Conflict Theorists

Conflict theory was developed from the concern that the structural functionalism theory neglected conflict in society and was politically conservative. This conflict theory also addressed the perceived failure of structural functionalism to account for change in society (Ritzer, 1992, p. 61). This theory has evolved to include elements of structural functionalism and traditional Marxist focus on dominant and subordinate groups.Conflict theory often depicts a: polarization of the forces of â€Å"law and order† on the one hand and left wing political activists and minority group members reacting to what they saw as excessive police repression of political protests and urban riots on the other (Giffen, et al. , 1991, pp. 8-9) This aspect of conflict theory assumes, however, that the dominant and subordinate groups are more or less homogenous in nature.Most research in the field of drug policy recently, however, deals with power being located in â€Å"institutional structures in society such as economic, governmental and religious institutions (Giffen, et al. 1991, p. 10)† that do not presuppose homogenous groups. An example of this would be the comment of Riley after attending a conference on drug issues in the United States, where he remarked that â€Å"many researchers felt the real reason for the war on drugs in that country was that it helped to suppress blacks and minorities. (Riley 1994b)† One of the failings of conflict theory becomes apparent when researchers in the history of this legislation find little in the actual discussion of the laws that pertains to race.Giffen, et al. (1991) write that the early legislation's principle proponents had the â€Å"altruistic aims of supporting the international anti-opium movement† despite the anti-Chinese sentiment of the times (p. 525). The fact that the laws were used solely against the Chinese at first is indicative of this anti-Chinese sentiment, and not the creation of the laws thems elves. Later legislation was driven mainly by enforcement officials, as there was little in the way of public outcry for more rigorous anti-opium legislation (p. 525).Johns (1991) under the heading â€Å"Race: The Creation of an Enemy Class,† writes bluntly: â€Å"The enforcement tactics of the War on Drugs are focused on minority populations† (p. 155). In her paper, Johns (1991) posits that the War on Drugs takes attention away from the factors which underlie the problems of drugs and trafficking, partly because the â€Å"more powerful segments in society† (p. 150) do not want attention focused the poor job they are doing to cure the ills of society. Johns also expands the group being oppressed to include the poor, who have been hit with massive housing and health care cuts under the Republican Presidencies.The dichotomy between those in power and minorities and the poor is self-perpetuating, in that these groups have a limited upward mobility (and, therefore c rimes like trafficking in illicit drugs becomes appealing), and when they do try to increase their wealth through illicit means, those in power see that as justification for minorities and the poor being in the position they are in. The conflict theory is problematic in describing why there is a war on drugs. It may help to explain (as Johns (1991) successfully does) why a War on Drugs continues in the U. S. , but leaves unanswered questions when applied to other situations.

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The Physics Behind Aircraft Wing Design Essay Example | Topics and Well Written Essays - 1500 words

The Physics Behind Aircraft Wing Design - Essay Example Birds alter the form of their wings to revolve and maneuver. They supposed that they could utilize this methodology to get spinning control by bending or alteration of shape, of a part of the wing. So, the brothers designed numerous gliders (Reals, 2012). These gliders were soared as unmanned, as well as piloted. They referred to the discoveries of Cayley and Langley, and the suspended gliding flights of Lilienthal. They matched with Chanute with regard to some of their ideologies. They distinguished that management of the flying airplanes would be the most vital and intricate quandaries to resolve. Following these triumphant glider examinations, the Wrights designed and scrutinized their test location due to its windy weather, sand, hilly topography and remote position. The Wrights victoriously analyzed their novel fifty pound duo plane glider in 1900. It had a seventeen feet wing width and wing arching machinery. The construction of gliders was an upshot of a gradual technique of d iscerning aerodynamics and then managing the flights, building and scrutinizing numerous glider designs. These gliders operated but not to the anticipations of the Wrights based on the researches and dissertations of their forerunners. Their foremost glider initiated in 1900 only has approximately half the expected lift. Moreover, their subsequent glider performed worse. However, they continued their endeavor and built a wind tunnel, as well as numerous, complex apparatus to calculate lift and drag on the two hundred wing plan they examined. As a consequence, they resolved their previous quandaries in computations concerning drag as well as lift. Their examination and computations generated another glider with an enormous aspect proportions and accurate three axes management. They flew it victoriously in numerous instances thorough the structure of designing, wind channel scrutinized of airfoils and flight examination of whole size models. They resolved the quandaries of power and m anagement of an airplane. These resolutions were through invention of wing arching for spin management; together with concurrent yaw management with a steerable back controls (Crouch, 2008). History and Improvements of Aircrafts Wing Designs and the Physics of its Operation A fluid passing through the facade of a body applies surface energy on that surface. Lift is the constituent of this force that is vertical to the approaching flow course. It differs from drag force, which is the constituent of the facade force corresponding to the flow course. If the substance is air, the energy is the aerodynamic energy. There is an enormous interconnection if lift and the wing of a fixed wing airplane, though lift also emanates from propeller. When an airplane is soaring straightforwardly and level, most of the lift contests gravity. Nonetheless, when an airplane is ascending or descending in a twirl, the lift tilts with regard to the vertical. Lift might also be totally downwards in some aero batic movements, or the wing on racing vehicle. Streamline form of enables aircrafts to produce considerable lift as contrasted to drag (Robert, 2012). The underpinning principles of lift in planes emanate from the concepts of physics. Firstly, there are Newton’s principles of movements, especially the second principle which interrelate the force on a component of air to its velocity of momentum alteration. Secondly, there is the preservation of mass, as

Financial resources and decisions management Essay

Financial resources and decisions management - Essay Example equity and debt, comes with their advantages and disadvantages. Several factors, such as statutory rules and requirements, terms and conditions imposed by the counter party and general economic conditions are analyzed before selecting one of the options. The downside of acquiring financing through issuance of equity is that the procedure is quite complicated as compared to acquiring funds by approaching any bank. In most cases, a loan is acquired from any bank or financial institution by filing an application for the sanctioning of the loan. The bank or any other financial institution, after evaluating the necessary details such as credit history, financial outlook for assessing the ability of the entity to repay the loans in future, and the purpose of the project for which the loan application was filed, sanctions the loan. Whereas in the case of raising finances through issuance of equity shares, the company has to fulfill several requirements such as issuing a predefined number of shares, issuing shares to the existing shareholder in proportion to their existing shares and appointing a financial advisor for conducting a due diligence of the entity’s operations. ... In contrast, in equity financing, the company has to wait for a considerable longer period of time for the funds to become available for their utilization. 1.2 The two modes of finance available to the company would be raising funds through issuance of equity or acquiring loan in the form of a mixture of a long term and short term debt. Let us assume that the total requirement of funding for Quality windows Ltd is for ? 100,000. As provided in the scenario, 40% of the funding requirement can be met through internally generated funds, whereas for the remaining 60% the company has to decide about the mode of funding. Thus the amount of fund required to issue is ? 60,000. Option 1: Raising the fund through the issuance of shares The company decides to issue 6,000 shares at ? 12 (par value is ? 10 and premium is ? 2). As per the current market knowledge, the issuance cost per share is ? 1. Other administrative cost pertaining to the issuance of share is ? 5,000 in total which relates to publishing prospectus and appointing an under-writing agent. Thus the total cash inflows to the company for the first financial year would be as under: Particulars Amount in ? Shares issued 72,000 Issuance cost (6,000) Other costs (5,000) Total inflow 61,000 Option 1: Acquiring loan from a financial institution The company decides to acquire loan from a financial institution amounting to ? 70,000. The principal repayment will start two years from the end of the current financial year. In return, the financial institution will charge interest rate at the rate of 12%. Thus, following is the net cash inflow at the end of the financial year: Particulars Amount in ? Loan acquired 70,000 Interest cost (8,400) Total inflow 61,600 Thus it is apparent from the above analysis, that acquiring

In what ways would understanding the causes of crime aid offenders in Research Paper

In what ways would understanding the causes of crime aid offenders in the restorative-justice process - Research Paper Example includes: priority should be given to victim needs either through financial, material, emotional or social means; to avoid re-offending by re-integrating the offenders into the community; to make offenders accept their responsibility for their actions; and rehabilitation of offenders and victims by recreating the working community in order to avoid crime and to avoid the long process of the justice system and the related costs and delays. (Marshall, 1991). The Restorative Justice process is based upon the following assumptions: the root cause of a crime could be social conditions, the responsibility has to be shared by the local government and communities because they are also responsible for such social conditions, and in order to rectify it they have to accept some responsibility; collaboration of all parties in order to solve the problems present and to achieve resolution, along with including the collaboration among victims, offenders and community are essential factors that lead toward effectiveness and efficiency; The conditions of the legal outcome should be flexible enough in order to take proactive actions in any case; and lastly, justice is meant to represent balance between both the parties and no single justification will be allowed to dominate other. (Marshall, 1991). The process of victim-offender mediation provides the victims and offenders a safe and secure place with the purpose of holding the offender accountable for their misconduct. It is a platform provided to the victims where they can, with the help of a mediator, let the offender know how much they have been affected by the crime, to receive compensation and to get some answers to their questions. Victims can be a part of the restitution plan and may claim their losses directly. Therefore, the offender has to accept their responsibility for their behavior and assist within the process to develop a plan that provides compensation for the victim. It is through this stage that the

Police Recorded Crime and British Crime Survey Essay

Police Recorded Crime and British Crime Survey - Essay Example Different types of graphs that can be created to view tendencies within the data are pie charts, bar charts and scatter plots. Governmental units utilize graphs a lot to inform the general public information. The police department is a governmental unit that uses statistical graphical applications a lot. This paper analyzes two graphs created by that illustrate the amount of crime that occurred in England and Wales last year to determine the discrepancies in the reported crime between the two graphical illustrations. The two graphs studied in this paper reflected the amount of crime and the type of crimes committed in the England and Wales regions. The two sources that reported the crime and created the two graphical illustrations are the police department and the British Crime Survey. Any observer looking at these two graphs would immediately notice that they portrayed two completely different perceptions of the crimes committed in the area. There are multiple reasons why these two graphs show different perceptions of crimes committed in this area. The first reason for the discrepancy is that the two graphs divided crime in totally different categories. The British Crime Survey divides crime in six categories while the police department divides crimes in eleven different categories. There are only three categories in the British Crime Surveys that match the police crimes categories which are burglary, vehicle theft and other theft. The British Crime Survey does not report crimes such as murders, drug procession, commercial crimes or crimes against children. The British Crime Survey data is a yearly survey performed by this organization among the population in the England Wales Region. The survey is realized utilizing a large pool of participants. The survey is sent to approximately 47,000 households and the response rate of the participants is about 75%. The time period of the data collection of the

Macroeconomics. The oils price Essay Example | Topics and Well Written Essays - 2500 words

Macroeconomics. The oils price - Essay Example However, how the various economic indicators behave during this short period of 'supply shock' and how they forecast performance or health of the economy in the coming period is the moot question. Inflation may be defined as "state of economy, where there is a general and abnormal rise in price of all goods and services". Recession is a state of economy where there is a "slump in Gross Domestic Product in two or three successive quarters of a year with general price rise or fall". In the short run, when a price of a product which is consumed every sector of the economy which contribute to GDP have suddenly risen, other things remain the same, lead to rising prices all commodities and services, fall in real value of money and slow down of economic growth. This phenomenon is attributed to 'supply shock'. Built-in inflation - induced by adaptive expectations, often linked to the "price/wage spiral" because it involves workers trying to keep their wages up with prices and then employers passing higher costs on to consumers as higher prices as part of a "vicious circle". Built-in inflation reflects events in the past, and so might be seen as hangover inflation. It is also known as "inertial" inflation, "inflationary momentum", and even "structural inflation. Cost Push inflation or Supply... Built-in inflation - induced by adaptive expectations, often linked to the "price/wage spiral" because it involves workers trying to keep their wages up with prices and then employers passing higher costs on to consumers as higher prices as part of a "vicious circle". Built-in inflation reflects events in the past, and so might be seen as hangover inflation. It is also known as "inertial" inflation, "inflationary momentum", and even "structural inflation. SUPPLY SHOCK INFLATION OR COST PUSH INFLATION: Cost Push inflation or Supply Shock inflation is caused by the rise in price of an important commodity for which there was no alternative, and consequent of which there was a general rise in price of all commodities and services. While the examples for cost push inflation are many viz., failure of monsoon/draught in an agrobased economy which would shoot up inflation etc.,. the best example in the modern industrialised countries, is rise in prices of petroleum prodoucts. Dependence to petroleum products in any economy need not be emphasised and it may not be forgotten that the crisis faced by the world in the year 1970 is attributed to the rise in oil prices all over the world. Since, petroluem is important for moving the economy in all industrial including agricultural dependent countries, any upward movement in the price will cause a cascading movement in the price of all commodities and services and it will have persistant effect. However, there are different school of thought which opine, that the reduction in oil price after 1970 have not contributed in reduction in general price level, hence, rise in oil prices have not directly caused inflation in 1970. However, Keynesian economists argue that many prices are 'sticky

Legal requirements and potential issues associated with HIV positive Essay

Legal requirements and potential issues associated with HIV positive workers - Essay Example Thus under this law, persons with HIV or AIDS are safeguarded from discrimination on the basis of their condition (U.S department of Health & Human Services Office for Civil Rights, 2014). 2. Relevance and importance: Well supported explanation of how and why this legal issue is important for you and your colleagues as current or future managers? The relevance of these laws is the fact that they assure equal opportunity for persons with disabilities in public accommodations, employment, transportation, local and state government services as well as telecommunications. Normally, discrimination takes place when an entity leaves out a person with HIV from taking part in a service or disallows a person a benefit. Instances of discriminatory acts toward persons with HIV/AIDS comprise of refusing access to medical treatment and/or social services or having treatment and/or services deferred for the sole reason of one having HIV/AIDS(Webber,2007). Thus, if the person with HIV fulfils the necessary eligibility needs for the benefit or service, the entity may be needed to make a rational accommodation to facilitate the person to take part. These laws are crucial in that it is unlawful to discriminate against individuals having or are believed to be having HIV/AIDS in regard to; employment, rental, acquisition or sale of apartment, real estate, or house, public accommodation places (theaters, restaurants etc), health care, home repairs, legal services or other various services available to the public generally, application of a credit card or loan, or other credit transactions as well as particular transactions in insurance. On the other hand, employers are obligated to offer and sustain a discrimination-free working place, in addition to ensuring that those with HIV face no intimidation or harassment. It is actually the responsibility of everyone to not only know and understand

Defense of leisure Essay Example | Topics and Well Written Essays - 1250 words

Defense of leisure - Essay Example This becomes the only path to reach man’s full potential. The human mind has the capacity to think and reason out, thus, allowing it to reflect on what happened and what is happening in his world, in his life. Leisure’s purpose is to act on directing the human to the path of wholeness. It is not to be just relived from the limited functions that humanity can do or take but it is to â€Å"retain the faculty of grasping the world as a whole and realizing his full potentialities†¦to reach wholeness† (Pieper, 50). Leisure has been described as a ritual for humans to rejuvenate or replenish their strength to move forward and continue the things they do, or the task of their existence (Pieper 17). In comparison with the modern days, as development progress in human along with their cultures, leisure is now commonly known as a way to release stress and fatigue mentally and emotionally. The concept of how leisure eases the mind by contemplating on one’s existence and direction for wholeness has slowly diminished. One of the essentials for survival is performing the limited functions of being human, that is to work and to perform the tasks at hand in order to continue living. As simple minds become more sophisticated, the traditions and one’s outlook on work has changed. This is in relation to utilitarianism, in which, humans focus on the means to live rather than on the purpose of one’s existence. Pieper quoted Weber saying one does not work to live, but rather one lives to work (Pieper p.20). And today that phrase is not very hard to understand, or rather the human race has come to accept it as part of our existence and use leisure to take away every worry that we have and temporarily forget our problems. Leisure, in connection with this, is thought to be something that humans can do after finishing all the limited functions that one has in order to live. One then finishes work to be

Subjective assessment Essay Example for Free

Subjective assessment Essay The overall assessment of the market may not reflect its true form. But looking at it more closely, it would seem that the subjective part of the analysis agrees with the objective part for a number of reasons. Firstly, the market demand for Denmark is considerably large, couple this with the lower tax than any of the countries it would really seem to be of the best choice. Spain, on the other hand, poses the highest task, with a considerably large market loaded with substitutes that may hinder the proliferation of the products of the distillery in question. Northern Ireland would logically be a good choice, if we don’t take in mind that, the market would have to be loaded with competitors on each other’s throats. It would be market suicide to try to penetrate a market full of veteran companies that may swallow the company’s products in one of their marketing campaigns. Decision Alternatives †¢ The first alternative with the marketing of these products is to offer the public a bargain of selling two bottles at the same time with a slightly lower price of the two bottles. Accompanied by massive publicity that includes radio commercials and even television commercials, The method would work two-ways: point-of-sale, information that the consumers would get in the retail shops would be the same as the ones that would be heard in the radio and television. Secondly, information dissemination. With the massive publicity, the company gets to introduce their new products that would somehow make an impact on the viewers’ or listeners’ choice. †¢ Taking adavantage of your strengths would be the second alternative to be considered. Previous experience dictates that the market has treated the company nicely when it came to mail order purchases. If you would be able to make the necessary adjustments in your mail order service in the host country, then you would be able to gain a share of the market. Advertising is important and should therefore be taken into consideration. The mail orders can be accompanied by leaflets that tell of the company’s other products and invitations to go to the distillery and see the Visitor’s Shop. The Visitor’s Shop was the provider of the profits before, it might the provider again. †¢ Discount offerings would be the third alternative. This allows us to offer the products in slightly lower introductory prices so that the public may taste your product for lower prices. This proves to be a good and bad mood at the same time. The competitors may lower their price, forcing you to lower your price even more, at profits’ expense. Or you could get a significant share of the market if the public gets satisfied with the offer, and especially satisfied with the products offered. †¢ The final alternative would be advertise heavily and rely solely on the marketing effects of your strategies. This would be a big gamble on the company’s part as they would spend and incur costs at an increasing rate. The rate of spending, however, may not be accompanied with positive results as the distillery would then depend on the reactions of the public on their advertisement. A positive effect would be a partial gain in your market share and hopefully improve your market position. A negative effect, however, would entail additional costs without pay-offs. Although advertising is never a bad investment, for a small company like that of The Olde Distillerie cannot afford to lose largely on its first attempt to enter the market. Final Decision The alternative that should be considered is that of the second alternative. This alternative not only gives profit for the organization but this could be the initial move that a new company may have. This initial move can be the first of the series of ideas that may spawn from the initial move. Improvements on the idea, additional perks, or just plain advertising would be the next moves that would be coupled with the strengths of the company. Furthermore, the strengths of the company are the same strengths that may help them survive in a new market. A different market that may hurt or help them. Contingency Plan If ever that the final decision alternative would not be effective, The Olde Distillerie may take the first alternative, the bargain of two bottles. This gives a sense of cheapness to the drink but would also be seen as an opportunity for the public to try something new. A small discount on this account may be explored further so that they may find new ideas to their marketing strategies. BIBLIOGRAPHIES Amerique, Remy (2006), The Macallan(R) Single Highland Malt Breaks New Global Advertising Creative Campaign, http://www.marketwire. com/mw/release_printer_friendly? release_id=58696category=, accessed November 7. Anonymous (2006), The History of Scotch Whisky In More Detail, http://www. scotch-whisky. org. uk/, accessed November 17. Anonymous (2006), Ireland (Information on the Irish State) : Land and People, http://www. irlgov. ie/aboutireland/eng/landandpeople. asp, accessed November 7. Anonymous (2006), Press Release: SUCCESSFUL WHISKY INDUSTRY VITAL TO SCOTTISH ECONOMY SAYS HENRY MCLEISH, http://www. scotland. gov. uk/news/1999/11/se1461. asp, accessed November 17. Anonymous (2006), The Scotch Experience: Scotch Whisky Statistics, http://www. scotchdoc. com/tsd/education/stats. html, accessed November 7. Anonymous (2006), Whiskys reputation under threat, http://www. scotchwhisky. net/news/threat. php, accessed November 7. Anonymous (2006), whiskey, http://www. britannica. com/eb/article-9076785/whiskey, accessed November 17. Anonymous (2006), Whiskey Rebellion, http://www. britannica. com/ebi/article-9340307, accessed November 17. Anonymous (2006), Denmark, http://www. britannica. com/eb/article-33928/Denmark, accessed November 17.

Basics of Topological Solutons

Basics of Topological Solutons Research into topological solitons began in the 1960s, when the fully nonlinear form of the classical field equations, were being thoroughly explored by mathematicians and theoretical physicists. Topological solitons were first examined when the solutions to these equations were interpreted as candidates for particles of the theory [1]. The particles that were observed from the results were different from the usual elementary particles. Topological solitons appeared to behave like normal particles in the sense that they were found to be localised and have finite energy [4]. However, the solitons topological structure distinguished them from the other particles. Topological solitons carry a topological charge (also known as the winding number), which results in these particlelike objects being stable. The topological charge is usually denoted by a single integer, N; it is a conserved quantity, i.e. it is constant unless a collision occurs, and it is equal to the total number of partic les, which means as |N| increases, the energy also increases. The conservation of the topological charge is due to the topological structure of the target space in which the soliton is defined. The most basic example of soliton has topological charge, N = 1, which is a stable solution, due to the fact a single soliton is unable to decay. 3 If the solution to a nonlinear classical field equation has the properties of being particle-like, stable, have finite mass; and the energy density is localised to a finite region of space, with a smooth structure; then this solution is a topological soliton. In addition to solitons existing with topological charge, N, there also exist antisolitons with -N. In the event of a collision between a soliton and an antisoliton, it is possible for them to annihilate each other or be pair-produced [1]. It is also possible for multi-soliton states to exist. Any field composition where N > 1, is known as a multi-soliton state. Likewise, multi-solitons also carry a topological charge which again means they are stable. Multi-state solitons either decay into N well separated charge 1 solitons or they can relax to a classical bound state of N solitons [1]. The energy and length scale [1] (a particular length which is determined to one order of magnitude.) the constant in the Lagrangian and field equations which represents the strength of the interaction between the particle and the field, also known as the coupling constant. The energy of a topological soliton is equal to its rest mass in a Lorentz invariant theory. [5] [6] Lorentz invariant: A quantity that does not change due to a transformation relating the space-time coordinates of one frame of reference to another in special relativity; a quantity that is independent of the inertial frame. In contrast to the topological soliton, the elementary particles mass is proportional to Plancks constant, ~. In the limit ~ à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ 0, the elementary particles mass goes to zero where as the topological solitons mass is finite. The quantization of the wave-like fields which satisfy the linearized field equations [1] determines the elementary particle states, where the interactions between the particles are determined by the nonlinear terms A fundamental discovery in supporting the research of topological solitons is that, given the coupling constants take special values, then the field equations can be reduced from second order to first order partial differential equations.[1] In general, the resulting first order equations are known as Bogomolny equations. These equations do not involve any time derivatives, and their solutions are either static soliton or multi-soliton configurations. [1] In these given field theories, if the field satisfies the Bogomolny equation then the energy is bounded below by a numerical multiple of the modulus of the topological charge, N, so the solutions of a Bogomolny equation with a certain 4 charge will all have the same energy value. [1] The solutions of the Bogomolny equations are automatically stable [1] because the fields minimize the energy [1]. As well as this they naturally satisfy the Euler-Lagrange equations of motion, which implies the static solutions are a stationary point of the energy. [1] Kinks are solutions to the first-order Bogomolny equation which we shall see in the following chapter Figure 2.2 shows a model of an infinite pendulum strip, with the angle à Ã¢â‚¬   being the angle to the downward vertical [3]. The energy (with all constraints set to 1) is E = Z à ¢Ã‹â€ Ã… ¾ à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã… ¾Ãƒâ€šÃ‚   1 2 à Ã¢â‚¬   02 + 1 à ¢Ã‹â€ Ã¢â‚¬â„¢ cos à Ã¢â‚¬  Ãƒâ€šÃ‚   dx (2.1) where à Ã¢â‚¬   0 = dà Ã¢â‚¬   dx . For the energy density to be finite this requires à Ã¢â‚¬   à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ 2à Ã¢â€š ¬nà ¢Ã‹â€ Ã¢â‚¬â„¢ as x à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã… ¾ and à Ã¢â‚¬   à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ 2à Ã¢â€š ¬n+ as x à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ à ¢Ã‹â€ Ã… ¾, where n ± à ¢Ã‹â€ Ã‹â€  Z. To find the number of twists, N, this is simply N = n+ à ¢Ã‹â€ Ã¢â‚¬â„¢ nà ¢Ã‹â€ Ã¢â‚¬â„¢ = à Ã¢â‚¬   (à ¢Ã‹â€ Ã… ¾) à ¢Ã‹â€ Ã¢â‚¬â„¢ à Ã¢â‚¬   (à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã… ¾) 2à Ã¢â€š ¬ = 1 2à Ã¢â€š ¬ Z à ¢Ã‹â€ Ã… ¾ à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã… ¾ à Ã¢â‚¬   0 dx à ¢Ã‹â€ Ã‹â€  Z This is the equation for the topological charge or the winding number. If we set nà ¢Ã‹â€ Ã¢â‚¬â„¢ = 0 and n+ = 1 then N = 1, this gives the lowest possible energy for a topological soliton. This is called a kink, and it is the term we use for the one spatial dimension soliton with a single scalar field. The name kink is due to the shape of the scalar field when plotted as a function of x [1]. Knowing that a kink gives the minimum of the energy, it is possible to apply the calculus of variations to derive a differential equation à Ã¢â‚¬  (x) and then solve it[3] to give the shape of the kink. Given a differentiable function on the real line, f(x), it is possible to find the minimum of f(x) by finding the solutions of f 0 (x) = 0, i.e. by finding the stationary points of f(x) [3]. It is achievable to derive this differential equation, f(x), by making a small change to x, i.e. x à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ x + ÃŽÂ ´x, and from this calculate the change in the value of the function to lea ding order in the variaton ÃŽÂ ´x [3]. ÃŽÂ ´f(x) = f(x + ÃŽÂ ´x) à ¢Ã‹â€ Ã¢â‚¬â„¢ f(x) = f(x) + ÃŽÂ ´xf0 (x) + à ¢Ã‹â€ Ã¢â‚¬â„¢ f(x) = f 0 (x)ÃŽÂ ´x + If f 0 (x) 0. If f 0 (x) > 0 then we can make ÃŽÂ ´f(x) The term [à Ã¢â‚¬   0 ÃŽÂ ´Ãƒ Ã¢â‚¬  ] à ¢Ã‹â€ Ã… ¾ à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã… ¾ equates to zero on the boundary because it must satisfy ÃŽÂ ´Ãƒ Ã¢â‚¬  ( ±Ãƒ ¢Ã‹â€ Ã… ¾) = 0 as we cannot change the boundary conditions, so ÃŽÂ ´E = Z à ¢Ã‹â€ Ã… ¾ à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã… ¾ {(à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ Ã¢â‚¬   00 + sin à Ã¢â‚¬  )ÃŽÂ ´Ãƒ Ã¢â‚¬  } dx (2.6) This equation can be minimised minimised further to the second order nonlinear differential equation, à Ã¢â‚¬   00 = sin à Ã¢â‚¬   (2.7) The solution of this differential equation with the boundary conditions, à Ã¢â‚¬  (à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã… ¾) = 0 and à Ã¢â‚¬  (à ¢Ã‹â€ Ã… ¾) = 2à Ã¢â€š ¬ is the kink. Therefore the kink solution is, à Ã¢â‚¬  (x) = 4 tanà ¢Ã‹â€ Ã¢â‚¬â„¢1 e xà ¢Ã‹â€ Ã¢â‚¬â„¢a (2.8) where a is an arbitrary constant. When x = a, this is the position of the kink (à Ã¢â‚¬  (a) = à Ã¢â€š ¬). It is clear to see à Ã¢â‚¬   = 0 is also a solution to the differential equation , however, it does not satisfy the boundary conditions. It is possible to find a lower bound on the kink energy without solving a differential equation [3]. First of all we need to rewrite the energy equation (2.1), using the double angle formula the equation becomes, E = 1 2 Z à ¢Ã‹â€ Ã… ¾ à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã… ¾Ãƒâ€šÃ‚   à Ã¢â‚¬   02 + 4 sin2   à Ã¢â‚¬   2   dx (2.9) By completing the square the equation becomes, E = 1 2 Z à ¢Ã‹â€ Ã… ¾ à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã… ¾Ãƒâ€šÃ‚   à Ã¢â‚¬   0 à ¢Ã‹â€ Ã¢â‚¬â„¢ 2 sin   à Ã¢â‚¬   2 2 + 4à Ã¢â‚¬   0 sin   à Ã¢â‚¬   2 dx (2.10) Therefore the energy satisfies the inequality, E > 2 Z à ¢Ã‹â€ Ã… ¾ à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã… ¾ à Ã¢â‚¬   0 sin   à Ã¢â‚¬   2   dx = 2 Z à ¢Ã‹â€ Ã… ¾ à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã… ¾ sin   à Ã¢â‚¬   2   dà Ã¢â‚¬   dxdx = 2 Z 2à Ã¢â€š ¬ 0 sin   à Ã¢â‚¬   2   dà Ã¢â‚¬   = à ¢Ã‹â€ Ã¢â‚¬â„¢4   cos   à Ã¢â‚¬   2 2à Ã¢â€š ¬ 0 = 8 (2.11) In order to obtain the solution which is exactly 8, the term à Ã¢â‚¬   0 à ¢Ã‹â€ Ã¢â‚¬â„¢ 2 sin à Ã¢â‚¬   2 2 would have to be exactly 0. Therefore the lower bound on the kink energy is calculated by the solution to the equation, à Ã¢â‚¬   0 = 2 sin   à Ã¢â‚¬   2   (2.12) This is a first order Bogomolny equation. Taking this Bogomolny equation and differentiating with respect to à Ã¢â‚¬   0 gives, à Ã¢â‚¬   00 = cos   à Ã¢â‚¬   2   à Ã¢â‚¬   0 = cos   à Ã¢â‚¬   2   2 sin   à Ã¢â‚¬   2   = sin à Ã¢â‚¬   (2.13) This shows that a solution of the Bogomo lny equation (2.12) gives the output of the kink solution (2.7). To calculate the energy density ÃŽÂ µ, equation (2.1), we need to use the fact that the Bogomolny equation shows that ÃŽÂ µ = à Ã¢â‚¬   02 . From equation (2.8) we have, tan à Ã¢â‚¬   4   = e xà ¢Ã‹â€ Ã¢â‚¬â„¢a , therefore 1 4 à Ã¢â‚¬   0 sec2   à Ã¢â‚¬   4 = e xà ¢Ã‹â€ Ã¢â‚¬â„¢a This equation gives, à Ã¢â‚¬   0 = 4 e xà ¢Ã‹â€ Ã¢â‚¬â„¢a 1 + tan2 à Ã¢â‚¬   4   = 4e xà ¢Ã‹â€ Ã¢â‚¬â„¢a 1 + e 2(xà ¢Ã‹â€ Ã¢â‚¬â„¢a) = 2 cosh (x à ¢Ã‹â€ Ã¢â‚¬â„¢ a) = 2 (x à ¢Ã‹â€ Ã¢â‚¬â„¢ a) (2.15) Therefore it can be seen that the energy density is given by ÃŽÂ µ = 42 (x à ¢Ã‹â€ Ã¢â‚¬â„¢ a) From this we get the solution of a lump with a maximal value of 4 when x = a. This maximal value is the position of the kink. The position of the kink is also the position of the pendulum strip when it is exactly upside down, this is due to the fact à Ã¢â‚¬  (a) = à Ã¢â€š ¬ [3]. Using this interpretation for the energy density, it can be verified that the energy is equal to the lower bound E = Z à ¢Ã‹â€ Ã… ¾ à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã… ¾ ÃŽÂ µdx = 4 Z à ¢Ã‹â€ Ã… ¾ à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã… ¾ 2 (x à ¢Ã‹â€ Ã¢â‚¬â„¢ a) dx = 4 [tanh (x à ¢Ã‹â€ Ã¢â‚¬â„¢ a)]à ¢Ã‹â€ Ã… ¾ à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã… ¾ = 8 (2.16) For N > 1 i.e. more than one kink, E > 8|N|. In order t o obtain the lower bound of N > 1 kinks, the kinks must be infinitely apart to create N infinitely separated kinks. This means there must be a repulsive force between kinks. We shall now look at applying Derricks theorem [3] to kinks to show that it does not rule out the existence of topological solitons. Derricks Theorem: If the energy E has no stationary points with respect to spatial rescaling then it has no solutions with 0 Derricks theorem can only be applied to an infinite domain. Firstly, the energy terms need to be split according to the powers of the derivative, E = E2 + E0 = Z à ¢Ã‹â€ Ã… ¾ à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã… ¾ 1 2 à Ã¢â‚¬   02 dx + Z à ¢Ã‹â€ Ã… ¾ à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã… ¾ (1 à ¢Ã‹â€ Ã¢â‚¬â„¢ cos à Ã¢â‚¬  ) dx (2.17) Now consider the spatial rescaling x 7à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ x ÃŽÂ » = X, so that à Ã¢â‚¬   (x) 7à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ à Ã¢â‚¬   (X), with dx = ÃŽÂ »dX, d dx = 1 ÃŽÂ » d dX . Under this rescaling the energy becomes E (ÃŽÂ »), E(ÃŽÂ ») = Z à ¢Ã‹â€ Ã… ¾ à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã… ¾ 1 2 ( 1 ÃŽÂ » dà Ã¢â‚¬   dX ) 2ÃŽÂ »dX + Z à ¢Ã‹â€ Ã… ¾ à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã… ¾ (1 à ¢Ã‹â€ Ã¢â‚¬â„¢ cos à Ã¢â‚¬  ) ÃŽÂ »dX = 1 ÃŽÂ » E2 + ÃŽÂ »E0 (2.18) It is now important to see whether E(ÃŽÂ ») has a stationary point with respect to ÃŽÂ », dE (ÃŽÂ ») dÃŽÂ » = à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 ÃŽÂ » 2 E2 + E0 = 0 (2.19) if ÃŽÂ » = qE2 E0 , where ÃŽÂ » equals the size of the soliton. From this we can see a stationary point exists, so by Derricks theorem we cannot rule out the possibility of a topological soliton solution existing. We already know this is the case due to already finding the kink solution earlier. If it is found that à Ã¢â‚¬  (x) is a solution then the stationary point corresponds to no rescaling [3], so ÃŽÂ » = 1, meaning E2 = E0. This is known as a virial relation. In order to extend the kink example to higher spatial dimensions, we will rewrite it using different variables. If we let à Ã¢â‚¬   = (à Ã¢â‚¬  1, à Ã¢â‚¬  2) be a two-component unit vector, where à Ã¢â‚¬    · à Ã¢â‚¬   = |à Ã¢â‚¬  | 2 = 1. By writing à Ã¢â‚¬   = (sin à Ã¢â‚¬  , cos à Ã¢â‚¬  ), the energy from (2.1) can be rewritten as E = Z à ¢Ã‹â€ Ã… ¾ à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã… ¾ ( 1 2  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚   dà Ã¢â‚¬   dx  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚   2 à ¢Ã‹â€ Ã¢â‚¬â„¢ H  · à Ã¢â‚¬   + |H| ) dx (2.20) where H = (0, 1). [3] In this new formulation à Ã¢â‚¬   represents the direction of the local magnetization (restricted to the plane) in a ferromagnetic medium [3] and H represents the constant background magnetic field which is also restricted to lie within the same plane as à Ã¢â‚¬  . There is only one point in which the systems ground state is equal to zero in terms of à Ã¢â‚¬  , which is à Ã¢â‚¬   = H |H| = (0, 1 ). Any structure with finite energy has to approach this zero energy ground state at spatial infinity, therefore the boundary conditions are à Ã¢â‚¬   à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ (0, 1) as x à ¢Ã¢â‚¬  Ã¢â‚¬â„¢  ±Ãƒ ¢Ã‹â€ Ã… ¾. As à Ã¢â‚¬   takes the same value at x = à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã… ¾ and x = +à ¢Ã‹â€ Ã… ¾, then these points can be identified so the target space, which is the real line R, topologically becomes a circle, S 1 of infinite radius. Therefore we have the mapping à Ã¢â‚¬   : S 1 7à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ S 1 between circles, because à Ã¢â‚¬   is a two-component vector so it also lies on a circle of unit radius. [3] The mapping between circles has a topological charge (winding number), N, which counts the number of times à Ã¢â‚¬   winds around the unit circle as x varies over the whole real line. [3] The topological charge is equal to the equation defined earlier in (2.2), but using the new variables it is given by the expression N = 1 2à Ã¢â€š ¬ Z à ¢ 蠁 ¾ à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã… ¾Ãƒâ€šÃ‚   dà Ã¢â‚¬  1 dx à Ã¢â‚¬  2 à ¢Ã‹â€ Ã¢â‚¬â„¢ dà Ã¢â‚¬  2 dx à Ã¢â‚¬  1   dx (2.21) If we consider a restricted ferromagnetic system in which there is the absence of a background magnetic field (H = 0); it is still possible for a topological soliton to exist if there is an easy axis anisotropy. [3] Magnetic anisotropy is the directional dependence of a materials magnetic property, and the easy axis is a energetically favorable direction if spontaneous magnetization occurs.[7] The energy for this system is E = Z à ¢Ã‹â€ Ã… ¾ à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã… ¾ ( 1 2  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚   dà Ã¢â‚¬   dx  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚  Ãƒâ€šÃ‚   2 + A 1 à ¢Ã‹â€ Ã¢â‚¬â„¢ (à Ã¢â‚¬    · k) 2   ) dx (2.22) where A > 0 is the anisotropy constant and k is the unit vector which specifies the easy axis. [3] For this type of system there are two zero energy ground states, à Ã¢â‚¬   =  ±k. The kink in t his system, also called a domain wall, interpolates between the two zero energy ground states and has boundary conditions à Ã¢â‚¬   à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ k as x à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã… ¾ and à Ã¢â‚¬   à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ à ¢Ã‹â€ Ã¢â‚¬â„¢k 15 as x à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ +à ¢Ã‹â€ Ã… ¾. Therefore the domain wall does not have a full twist of a kink and only has a half-twist. It is possible to map this system to our original kink example by a change of variables. If we set k = (0, 1) for convenience, and choose A = 1 2 . Setting à Ã¢â‚¬   = sin à Ã¢â‚¬   2   , cos à Ã¢â‚¬   2 , then the energy equation becomes E = 1 4 Z à ¢Ã‹â€ Ã… ¾ à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã… ¾Ãƒâ€šÃ‚   1 2 à Ã¢â‚¬   02 + 1 à ¢Ã‹â€ Ã¢â‚¬â„¢ cos à Ã¢â‚¬  Ãƒâ€šÃ‚   dx (2.23) which is equal to the energy equation (2.1) but with a normalization factor of 1 4 . The domain wall boundaries are à Ã¢â‚¬   à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ (0,  ±1) as x à ¢Ã‹â€ Ã¢â‚¬Å" à ¢Ã‹â€ Ã… ¾ are exactly the kink boundary conditions à Ã¢â‚¬   (à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã… ¾) = 0 and à Ã¢â‚¬   (à ¢Ã‹â€ Ã… ¾) = 2à Ã¢â€š ¬. [1] This chapter will focus on topological solitons in (2+1) spatial dimensions. It would be incorrect to use the term soliton for these solutions due to their lack of stability, instead they are often referred to as lumps. The solutions for these lumps are given explicitly by rational maps between Riemann spheres. [1] For this chapter we shall be looking at one of the simplest Lorentz invariant sigma models in (2+1) spatial dimensions which renders static topological soliton solutions; the O(3) sigma model in the plane. [1] A sigma model is a nonlinear scalar field theory, where the field takes values in a target space which is a curved Riemannian manifold, usually with large symmetry. [1] For the O(3) sigma model the target space is the unit 2-sphere, S 2 . This model uses three real scalar fields, ÃŽÂ ¦ = (à Ã¢â‚¬  1, à Ã¢â‚¬  2, à Ã¢â‚¬  3), which are functions of the space-time coordinates (t, x, y) in (2+1) spatial dimensions. [2] The O(3) model is defined by the Lagrangia n density L = 1 4 (à ¢Ã‹â€ Ã¢â‚¬Å¡Ãƒâ€šÃ‚ µÃƒÅ½Ã‚ ¦)  · (à ¢Ã‹â€ Ã¢â‚¬Å¡  µÃƒÅ½Ã‚ ¦)  with the constraint ÃŽÂ ¦  · ÃŽÂ ¦ = 1. For this equation the indices represent the space-time coordinates and take the values 0, 1, 2, and à ¢Ã‹â€ Ã¢â‚¬Å¡Ãƒâ€šÃ‚ µ is partial differentiation with respect to X µ . [2] From (3.1), the Euler-Lagrange equation can be derived, which is à ¢Ã‹â€ Ã¢â‚¬Å¡Ãƒâ€šÃ‚ µÃƒ ¢Ã‹â€ Ã¢â‚¬Å¡  µÃƒÅ½Ã‚ ¦ + (à ¢Ã‹â€ Ã¢â‚¬Å¡Ãƒâ€šÃ‚ µÃƒÅ½Ã‚ ¦  · à ¢Ã‹â€ Ã¢â‚¬Å¡  µÃƒÅ½Ã‚ ¦) ÃŽÂ ¦ = 0 (3.2) Due to the dot product in à ¢Ã‹â€ Ã¢â‚¬Å¡Ãƒâ€šÃ‚ µÃƒÅ½Ã‚ ¦  · à ¢Ã‹â€ Ã¢â‚¬Å¡  µÃƒÅ½Ã‚ ¦, this shows that the Euclidean metric of R 3 is being used, and this becomes the standard metric on the target space S 2 when the constraint ÃŽÂ ¦  · ÃŽÂ ¦ = 1 is being imposed. [1] For the sigma model we are exploring, the O(3) represents the global symmetry in the target space corresponding to the rotation s: ÃŽÂ ¦ 7à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ MÃŽÂ ¦ Where M à ¢Ã‹â€ Ã‹â€  O(3) is a constant matrix. [1] The sigma in the models name represents the fields (à Ã¢â‚¬  1, à Ã¢â‚¬  2, à Ã†â€™), where à Ã¢â‚¬  1 and à Ã¢â‚¬  2 are locally unconstrained [1] and à Ã†â€™ = p 1 à ¢Ã‹â€ Ã¢â‚¬â„¢ à Ã¢â‚¬   2 1 à ¢Ã‹â€ Ã¢â‚¬â„¢ à Ã¢â‚¬   2 2 is dependent on à Ã¢â‚¬  1 and à Ã¢â‚¬  2. The energy for the O(3) sigma model is E = 1 4 Z à ¢Ã‹â€ Ã¢â‚¬Å¡iÃŽÂ ¦  · à ¢Ã‹â€ Ã¢â‚¬Å¡iÃŽÂ ¦d 2x (3.3) where i = 1, 2 runs over the spatial indices. In order for the energy to be finite, ÃŽÂ ¦ has to tend to a constant vector at spatial infinity, so without loss of generality we are able to set the boundary condition ÃŽÂ ¦ à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ (0, 0, 1) as x 2 + y 2 à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ à ¢Ã‹â€ Ã… ¾. Topologically we have R 2 à ¢Ã‹â€ Ã‚ ª {à ¢Ã‹â€ Ã… ¾}, which is interpreted as a sphere S 2 via the stereographic projection. (The sphere itself has finite radius.) Therefore we are considering mapping between spheres ÃŽÂ ¦ : S 2 7à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ S 2 . Just like in our kink example, mapping between spheres means there exists a topological charge, which can be found using N = 1 4à Ã¢â€š ¬ Z ÃŽÂ ¦  · (à ¢Ã‹â€ Ã¢â‚¬Å¡1ÃŽÂ ¦ ÃÆ'- à ¢Ã‹â€ Ã¢â‚¬Å¡2ÃŽÂ ¦) d 2x (3.4) The topological charge represents the number of lumps in the field configuration [1], since generally there are N well-separated, localized areas where the energy density is concentrated and each area has one unit of charge. However, as the lumps approach each other this is no longer the case. In order to apply Derricks theorem to the energy (3.3), we would need to consider the scaling x 7à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ x ÃŽÂ » = X and y 7à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ y ÃŽÂ » = Y which would give E (ÃŽÂ ») = E. The energy is independent of ÃŽÂ », therefore any value of ÃŽÂ » is a stationary point since the energy does not change from spatial rescaling. If we integrate the inequality  (à ¢Ã‹â€ Ã¢â‚¬Å¡iÃŽÂ ¦  ± ÃŽÂ µijÃŽÂ ¦ ÃÆ'- à ¢Ã‹â€ Ã¢â‚¬Å¡jÃŽÂ ¦)  · (à ¢Ã‹â€ Ã¢â‚¬Å¡iÃŽÂ ¦  ± ÃŽÂ µikÃŽÂ ¦ ÃÆ'- à ¢Ã‹â€ Ã¢â‚¬Å¡kÃŽÂ ¦) à ¢Ã¢â‚¬ °Ã‚ ¥ 0 (3.5) over the plane and use the equations (3.3) and (3.4) for the energy density and the topological charge respectively [1], then we get the Bogomolny bound E à ¢Ã¢â‚¬ °Ã‚ ¥ 2à Ã¢â€š ¬ |N| (3.6) This Bogomolny bound is the lower bound of the energy in terms of lumps. [1] If the field is a solution to one of the first-order Bogomolny equations à ¢Ã‹â€ Ã¢â‚¬Å¡iÃŽÂ ¦  ± ÃŽÂ µijÃŽÂ ¦ ÃÆ'- à ¢Ã‹â€ Ã¢â‚¬Å¡jÃŽÂ ¦ = 0 (3.7) then the energy is equal to the Bogomolny bound. In order to analyse the Bogomolny equations it is best to make the following changes of variables. For the first change in variable let X = (X1, X2, X3) denote the Cartesian coordinates in R 3 and take X = ÃŽÂ ¦ to be a point on the unit sphere, (X2 1 , X2 2 , X2 3 ) = 1. Let L be the line going through X = (0, 0, à ¢Ã‹â€ Ã¢â‚¬â„¢1) and ÃŽÂ ¦ and set W = X1 + iX2 to be the complex coordinate of the point where L intersects the plane at X3 = 0. We then get W = (à Ã¢â‚¬  1 + ià Ã¢â‚¬  2) (1 + à Ã¢â‚¬  3) (3.8) where à Ã¢â‚¬  1 =   W + W 1 + |W| 2   , à Ã¢â‚¬  2 = i   W à ¢Ã‹â€ Ã¢â‚¬â„¢ W 1 + |W| 2   , à Ã¢â‚¬  3 = 1 à ¢Ã‹â€ Ã¢â‚¬â„¢ |W| 2 1 + |W| 2 ! (3.9) As ÃŽÂ ¦ tends to the point (0, 0, à ¢Ã‹â€ Ã¢â‚¬â„¢1) then L only intersects X3 = 0 at à ¢Ã‹â€ Ã… ¾, therefore the point (0, 0, à ¢Ã‹â€ Ã¢â‚¬â„¢1) maps to the point W = à ¢Ã‹â€ Ã… ¾. This method of assigning each point on the sphere to a point in C à ¢Ã‹â€ Ã‚ ª {à ¢Ã‹â€ Ã… ¾} is called stereographic projection as seen in Figure 3.1.[3] The next change in variable comes from using a complex coordinate in the (x, y) plane by letting z = x + iy. Using this formation it is possible to rewrite the Lagrangian density, from (3.1) L = 1 4 ( à ¢Ã‹â€ Ã¢â‚¬Å¡Ãƒâ€šÃ‚ µÃƒ Ã¢â‚¬  1) 2 + (à ¢Ã‹â€ Ã¢â‚¬Å¡Ãƒâ€šÃ‚ µÃƒ Ã¢â‚¬  2) 2 + (à ¢Ã‹â€ Ã¢â‚¬Å¡Ãƒâ€šÃ‚ µÃƒ Ã¢â‚¬  3) 2   . Firstly we need to partially differentiate à Ã¢â‚¬  1, à Ã¢â‚¬  2, à Ã¢â‚¬  3, giving à ¢Ã‹â€ Ã¢â‚¬Å¡Ãƒâ€šÃ‚ µÃƒ Ã¢â‚¬  1 = à ¢Ã‹â€ Ã¢â‚¬Å¡Ãƒâ€šÃ‚ µW + à ¢Ã‹â€ Ã¢â‚¬Å¡Ãƒâ€šÃ‚ µW 1 + |W| 2 à ¢Ã‹â€ Ã¢â‚¬â„¢ (à ¢Ã‹â€ Ã¢â‚¬Å¡Ãƒâ€šÃ‚ µW) W + W à ¢Ã‹â€ Ã¢â‚¬Å¡Ãƒâ€šÃ‚ µW   1 + |W| 2 2 W + W   (3.10) à ¢Ã‹â€ Ã¢â‚¬Å¡Ãƒâ€šÃ‚ µÃƒ Ã¢â‚¬  2 = i à ¢Ã‹â€ Ã¢â‚¬Å¡Ãƒâ€šÃ‚ µW à ¢Ã‹â€ Ã¢â‚¬â„¢ à ¢Ã‹â€ Ã¢â‚¬Å¡Ãƒâ€šÃ‚ µW 1 + |W| 2 à ¢Ã‹â€ Ã¢â‚¬â„¢ (à ¢Ã‹â€ Ã¢â‚¬Å¡Ãƒâ€šÃ‚ µW) W + W à ¢Ã‹â€ Ã¢â‚¬Å¡Ãƒâ€šÃ‚ µW   1 + |W| 2 2 W à ¢Ã‹â€ Ã¢â‚¬â„¢ W Finally, from simplifying (3.37) we get the equation for the topological charge in the new formulation to be N = 1 4à Ã¢â€š ¬ Z 4 1 + |W| 2 2 à ¢Ã‹â€ Ã¢â‚¬Å¡zW à ¢Ã‹â€ Ã¢â‚¬Å¡zW à ¢Ã‹â€ Ã¢â‚¬â„¢ à ¢Ã‹â€ Ã¢â‚¬Å¡zW à ¢Ã‹â€ Ã¢â‚¬Å¡zW   d 2x = 1 à Ã¢â€š ¬ Z |à ¢Ã‹â€ Ã¢â‚¬Å¡zW| 2 à ¢Ã‹â€ Ã¢â‚¬â„¢ |à ¢Ã‹â€ Ã¢â‚¬Å¡zW| 2   1 + |W| 2 2 d 2x (3.38) In this formulation it is clear to see E à ¢Ã¢â‚¬ °Ã‚ ¥ 2à Ã¢â€š ¬ |N|, with equality if and only if Bogomolny equation is satisfied à ¢Ã‹â€ Ã¢â‚¬Å¡W à ¢Ã‹â€ Ã¢â‚¬Å¡z = 0 (3.39) This equation shows that W is a holomorphic function of z only. [4] Due to the requirement that the total energy is finite, together with the boundary condition [4] W à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ 0 as |z| à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ à ¢Ã‹â€ Ã… ¾, this means that N is finite. [3] The simplest solution for the Bogomolny equation would be W = ÃŽÂ » z , where ÃŽÂ » is a real and positive constant. Applying this to the equation (3.9) yields the solution for t he N = 1 solution ÃŽÂ ¦ =   2 ÃŽÂ » 2 + x 2 + y 2 , à ¢Ã‹â€ Ã¢â‚¬â„¢2 ÃŽÂ » 2 + x 2 + y 2 , x 2 + y 2 à ¢Ã‹â€ Ã¢â‚¬â„¢ ÃŽÂ » 2 ÃŽÂ » 2 + x 2 + y 2 (3.40) If we change the negative sign in the second component to a positive sign then we get the solution of the anti-Bogomolny equation (3.7) (with the minus sign), which also has E = 2à Ã¢â€š ¬ but has N = à ¢Ã‹â€ Ã¢â‚¬â„¢1. This soliton is located at thee origin because W(0) = à ¢Ã‹â€ Ã… ¾. [3] The N = 1 general solution has 4 real parameters and is given by the Bogomolny solution W = ÃŽÂ »eiÃŽÂ ¸ z à ¢Ã‹â€ Ã¢â‚¬â„¢ a (3.41) where ÃŽÂ » is the size of the soliton, ÃŽÂ ¸ is the constant angle of rotation in the (à Ã¢â‚¬  1, à Ã¢â‚¬  2) plane and a à ¢Ã‹â€ Ã‹â€  C is the position of the soliton in the complex plane, z = x + iy. The O(3) sigma model can be modified to stabilise a lump, and the simplest way in doing this is by introducing extra terms into the Lagrangian which break the conformal invariance of the static energy. [1] These new terms must scale as negative and positive powers of a spatial dilation factor. [1] An example of this is the Baby Skyrme model which is given by the Lagrangian L = 1 4 à ¢Ã‹â€ Ã¢â‚¬Å¡Ãƒâ€šÃ‚ µÃƒÅ½Ã‚ ¦  · à ¢Ã‹â€ Ã¢â‚¬Å¡  µÃƒÅ½Ã‚ ¦ à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 8 (à ¢Ã‹â€ Ã¢â‚¬Å¡Ãƒâ€šÃ‚ µÃƒÅ½Ã‚ ¦ ÃÆ'- à ¢Ã‹â€ Ã¢â‚¬Å¡ÃƒÅ½Ã‚ ½ÃƒÅ½Ã‚ ¦)  · (à ¢Ã‹â€ Ã¢â‚¬Å¡  µÃƒÅ½Ã‚ ¦ ÃÆ'- à ¢Ã‹â€ Ã¢â‚¬Å¡ ÃŽÂ ½ÃƒÅ½Ã‚ ¦) à ¢Ã‹â€ Ã¢â‚¬â„¢ m2 2 (1 à ¢Ã‹â€ Ã¢â‚¬â„¢ à Ã¢â‚¬  3) (3.42) where the constraint ÃŽÂ ¦  · ÃŽÂ ¦ = 1 is implied. As we can see the first term in this Lagrangian is simply that of the O(3) sigma model. The second term in (3.42), is known as the Skyrme term and the final term in this Lagrangian is the mass term. The complete understanding of topological solitons is unknown and there are very limited experimental tests of many of the theories of topological solitons and their mathematical results. However, there is evidence of topological solitons existing in some physical systems, for example in one-dimensional systems they exist in optical fibres and narrow water channels. [1] Topological solitons can be applied to a range of different areas including particle physics, condensed matter physics, nuclear physics and cosmology. They also can be applied within technology, which involves using topological solitons in the design for the next generation of data storage devices. [3] In August 2016, a 7 million pound research programme, being led by Durham University, was announced into looking at how magnetic skyrmions can be used in creating efficient ways to store data. [10] Magnetic skyrmions are a theoretical particle in three spatial dimensions which have been observed experimentally in condensed matter systems. [11] This type of soliton was first predicted by scientists back in 1962, but was first observed experimentally in 2009. [10] In certain types of magnetic material it is possible for these magnetic skyrmions to be created,manipulated and controlled[10], and because of their size and structure it is possible for them to be tightly packed together. The structure inside the skyrmions [10] Due to this and the force which locks the magnetic field into the skyrmion arrangement, any magnetic information which is encoded by skyrmions is very robust. [10] It is thought that it will be possible to move these magnetic skyrmions with a lot less energy than the ferromagnetic domain being used in current data storage devices of smartphones and computers. Therefore, these magnetic skyrmions could revolutionise data storage devices, as the devices could be created on a smaller scale and use a lot less energy, meaning they would be more cost effective and would generate less heat. This project has given an insight into the very basics of topological solutons by analysing the energy and topological charge equations for kinks in one spatial dimension and lumps in (2+1) spatial dimensions. From the energy equation for a kink, we could derive the solution of a kink and find the lower energy bound. From the lump model, we successfully changed the variables for the energy, topological charge and the Lagrange equation for a lump to be able to analyse the Bogomolny equation. From this change of variables of the Lagrange equation we successfully solved the Euler-Lagrange equations of motion for the lump model. This research project has been captivating and has given me an insight into how the complex mathematics we learn is applied to real world situations. I first became interested in this topic after attending the London Mathematical Societys summer 33 school in 2016, where I had the privilege of attending a few lectures given by Dr Paul Sutcliffe, one of the authors of the book on Topological Solitons. It was in these few lectures where I first learnt about topological solitons and some of their applications, and this inspired me for my research project as I wanted to study the topic further. Although this project has been thoroughly enjoyable, it came with challenging aspects, due to its complex mathematics in such a specialised subject. As a result of this topic being so specific, I was very limited in the resources I had for my research, my main resource being the book on topological solitons by Dr Paul Sutcliffe and Dr Nicholas Manton. I have gained a lot of new skills from this research project and it has given me an opportunity to apply my current mathematical knowledge. There is an endless amount of research that can be continued within this subject. I, for example, would have liked to do some further research into the (2+1) spatial dimension model of the Baby Skyrmion and, like the lump example, solve the EulerLagrange equations motion . As well as this, I would have liked to input the equations of motion I solved for the lump model in Maple, so it was possible to simulate two lumps colliding and from this graph the energy density. It would have been really interesting to research further into topological solitons in three spatial dimensions, specifically Skyrmions, to learn further about their technological applications. However, the mathematics used for this model is very challenging and specialised, and goes beyond my understanding and knowledge.