ u, v à Ã‚ µ G it holds that f(u*v)=f(u) °f(v) 5.Commutative diagrams A commutative diagram is showing the composition of maps represented by arrows. The fundament operation of Hopf algebras is the tensor product. A tensor product is a multiplication of vector spaces V and W with a result a single vector space, denoted as V    W. Definition 0.1 Let V and W be Ã’Å“-vector spaces with bases {ei } and {fj } respectively. The tensor product V and W is a new Ã’Å“-vector space,  Ãƒâ€šÃ‚   V      W with basis { ei fj }, is the set of all elements v    w= à ¢Ã‹â€ Ã¢â‚¬Ëœ (ci,j ei    fj ). ci,j à Ã‚ µÃƒâ€™Ã…“ are scalars. Also tensor products obey to distributive and scalar multiplication laws. The dimension of the tensor product of two vector spaces is: Dim(V   W)=dim(V)dim(W) Theorem of Universal Property of Tensor products 0.2 Let V, W, U be vector spaces with map f: V x W à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ U is defined as f: (v, w) à ¢Ã¢â‚¬  Ã¢â‚¬â„¢vw. There exists a bilinear mapping b: V x W à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ V   W , (v,w) à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ v   Ãƒâ€šÃ‚   w If f: V x W à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ U is bilinear, then there exist a unique function, f: V   Wà ¢Ã¢â‚¬  Ã¢â‚¬â„¢U with f=f °b   Extension of Tensor Products0.3 The definition of Tensor products can be extended for more than two vectors such as; V1 à ¢Ã…  -   V2à ¢Ã…  -  Ãƒâ€šÃ‚   V3 à ¢Ã…  -   à ¢Ã¢â€š ¬Ã‚ ¦..à ¢Ã…  -   VN = à ¢Ã‹â€ Ã¢â‚¬Ëœ( biv1à ¢Ã…  -   v2à ¢Ã…  -   à ¢Ã¢â€š ¬Ã‚ ¦.à ¢Ã…  -   vn )   (Becca 2014) Definition0.4 Let U,V be vector spacers over a field k and ÃŽÂ ½ à Ã‚ µ Uà ¢Ã‚ ¨Ã¢â‚¬Å¡V. If ÃŽÂ ½=0 then Rank (ÃŽÂ ½) =0. If ÃŽÂ ½Ãƒ ¢Ã¢â‚¬ °Ã‚  0 then rank (ÃŽÂ ½) is equal to the smallest positive integer r arising from the representations of ÃŽÂ ½= à ¢Ã‹â€ Ã¢â‚¬Ëœui à ¢Ã‚ ¨Ã¢â‚¬Å¡ vi à Ã‚ µUà ¢Ã‚ ¨Ã¢â‚¬Å¡V for i=1,2,à ¢Ã¢â€š ¬Ã‚ ¦,r. Definition0.5 Let U be a finite dimensional vector space over the field k with basis {u1,à ¢Ã¢â€š ¬Ã‚ ¦.,un}   be a basis for U. the dual basis for U*is {u1,à ¢Ã¢â€š ¬Ã‚ ¦.,un} where ui(uj)= ÃŽÂ ´ij for 1à ¢Ã¢â‚¬ °Ã‚ ¤I,jà ¢Ã¢â‚¬ °Ã‚ ¤n. Dual Pair0.6 A dual pair is a 3 -tuple (X,Y,) consisting two vector spaces X,Y over the same field K and a bilinear map, : X x Yà ¢Ã¢â‚¬  Ã¢â‚¬â„¢K with à ªÃ¢â‚¬Å" ¯x à Ã‚ µ X{0} yà Ã‚ µY: 0 and à ªÃ¢â‚¬Å" ¯y à Ã‚ µ Y{0} xà Ã‚ µX: 0 Definition0.7 The wedge product is the product in an exterior algebra. If ÃŽÂ ±, ÃŽÂ ² are differential k-forms of degree p, g respectively, then   ÃƒÅ½Ã‚ ±Ãƒ ¢Ã‹â€ Ã‚ §ÃƒÅ½Ã‚ ²=(-1)pq ÃŽÂ ²Ãƒ ¢Ã‹â€ Ã‚ §ÃƒÅ½Ã‚ ±, is not in general commutative, but is associative, (ÃŽÂ ±Ãƒ ¢Ã‹â€ Ã‚ §ÃƒÅ½Ã‚ ²)à ¢Ã‹â€ Ã‚ §u= ÃŽÂ ±Ãƒ ¢Ã‹â€ Ã‚ §(ÃŽÂ ²Ãƒ ¢Ã‹â€ Ã‚ §u) and bilinear (c1 ÃŽÂ ±1+c2 ÃŽÂ ±2)à ¢Ã‹â€ Ã‚ § ÃŽÂ ²= c1( ÃŽÂ ±1à ¢Ã‹â€ Ã‚ § ÃŽÂ ²) + c2( ÃŽÂ ±2à ¢Ã‹â€ Ã‚ § ÃŽÂ ²) ÃŽÂ ±Ãƒ ¢Ã‹â€ Ã‚ §( c1 ÃŽÂ ²1+c2 ÃŽÂ ²2)= c1( ÃŽÂ ±Ãƒ ¢Ã‹â€ Ã‚ § ÃŽÂ ²1) + c2( ÃŽÂ ±Ãƒ ¢Ã‹â€ Ã‚ § ÃŽÂ ²2).  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚   (Becca 2014) Chapter 1 Definition1.1 Let (A, m, ÃŽÂ ·) be an algebra over k and write mop (ab) = ab à ªÃ¢â‚¬Å" ¯ a, bà Ã‚ µ A where mop=mà Ã¢â‚¬Å¾ÃƒÅ½Ã¢â‚¬Ëœ,Α. Thus ab=ba à ªÃ¢â‚¬Å" ¯a, b à Ã‚ µA. The (A, mop, ÃŽÂ ·) is the opposite algebra. Definition1.2 A co-algebra C is A vector space over K A map Ά: Cà ¢Ã¢â‚¬  Ã¢â‚¬â„¢C à ¢Ã…  -   C which is coassociative in the sense of à ¢Ã‹â€ Ã¢â‚¬Ëœ (c(1)(1) à ¢Ã…  -  Ãƒâ€šÃ‚   c(1)(2) à ¢Ã…  -   c(2))= à ¢Ã‹â€ Ã¢â‚¬Ëœ (c(1) à ¢Ã…  -  Ãƒâ€šÃ‚   c(2)(1) à ¢Ã…  -   c(2)c(2) )  Ãƒâ€šÃ‚   à ªÃ¢â‚¬Å" ¯ cà Ã‚ µC (Ά called the co-product) A map ÃŽÂ µ: Cà ¢Ã¢â‚¬  Ã¢â‚¬â„¢ k obeying à ¢Ã‹â€ Ã¢â‚¬Ëœ[ÃŽÂ µ((c(1))c(2))]=c= à ¢Ã‹â€ Ã¢â‚¬Ëœ[(c(1)) ÃŽÂ µc(2))] à ªÃ¢â‚¬Å" ¯ cà Ã‚ µC ( ÃŽÂ µ called the counit) Co-associativity and co-unit element can be expressed as commutative diagrams as follow: Figure 1: Co-associativity map Ά Figure 2: co-unit element map ÃŽÂ µ Definition1.3 A bi-algebra H is An algebra (H, m ,ÃŽÂ ·) A co-algebra (H, Ά, ÃŽÂ µ) Ά,ÃŽÂ µ are algebra maps, where Hà ¢Ã…  -   H has the tensor product algebra structure (hà ¢Ã…  - g)(hà ¢Ã…  -   g)= hhà ¢Ã…  -  Ãƒâ€šÃ‚   gg à ªÃ¢â‚¬Å" ¯h, h, g, g à Ã‚ µH. A representation of Hopf algebras as diagrams is the following: Definition1.4 A Hopf Algebra H is A bi-algebra H, Ά, ÃŽÂ µ, m, ÃŽÂ · A map S : Hà ¢Ã¢â‚¬  Ã¢â‚¬â„¢ H such that à ¢Ã‹â€ Ã¢â‚¬Ëœ [(Sh(1))h(2) ]= ÃŽÂ µ(h)= à ¢Ã‹â€ Ã¢â‚¬Ëœ [h(1)Sh(2) ]à ªÃ¢â‚¬Å" ¯ hà Ã‚ µH The axioms that make a simultaneous algebra and co-algebra into Hopf algebra is à Ã¢â‚¬Å¾:   Hà ¢Ã…  - Hà ¢Ã¢â‚¬  Ã¢â‚¬â„¢Hà ¢Ã…  -H Is the map à Ã¢â‚¬Å¾(hà ¢Ã…  -g)=gà ¢Ã…  -h called the flip map à ªÃ¢â‚¬Å" ¯ h, g à Ã‚ µ H. Definition1.5 Hopf Algebra is commutative if its commutative as algebra. It is co-commutative if its co-commutative as a co-algebra, à Ã¢â‚¬Å¾ÃƒÅ½Ã¢â‚¬ =Ά. It can be defined as S2=id. A commutative algebra over K is an algebra (A, m, ÃŽÂ ·) over k such that m=mop. Definition1.6 Two Hopf algebras H,H are dually paired by a map : H H à ¢Ã¢â‚¬  Ã¢â‚¬â„¢k if, =à Ã‹â€ ,Άh>, =ÃŽÂ µ(h) g   >=, ÃŽÂ µ(à Ã¢â‚¬  )= = à ªÃ¢â‚¬Å" ¯ à Ã¢â‚¬  , à Ã‹â€ Ãƒ Ã‚ µ H and h, g à Ã‚ µH. Let (C, Ά,ÃŽÂ µ) be a co-algebra over k. The co-algebra (C, Άcop, ÃŽÂ µ) is the opposite co-algebra. A co-commutative co-algebra over k is a co-algebra (C, Ά, ÃŽÂ µ) over k such that Ά= Άcop. Definition1.7 A bi-algebra or Hopf algebra H acts on algebra A (called H-module algebra) if: H acts on A as a vector space. The product map m: AAà ¢Ã¢â‚¬  Ã¢â‚¬â„¢A commutes with the action of H The unit map ÃŽÂ ·: kà ¢Ã¢â‚¬  Ã¢â‚¬â„¢ A commutes with the action of H. From b,c we come to the next action hà ¢Ã…  Ã‚ ³(ab)=à ¢Ã‹â€ Ã¢â‚¬Ëœ(h(1)à ¢Ã…  Ã‚ ³a)(h(2)à ¢Ã…  Ã‚ ³b), hà ¢Ã…  Ã‚ ³1= ÃŽÂ µ(h)1, à ªÃ¢â‚¬Å" ¯a, b à Ã‚ µ A, h à Ã‚ µ H This is the left action. Definition1.8 Let (A, m, ÃŽÂ ·) be algebra over k and is a left H- module along with a linear map m: Aà ¢Ã…  -Aà ¢Ã¢â‚¬  Ã¢â‚¬â„¢A and a scalar multiplication ÃŽÂ ·: k à ¢Ã…  - Aà ¢Ã¢â‚¬  Ã¢â‚¬â„¢A if the following diagrams commute. Figure 3: Left Module map Definition1.9 Co-algebra (C, Ά, ÃŽÂ µ) is H-module co-algebra if: C is an H-module Ά: Cà ¢Ã¢â‚¬  Ã¢â‚¬â„¢CC and ÃŽÂ µ: Cà ¢Ã¢â‚¬  Ã¢â‚¬â„¢ k commutes with the action of H. (Is a right C- co-module). Explicitly, Ά(hà ¢Ã…  Ã‚ ³c)=à ¢Ã‹â€ Ã¢â‚¬Ëœh(1)à ¢Ã…  Ã‚ ³c(1)à ¢Ã‚ ¨Ã¢â‚¬Å¡h(2)à ¢Ã…  Ã‚ ³c(2), ÃŽÂ µ(hà ¢Ã…  Ã‚ ³c)= ÃŽÂ µ(h)ÃŽÂ µ(c), à ªÃ¢â‚¬Å" ¯h à Ã‚ µ H, c à Ã‚ µ C.   Definition1.10 A co-action of a co-algebra C on a vector space V is a map ÃŽÂ ²: Và ¢Ã¢â‚¬  Ã¢â‚¬â„¢Cà ¢Ã‚ ¨Ã¢â‚¬Å¡V such that, (idà ¢Ã‚ ¨Ã¢â‚¬Å¡ÃƒÅ½Ã‚ ²) à ¢Ã‹â€ Ã‹Å"ÃŽÂ ²=(ΆÃƒ ¢Ã‚ ¨Ã¢â‚¬Å¡ id )ÃŽÂ ²;   id =(ÃŽÂ µÃƒ ¢Ã‚ ¨Ã¢â‚¬Å¡id )à ¢Ã‹â€ Ã‹Å"ÃŽÂ ². Definition1.11 A bi-algebra or Hopf algebra H co-acts on an algebra A (an H- co-module algebra) if: A is an H- co-module The co-action ÃŽÂ ²: Aà ¢Ã¢â‚¬  Ã¢â‚¬â„¢ Hà ¢Ã‚ ¨Ã¢â‚¬Å¡A is an algebra homomorphism, where Hà ¢Ã‚ ¨Ã¢â‚¬Å¡A has the tensor product algebra structure. Definition1.12 Let C be co- algebra (C, Ά, ÃŽÂ µ), map ÃŽÂ ²: Aà ¢Ã¢â‚¬  Ã¢â‚¬â„¢ Hà ¢Ã‚ ¨Ã¢â‚¬Å¡A is a right C- co- module if the following diagrams commute. Figure 6:Co-algebra of a right co-module Sub-algebras, left ideals and right ideals of algebra have dual counter-parts in co-algebras. Let (A, m, ÃŽÂ ·) be algebra over k and suppose that V is a left ideal of A. Then m(Aà ¢Ã‚ ¨Ã¢â‚¬Å¡V)à ¢Ã…  Ã¢â‚¬  V. Thus the restriction of m to Aà ¢Ã‚ ¨Ã¢â‚¬Å¡V determines a map Aà ¢Ã‚ ¨Ã¢â‚¬Å¡Và ¢Ã¢â‚¬  Ã¢â‚¬â„¢V. Left co-ideal of a co-algebra C is a subspace V of C such that the co-product Ά restricts to a map Và ¢Ã¢â‚¬  Ã¢â‚¬â„¢Cà ¢Ã‚ ¨Ã¢â‚¬Å¡V. Definition1.13 Let V be a subspace of a co-algebra C over k. Then V is a sub-co-algebra of C if Ά(V)à ¢Ã…  Ã¢â‚¬  Và ¢Ã‚ ¨Ã¢â‚¬Å¡V, for left co-ideal Ά(V)à ¢Ã…  Ã¢â‚¬  Cà ¢Ã‚ ¨Ã¢â‚¬Å¡V and for right co-ideal Ά(V)à ¢Ã…  Ã¢â‚¬  Và ¢Ã‚ ¨Ã¢â‚¬Å¡C. Definition1.14 Let V be a subspace of a co-algebra C over k. The unique minimal sub-co-algebra of C which contains V is the sub-co-algebra of C generated by V. Definition1.15 A simple co-algebra is a co-algebra which has two sub-co-algebras. Definition1.16 Let C be co-algebra over k. A group-like element of C is c à Ã‚ µC with satisfies, Ά(s)=sà ¢Ã‚ ¨Ã¢â‚¬Å¡s   and ÃŽÂ µ(s)=1 à ªÃ¢â‚¬Å" ¯ s à Ã‚ µS. The set of group-like elements of C is denoted G(C). Definition1.17 Let S be a set. The co-algebra k[S] has a co-algebra structure determined by Ά(s)=sà ¢Ã‚ ¨Ã¢â‚¬Å¡s   and ÃŽÂ µ(s)=1 à ªÃ¢â‚¬Å" ¯ s à Ã‚ µS. If S=à ¢Ã‹â€ Ã¢â‚¬ ¦ we set C=k[à ¢Ã‹â€ Ã¢â‚¬ ¦]=0. Is the group-like co-algebra of S over k. Definition1.18 The co-algebra C over k with basis {co, c1, c2,à ¢Ã¢â€š ¬Ã‚ ¦..} whose co-product and co-unit is satisfy by Ά(cn)= à ¢Ã‹â€ Ã¢â‚¬Ëœcn-là ¢Ã‚ ¨Ã¢â‚¬Å¡cl and ÃŽÂ µ(cn)=ÃŽÂ ´n,0 for l=1,à ¢Ã¢â€š ¬Ã‚ ¦.,n and for all nà ¢Ã¢â‚¬ °Ã‚ ¥0. Is denoted by Pà ¢Ã‹â€ Ã… ¾(k). The sub-co-algebra which is the span of co, c1, c2,à ¢Ã¢â€š ¬Ã‚ ¦,cn is denoted Pn(k). Definition1.19 A co-matrix co-algebra over k is a co-algebra over k isomorphic to Cs(k) for some finite set S. The co-matrix identities are: Ά(ei, j)= à ¢Ã‹â€ Ã¢â‚¬Ëœei, là ¢Ã‚ ¨Ã¢â‚¬Å¡el, j ÃŽÂ µ(ei, j)=ÃŽÂ ´i, j à ¢Ã‹â€ Ã¢â€š ¬ i, j à Ã‚ µS. Set Cà ¢Ã‹â€ Ã¢â‚¬ ¦(k)=(0). Definition1.20 Let S be a non-empty finite set. A standard basis for Cs(k) is a basis {c i ,j}I, j à Ã‚ µS for Cs(k) which satisfies the co-matrix identities. Definition1.21 Let (C, Άc, ÃŽÂ µc) and (D, ΆD, ÃŽÂ µD) be co-algebras over the field k. A co-algebra map f: Cà ¢Ã¢â‚¬  Ã¢â‚¬â„¢D is a linear map of underlying vector spaces such that ΆDà ¢Ã‹â€ Ã‹Å"f=(fà ¢Ã‚ ¨Ã¢â‚¬Å¡f)à ¢Ã‹â€ Ã‹Å" Άc and ÃŽÂ µDà ¢Ã‹â€ Ã‹Å"f= ÃŽÂ µc. An isomorphism of co-algebras is a co-algebra map which is a linear isomorphism. Definition1.22 Let C be co-algebra over the field k. A co-ideal of C is a subspace I of C such that ÃŽÂ µ (I) = (0) and Ά (ÃŽâ„ ¢) à ¢Ã…  Ã¢â‚¬   Ià ¢Ã‚ ¨Ã¢â‚¬Å¡C+Cà ¢Ã‚ ¨Ã¢â‚¬Å¡I. Definition1.23 The co-ideal Ker (ÃŽÂ µ) of a co-algebra C over k is denoted by C+. Definition1.24 Let I be a co-ideal of co-algebra C over k. The unique co-algebra structure on C /I such that the projection à Ã¢â€š ¬: Cà ¢Ã¢â‚¬  Ã¢â‚¬â„¢ C/I is a co-algebra map, is the quotient co-algebra structure on C/I. Definition1.25 The tensor product of co-algebra has a natural co-algebra structure as the tensor product of vector space Cà ¢Ã…  -D is a co-algebra over k where Ά(c(1)à ¢Ã‚ ¨Ã¢â‚¬Å¡d(1))à ¢Ã‚ ¨Ã¢â‚¬Å¡( c(2)à ¢Ã‚ ¨Ã¢â‚¬Å¡d(2)) and ÃŽÂ µ(cà ¢Ã‚ ¨Ã¢â‚¬Å¡d)=ÃŽÂ µ(c)ÃŽÂ µ(d) à ¢Ã‹â€ Ã¢â€š ¬ c in C and d in D. Definition1.26 Let C be co-algebra over k. A skew-primitive element of C is a cà Ã‚ µC which satisfies Ά(c)= gà ¢Ã‚ ¨Ã¢â‚¬Å¡c +cà ¢Ã‚ ¨Ã¢â‚¬Å¡h, where c, h à Ã‚ µG(c). The set of g:h-skew primitive elements of C is denoted   by Pg,h (C). Definition1.27 Let C be co-algebra over a field k. A co-commutative element of C is cà Ã‚ µC such that Ά(c) = Άcop(c). The set of co-commutative elements of C is denoted by Cc(C). Cc(C) à ¢Ã…  Ã¢â‚¬  C. Definition1.28 The category whose objects are co-algebras over k and whose morphisms are co-algebra maps under function composition is denoted by k-Coalg. Definition1.29 The category whose objects are algebras over k and whose morphisms are co-algebra maps under function composition is denoted by k-Alg. Definition1.30 Let (C, Ά, ÃŽÂ µ) be co-algebra over k. The algebra (Cà ¢Ã‹â€ -, m, ÃŽÂ ·) where m= ΆÃƒ ¢Ã‹â€ -| Cà ¢Ã‹â€ -à ¢Ã‚ ¨Ã¢â‚¬Å¡Cà ¢Ã‹â€ -, ÃŽÂ · (1) =ÃŽÂ µ, is the dual algebra of (C, Ά, ÃŽÂ µ). Definition1.31 Let A be algebra over the field k. A locally finite A-module is an A-module M whose finitely generated sub-modules are finite-dimensional. The left and right Cà ¢Ã‹â€ --module actions on C are locally finite. Definition1.32 Let A be algebra over the field k. A derivation of A is a linear endomorphism F of A such that F (ab) =F (a) b-aF(b) for all a, b à Ã‚ µA. For fixed b à Ã‚ µA note that F: Aà ¢Ã¢â‚¬  Ã¢â‚¬â„¢A defined by F(a)=[a, b]= ab- ba   for all a à Ã‚ µA is a derivation of A. Definition1.33 Let C be co-algebra over the field k. A co-derivation of C is a linear endomorphism f of C such that ΆÃƒ ¢Ã‹â€ Ã‹Å"f= (fà ¢Ã‚ ¨Ã¢â‚¬Å¡IC + IC à ¢Ã‚ ¨Ã¢â‚¬Å¡f) à ¢Ã‹â€ Ã‹Å"Ά. Definition1.34 Let A and B ne algebra over the field k. The tensor product algebra structure on Aà ¢Ã‚ ¨Ã¢â‚¬Å¡B is determined by (aà ¢Ã‚ ¨Ã¢â‚¬Å¡b)(aà ¢Ã‚ ¨Ã¢â‚¬Å¡b)= aaà ¢Ã‚ ¨Ã¢â‚¬Å¡bb à ªÃ¢â‚¬Å" ¯ a, aà Ã‚ µA and b, bà Ã‚ µB. Definition1.35 Let X, Y be non-empty subsets of an algebra A over the field k. The centralizer of Y in X is ZX(Y) = {xà Ã‚ µX|yx=xy à ªÃ¢â‚¬Å" ¯yà Ã‚ µY} For y à Ã‚ µA the centralizer of y in X is ZX(y) = ZX({y}). Definition1.36 The centre of an algebra A over the field Z (A) = ZA(A). Definition1.37 Let (S, à ¢Ã¢â‚¬ °Ã‚ ¤) be a partially ordered set which is locally finite, meaning that à ªÃ¢â‚¬Å" ¯, I, jà Ã‚ µS which satisfy ià ¢Ã¢â‚¬ °Ã‚ ¤j the interval [i, j] = {là Ã‚ µS|ià ¢Ã¢â‚¬ °Ã‚ ¤là ¢Ã¢â‚¬ °Ã‚ ¤j} is a finite set. Let S= {[i, j] |I, jà Ã‚ µS, ià ¢Ã¢â‚¬ °Ã‚ ¤j} and let A be the algebra which is the vector space of functions f: Sà ¢Ã¢â‚¬  Ã¢â‚¬â„¢k under point wise operations whose product is given by (fà ¢Ã¢â‚¬ ¹Ã¢â‚¬  g)([i, j])=f([i, l])g([l, j])   ià ¢Ã¢â‚¬ °Ã‚ ¤là ¢Ã¢â‚¬ °Ã‚ ¤j For all f, g à Ã‚ µA and [i, j]à Ã‚ µS and whose unit is given by 1([I,j])= ÃŽÂ ´i,j à ªÃ¢â‚¬Å" ¯[I,j]à Ã‚ µS. Definition1.38 The algebra of A over the k described above is the incidence algebra of the locally finite partially ordered set (S, à ¢Ã¢â‚¬ °Ã‚ ¤). Definition1.39 Lie co-algebra over k is a pair (C, ÃŽÂ ´), where C is a vector space over k and ÃŽÂ ´: Cà ¢Ã¢â‚¬  Ã¢â‚¬â„¢Cà ¢Ã‚ ¨Ã¢â‚¬Å¡C is a linear map, which satisfies: à Ã¢â‚¬Å¾Ãƒ ¢Ã‹â€ Ã‹Å"ÃŽÂ ´=0 and (ÃŽâ„ ¢+(à Ã¢â‚¬Å¾Ãƒ ¢Ã‚ ¨Ã¢â‚¬Å¡ÃƒÅ½Ã¢â€ž ¢)à ¢Ã‹â€ Ã‹Å"(ÃŽâ„ ¢Ãƒ ¢Ã‚ ¨Ã¢â‚¬Å¡Ãƒ Ã¢â‚¬Å¾)+(ÃŽâ„ ¢Ãƒ ¢Ã‚ ¨Ã¢â‚¬Å¡Ãƒ Ã¢â‚¬Å¾)à ¢Ã‹â€ Ã‹Å" (à Ã¢â‚¬Å¾Ãƒ ¢Ã‚ ¨Ã¢â‚¬Å¡ÃƒÅ½Ã¢â€ž ¢))à ¢Ã‹â€ Ã‹Å"(ÃŽâ„ ¢Ãƒ ¢Ã‚ ¨Ã¢â‚¬Å¡ÃƒÅ½Ã‚ ´)à ¢Ã‹â€ Ã‹Å"ÃŽÂ ´=0 à Ã¢â‚¬Å¾=à Ã¢â‚¬Å¾C,C and I is the appropriate identity map. Definition1.40 Suppose that C is co-algebra over the field k. The wedge product of subspaces U and V is Uà ¢Ã‹â€ Ã‚ §V = Ά-1(Uà ¢Ã‚ ¨Ã¢â‚¬Å¡C+ Cà ¢Ã‚ ¨Ã¢â‚¬Å¡V). Definition1.41 Let C be co-algebra over the field k. A saturated sub-co-algebra of C is a sub-co-algebra D of C such that Uà ¢Ã‹â€ Ã‚ §Và ¢Ã…  Ã¢â‚¬  D, à ªÃ¢â‚¬Å" ¯ U, V of D. Definition1.42 Let C be co-algebra over k and (N, à Ã‚ ) be a left co-module. Then Uà ¢Ã‹â€ Ã‚ §X= à Ã‚ -1(Uà ¢Ã‚ ¨Ã¢â‚¬Å¡N+ Cà ¢Ã‚ ¨Ã¢â‚¬Å¡X) is the wedge product of subspaces U of C and X of N. Definition1.43 Let C be co-algebra over k and U be a subspace of C. The unique minimal saturated sub-co-algebra of C containing U is the saturated closure of U in C. Definition1.44 Let (A, m, ÃŽÂ ·) be algebra over k. Then, Aà ¢Ã‹â€ Ã‹Å"=mà ¢Ã‹â€ 1(Aà ¢Ã‹â€ -à ¢Ã‚ ¨Ã¢â‚¬Å¡Aà ¢Ã‹â€ - ) (Aà ¢Ã‹â€ Ã‹Å", Ά, ÃŽÂ µ) is a co-algebra over k, where Ά= mà ¢Ã‹â€ -| Aà ¢Ã‹â€ Ã‹Å" and ÃŽÂ µ=ÃŽÂ ·Ãƒ ¢Ã‹â€ -. ÃŽÂ ¤he co-algebra (Aà ¢Ã‹â€ Ã‹Å", Ά, ÃŽÂ µ) is the dual co-algebra of (A, m, ÃŽÂ ·). Also we denote Aà ¢Ã‹â€ Ã‹Å" by aà ¢Ã‹â€ Ã‹Å" and ΆÃƒ ¢Ã‹â€ Ã‹Å"= aà ¢Ã‹â€ Ã‹Å"(1)à ¢Ã‚ ¨Ã¢â‚¬Å¡ aà ¢Ã‹â€ Ã‹Å"(2), à ªÃ¢â‚¬Å" ¯ aà ¢Ã‹â€ Ã‹Å" à Ã‚ µ Aà ¢Ã‹â€ Ã‹Å". Definition1.45 Let A be algebra over k. An ÃŽÂ ·:ÃŽÂ ¾- derivation of A is a linear map f: Aà ¢Ã¢â‚¬  Ã¢â‚¬â„¢k which satisfies f(ab)= ÃŽÂ ·(a)f(b)+f(a) ÃŽÂ ¾(b), à ªÃ¢â‚¬Å" ¯ a, bà Ã‚ µ A and ÃŽÂ ·, ÃŽÂ ¾ à Ã‚ µ Alg(A, k). Definition1.46 The full subcategory of k-Alg (respectively of k-Co-alg) whose objects are finite dimensional algebras (respectively co-algebras) over k is denoted k-Alg fd (respectively  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚   k-Co-alg fd). Definition1.47 A proper algebra over k is an algebra over k such that the intersection of the co-finite ideals of A is (0), or equivalently the algebra map jA:Aà ¢Ã¢â‚¬  Ã¢â‚¬â„¢(Aà ¢Ã‹â€ Ã‹Å")*, be linear map defined by jA(a)(aà ¢Ã‹â€ Ã‹Å")=aà ¢Ã‹â€ Ã‹Å"(a), a à Ã‚ µA and aà ¢Ã‹â€ Ã‹Å"à Ã‚ µAà ¢Ã‹â€ Ã‹Å". Then: jA:Aà ¢Ã¢â‚¬  Ã¢â‚¬â„¢(Aà ¢Ã‹â€ Ã‹Å")* is an algebra map Ker(jA) is the intersection of the co-finite ideals of A Im(jA) is a dense subspace of (Aà ¢Ã‹â€ Ã‹Å")*. Is one-to-one. Definition1.48 Let A (respectively C) be an algebra (respectively co-algebra ) over k. Then A (respectively C) is reflexive if jA:Aà ¢Ã¢â‚¬  Ã¢â‚¬â„¢(Aà ¢Ã‹â€ Ã‹Å")*, as defined before and jC:Cà ¢Ã¢â‚¬  Ã¢â‚¬â„¢(C*)à ¢Ã‹â€ Ã‹Å", defined as: jC(c)(c*)=c*(c), à ªÃ¢â‚¬Å" ¯ c*à Ã‚ µC* and cà Ã‚ µC. Then: Im(jC)à ¢Ã…  Ã¢â‚¬  (C*)à ¢Ã‹â€ Ã‹Å" and jC:Cà ¢Ã¢â‚¬  Ã¢â‚¬â„¢(C*)à ¢Ã‹â€ Ã‹Å" is a co-algebra map. jC is one-to-one. Im(jC) is the set of all aà Ã‚ µ(C*)* which vanish on a closed co-finite ideal of C*. Is an isomorphism. Definition1.49 Almost left noetherian algebra over k is an algebra over k whose co-finite left ideal are finitely generated. (M is called almost noetherian if every co-finite submodule of M is finitely generated). Definition1.50 Let f:Uà ¢Ã¢â‚¬  Ã¢â‚¬â„¢V be a map of vector spaces over k. Then f is an almost one-to-one linear map if ker(f) is finite-dimensional, f is an almost onto linear map if Im(f) is co-finite subspace of V and f is an almost isomorphism if f is an almost one-to-one and an almost linear map. Definition1.51 Let A be algebra over k and C be co-algebra over k. A pairing of A and C is a bilinear map   ÃƒÅ½Ã‚ ²: AÃÆ'-Cà ¢Ã¢â‚¬  Ã¢â‚¬â„¢k which satisfies, ÃŽÂ ²(ab,c)= ÃŽÂ ² (a, c(1))ÃŽÂ ² (b, c(2)) and ÃŽÂ ²(1, c) = ÃŽÂ µ(c), à ªÃ¢â‚¬Å" ¯ a, b à Ã‚ µ A and  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚   c à Ã‚ µC. Definition1.52 Let V be a vector space over k. A co-free co-algebra on V is a pair (à Ã¢â€š ¬, Tco(V)) such that: Tco(V) is a co-algebra over k and à Ã¢â€š ¬: Tco(V)à ¢Ã¢â‚¬  Ã¢â‚¬â„¢T is a linear map. If C is a co-algebra over k and f:Cà ¢Ã¢â‚¬  Ã¢â‚¬â„¢V is a linear map,à ¢Ã‹â€ Ã†â€™ a co-algebra map F: Cà ¢Ã¢â‚¬  Ã¢â‚¬â„¢ Tco(V) determined by à Ã¢â€š ¬Ãƒ ¢Ã‹â€ Ã‹Å"F=f. Definition1.53 Let V be a vector space over k. A co-free co-commutative co-algebra on V is any pair (à Ã¢â€š ¬, C(V)) which satisfies: C(V) is a co-commutative co-algebra over k and à Ã¢â€š ¬:C(V)à ¢Ã¢â‚¬  Ã¢â‚¬â„¢V is a linear map. If C is a co-commutative co-algebra over k and f: Cà ¢Ã¢â‚¬  Ã¢â‚¬â„¢V is linear map, à ¢Ã‹â€ Ã†â€™ co-algebra map F:C à ¢Ã¢â‚¬  Ã¢â‚¬â„¢C(V) determined by à Ã¢â€š ¬Ãƒ ¢Ã‹â€ Ã‹Å"F=f.   Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚   (Majid 2002, Radford David E) Chapter 2 Proposition (Anti-homomorphism property of antipodes) 2.1 The antipode of a Hopf algebra is unique and obey S(hg)=S(g)S(h), S(1)=1 and (Sà ¢Ã‚ ¨Ã¢â‚¬Å¡S)à ¢Ã‹â€ Ã‹Å"Άh=à Ã¢â‚¬Å¾Ãƒ ¢Ã‹â€ Ã‹Å"ΆÃƒ ¢Ã‹â€ Ã‹Å"Sh, ÃŽÂ µSh=ÃŽÂ µh, à ¢Ã‹â€ Ã¢â€š ¬h,g à ¢Ã‹â€ Ã‹â€  H.   Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚   (Majid 2002, Radford David E) Proof Let S and S1 be two antipodes for H. Then using properties of antipode, associativity of à Ã¢â‚¬Å¾ and co-associativity of Ά we get S= à Ã¢â‚¬Å¾Ãƒ ¢Ã‹â€ Ã‹Å"(Sà ¢Ã…  -[ à Ã¢â‚¬Å¾Ãƒ ¢Ã‹â€ Ã‹Å"(Idà ¢Ã…  -S1)à ¢Ã‹â€ Ã‹Å"Ά])à ¢Ã‹â€ Ã‹Å"Ά= à Ã¢â‚¬Å¾Ãƒ ¢Ã‹â€ Ã‹Å"(Idà ¢Ã…  - à Ã¢â‚¬Å¾)à ¢Ã‹â€ Ã‹Å"(Sà ¢Ã‚ ¨Ã¢â‚¬Å¡Idà ¢Ã…  -S1)à ¢Ã‹â€ Ã‹Å"(Id à ¢Ã…  -Ά)à ¢Ã‹â€ Ã‹Å"Ά= à Ã¢â‚¬Å¾Ãƒ ¢Ã‹â€ Ã‹Å"(à Ã¢â‚¬Å¾Ãƒ ¢Ã‚ ¨Ã¢â‚¬Å¡Id)à ¢Ã‹â€ Ã‹Å"(Sà ¢Ã‚ ¨Ã¢â‚¬Å¡Idà ¢Ã…  -S1)à ¢Ã‹â€ Ã‹Å"(Ά à ¢Ã…  -Id)à ¢Ã‹â€ Ã‹Å"Ά = à Ã¢â‚¬Å¾Ãƒ ¢Ã‹â€ Ã‹Å"( [à Ã¢â‚¬Å¾Ãƒ ¢Ã‹â€ Ã‹Å"(Sà ¢Ã‚ ¨Ã¢â‚¬Å¡Id)à ¢Ã‹â€ Ã‹Å"Ά]à ¢Ã‚ ¨Ã¢â‚¬Å¡S1)à ¢Ã‹â€ Ã‹Å" Ά=S1. So the antipode is unique. Let Sà ¢Ã‹â€ -id=ÃŽÂ µs idà ¢Ã‹â€ -S=ÃŽÂ µt To check that S is an algebra anti-homomorphism, we compute S(1)= S(1(1))1(2)S(1(3))= S(1(1)) ÃŽÂ µt (1(2))= ÃŽÂ µs(1)=1, S(hg)=S(h(1)g(1)) ÃŽÂ µt(h(2)g(2))= S(h(1)g(1))h(2) ÃŽÂ µt(g(2))S(h(3))=ÃŽÂ µs (h(1)g(1))S(g(2))S(h(2))= S(g(1)) ÃŽÂ µs(h(1)) ÃŽÂ µt (g(2))S(h(2))=S(g)S(h), à ¢Ã‹â€ Ã¢â€š ¬h,g à ¢Ã‹â€ Ã‹â€ H and we used ÃŽÂ µt(hg)= ÃŽÂ µt(h ÃŽÂ µt(g)) and ÃŽÂ µs(hg)= ÃŽÂ µt(ÃŽÂ µs(h)g). Dualizing the above we can show that S is also a co-algebra anti-homomorphism: ÃŽÂ µ(S(h))= ÃŽÂ µ(S(h(1) ÃŽÂ µt(h(2)))= ÃŽÂ µ(S(h(1)h(2))= ÃŽÂ µ(ÃŽÂ µt(h))= ÃŽÂ µ(h), Ά(S(h))= Ά(S(h(1) ÃŽÂ µt(h(2)))= Ά(S(h(1) ÃŽÂ µt(h(2))à ¢Ã‚ ¨Ã¢â‚¬Å¡1)= Ά(S(h(1) ))(h(2)S(h(4))à ¢Ã‚ ¨Ã¢â‚¬Å¡ ÃŽÂ µt (h(3))= Ά(ÃŽÂ µs(h(1))(S(h(3))à ¢Ã‚ ¨Ã¢â‚¬Å¡S(h(2)))=S(h(3))à ¢Ã‚ ¨Ã¢â‚¬Å¡ ÃŽÂ µs(h(1))S(h(2))=S(h(2))à ¢Ã‚ ¨Ã¢â‚¬Å¡ S(h(1)). (New directions) Example2.2 The Hopf Algebra H=Uq(b+) is generated by 1 and the elements X,g,g-1 with relations gg-1=1=g-1g and g X=q X g, where q   is a fixed invertible element of the field k. Here ΆX= Xà ¢Ã‚ ¨Ã¢â‚¬Å¡1 +g à ¢Ã‚ ¨Ã¢â‚¬Å¡ X, Άg=g à ¢Ã‚ ¨Ã¢â‚¬Å¡ g, Άg-1=g-1à ¢Ã‚ ¨Ã¢â‚¬Å¡g-1, ÃŽÂ µX=0, ÃŽÂ µg=1=ÃŽÂ µ g-1, SX=- g-1X, Sg= g-1, S g-1=g. S2X=q-1X. Proof We have Ά, ÃŽÂ µ on the generators and extended them multiplicatively to products of the generators. ΆgX=(Άg)( ΆX)=( gà ¢Ã‚ ¨Ã¢â‚¬Å¡g)( Xà ¢Ã‚ ¨Ã¢â‚¬Å¡1 +gà ¢