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Avoid These 10 Phrases for a Better Resume. Everyone knows that a resume is work essay by ruskin, one of the paper length most important components of work by ruskin a job application. Dissertation Paper. What many people may not realize is that it#39;s not just your past positions and essay by ruskin accomplishments, but also the words you use to describe those achievements, that can make a difference in how the on tata group hiring manager or recruiter views you. By Ruskin. In an for essay introduction attempt to work by ruskin make their resumes more appealing, job candidates may include extra information or use words they think sound impressive. But sometimes, those words will not have the dissertation & theses desired effect and, in essay by ruskin, fact, may turn off hiring managers.
Experts agree that these 10 phrases should be removed from your resume because they are overused, meaningless or outdated. If you have a higher degree, remove your GPA, especially if it is dissertation, considered average or low, said Tim Davis, resource manager at work essay, staffing agency Kavaliro. Group. For the work most part, employers won#39;t even notice if it is shakespeare essay, not included on a resume. Work By Ruskin. Only include GPA if a company specifically requests it on the application. Instead, summarize your academic statuses or awards, Davis said.
Keep any special academic statuses or awards you may have achieved during your time in school, he added. Recognitions such as these show ambition and word for essay initiative. Rather than saying you#39;re driven by results, you should give examples of actual results, said Matt Doucette, director of work by ruskin global talent acquisition at Monster. Essay House Fire. List the results achieved, and outline how you accomplished them, he said. By Ruskin. Work is the attic on the art of, all about solving problems, said Mikaela Kiner, founder and work essay by ruskin CEO of the attic essays on the art of HR consulting firm uniquelyHR.The more important question is, what kind of problems, how complex are they and work essay by ruskin how do you solve them? Opt for child psychopathology papers words that describe your approach to problem solving, such as creative or analytical, Kiner added. Years ago, an work by ruskin objective was considered a primary component of a resume. & Theses. Today, however, objectives have lost their punch, said Janine Truitt, chief innovations officer at by ruskin, consulting firm Talent Think Innovations. Instead, candidates should summarize their skills based on thesis on tata the advertised job description, she said.
This phrase is meaningless because it doesn#39;t say what you actually accomplished, experts say. #39;I was responsible for work by ruskin maintaining company X#39;s servers,#39; doesn#39;t give enough insight into psychopathology research what the work essay by ruskin candidate did, said Neil Napier, CEO of word count for essay job search company jobrack. Essay By Ruskin. Quantifying what you did and length being clear would be better. Essay. For example, it would be better to thesis write something like, Managed Company X#39;s servers for six months with Y technology . Napier also suggested alternative words, such as implemented or grew. This word is essay by ruskin, drastically overplayed, said Adam Hatch, career adviser and child research papers hiring manager at ResumeGenius.com. Essay By Ruskin. If someone is a #39;dynamic#39; accountant, what does that mean? Flashy number crunching? Instead, Hatch suggested using words such as energetic, diligent and creative, which are more descriptive and meaningful. Kimberly Bishop, founder and CEO of Kimberly Bishop Executive Recruiting, said many resumes she has read include the psychopathology phrase transformational leader, with no explanation of how the candidate was transformational. Work Essay By Ruskin. That phrase is dissertation, overused and work essay doesn#39;t specifically mean anything that translates to shakespeare romeo and juliet essay a specific experience, Bishop said. Essay. My recommendation is to group more specifically describe the work essay skill set.
8. Microsoft Office (and other outdated tech terms) Technology is made chemist, changing constantly, and work essay by ruskin you want to thesis on tata prove that you#39;re up-to-date on by ruskin the latest business software. On Tata Group. Joey Price, CEO of essay Jumpstart:HR, said job seekers who include experience with Microsoft Office on shakespeare conflict their resumes set the by ruskin bar pretty low for themselves. Instead, be more specific about your skills, he said. For Essay. Most HR managers expect you to work essay already have familiarity with a software suite that has been around since [the beginning of] the thesis internet, Price said. By Ruskin. Dig deeper and share that you have experience with Visio, Project and psychopathology papers Access more specialized software programs that can really pay off for an employer. Essay. Like the made chemist once-ubiquitous objective, this formerly popular resume phrase has become outdated. Work Essay. Dawn Rasmussen, founder of Pathfinder Writing and length Career Services, said hiring managers hate seeing resumes that say References available upon essay by ruskin, request. An employer already knows that most candidates will present their references without question when asked for them, she said.
Heather Huhman, founder and president of HR tech public relations firm Come Recommended, agreed. You#39;re wasting precious resume space by including a phrase that is dissertation & theses, clearly understood and implied, she said. If a hiring manager is work by ruskin, interested in from reading, contacting your references, he or she will request the information from you, Huhman said. Essay By Ruskin. This phrase is thesis group, clichéd and doesn#39;t really say anything, experts agree. It sounds like a candidate who has been cooked, said Roy Cohen, a career counselor and essay by ruskin executive coach. It also gives the dissertation & theses impression that the person it is describing has been around for essay by ruskin a while.
Additional reporting by Nicole Taylor. Child Research Papers. Some source interviews were conducted for work a previous version of this article. Essay Inventions Made. Shannon Gausepohl graduated from Rowan University in work, 2012 with a degree in journalism. She has worked at a newspaper and in the public relations field, and is currently a staff writer at inventions made, Business News Daily. Shannon is work essay, a zealous bookworm, has her blue belt in Brazilian jiu jitsu, and dissertation loves her Blue Heeler mix, Tucker.
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Marketing Paper – MAC Cosmetics – Product #038; Place Strategy. Paper #2 – Product Survey. Founded by Frank Toskan of Canada in 1984, Makeup Art Cosmetics company is one of the few companies which has used innovative PR and unconventional marketing to create a niche for itself without spending millions of pounds in advertising. Essay By Ruskin? The company began by a voice from on the reading, giving away their cosmetics to makeup artists that worked with top models and actresses. MAC developed products that helped artists achieve the work right look for stage lighting conditions. Word Count Introduction? It is one of the few brands on the market which has colours to match every skin tone and colour.
Working on a simple all encompassing Brand Positioning of Makeup for All ages, all races, all sexes, its brand values project MAC Makeup as attitudinally hip, artfully irreverent, dedicated fully to the art of self-decoration and essay by ruskin the realization of the Individual. The brand personality is cosmopolitan, innovative, irreverent, Original, which has almost created a Cult like following. Loved by women of all ages, it is one of the few fashion brands which truly transcend age. (Viral Marketing, November 9, 2007). Essay Inventions? MAC fits into the classification of work essay by ruskin consumer products known as “shopping products”. Dissertation? A shopping product is work essay by ruskin usually more expensive than a convenience product and is found in fewer stores. Consumers usually buy a shopping product only after comparing several brands or stores on & theses, style, practicality, price and lifestyle compar-ability. Need essay sample on Marketing Paper – MAC Cosmetics – Product #038; Place Strategy ? We will write a custom essay sample specifically for essay, you for only $12.90/page. They are willing to invest some effort into this process to a voice on the art of reading, get the desired benefits.
There are two types of shopping products: homogenous and heterogeneous. By Ruskin? MAC is a perceived as a heterogeneous product because consumers perceive it as being essentially different – for example, furniture, housing, clothing and universities. ( MKTG 200 – Student Edition, 2013. Lamb/Hair/McDaniel) As one of the 25 individual cosmetic brands that are owned by Estee Lauder cosmetics, MAC varies in use and performance and shakespeare and juliet conflict essay meets the needs of specific market of essay by ruskin women with its products. Essay Inventions? (MKTG 200 – Student Edition, 2013. Lamb/Hair/McDaneil). From the very beginning the marketing strategy was to created makeup for makeup artists for fashion, movies and theatre. Endorsement through word of mouth was generated by giving away cosmetics to makeup artists who worked with top models and work by ruskin actresses in the early years of the Franks venture.
The early adopters are still what gives MAC its solid foundation in the industry. As well over the years, MAC has built a reputation for itself by unconventional and innovative marketing, breaking away from the safe and obvious. This is where they captured the industry professional as well as the upwardly mobile woman who loves fashion and looking good. Through the made chemist maturity stage, MAC has transitions beautifully and marketing its brand where it is accepted and loved by women from essay by ruskin all social and economic classes. They have a keen sensitivity to pop culture and helps to guide what they do. Since its philosophy is to create makeup for all ages, sexes and races, it has never shied away from affiliations with controversial artists and personalities that are known the world over introduction, (Love to Know Makeup. Michalak, Jodie. Work Essay By Ruskin? www.lovetoknowmakeup.com). MAC is currently in its maturity stage.
It is a well received brand that is a voice the attic essays on the currently expanding its product presence into other countries outside of the United States. Essay By Ruskin? Many of these countries are impoverished, such as Nigeria, but it has not stopped MAC from dissertation & theses going against what is considered to be normal to create new clients and expand its market. Because the North America market has been struggling, they felt it was wise to take chance out of the country to capture new markets (Coleman-Lochner, Lauren. 2013). Paper #3 – Place Strategy. Place, or distribution, strategies are concerned with making products availanle when and where customers want them. A part of this P – place – is physical distribution, which involves all the work by ruskin business activities concerned with storing and transporting raw materials or finished products. Shakespeare? The goal is to make sure products arrive in usable condition at designated places when needed. (MKTG 200 – Student Edition, 2013. Lamb/Hair/McDaniel). MAC uses distinct placement strategies to distribute its product.
MAC Cosmetics intensive distribution takes place at mainly all MAC Pro stores, major department stores or international perfumeries, (Nordstorm’s, Dillard’s, Macy’s, Bloomingdales, Neiman Marcus). Essay By Ruskin? This allows them maximum coverage and keeps the product available in every outlet where the potential customers might want to buy it. If buyers are unwilling to search for a product, it’s very accessible. Dissertation? They accomplished this by placing it in essay by ruskin, the types of dissertation paper stores that their target market frequents. (MKTG 200 – Student Edition, 2013. Lamb/Hair/McDaniel). The only difference between the MAC Pro Stores and the department store counters is the variety of products that are available. Because MAC was first created to by ruskin, cater to word count for essay introduction, the needs of the make-up industry professionals, it’s lines within the PRO stores cater to those needs. There you will find, pigments, palettes, some primers, air brush machines along with old and work new limited edition items that are specific to word count introduction, their store and work essay by ruskin will not be found at the department store counters. This is the only distribution that MAC uses. They do not use wholesalers, and although many inquire about becoming a wholesalers, they currently only essay chemist use the aforementioned type of placement for their products. ( MAC Cosmetics Frequently Asked Questions. www.maccosmetics.com) The retail classifications that MAC uses are department and work essay specialty stores.
The department stores carry a wide variety of word for essay shopping and specialty goods, including apparel, cosmetics, housewares, electronics and sometime furniture. Macy’s, Nordstrom’s, Neiman Marcus, and some of the larger US department stores that carry MAC. On an work, international level they are carried in perfumeries. From? (McKelney, Lauren. www.wordpress.com, April 30, 2012. MAC Promotion Proposal) (MKTG – 200, 2013. Student Edition. Lamb/Hair/McDaniel) After much research I was unable to find the shipping information for MAC. I do know that currently they are unable to work, ship cosmetics out of the United States to foreign countries and they ask that clients go to the stores nearest to them in their country. Outside of the stores, MAC has a strong on-line presence with clients purchasing products. You can ship same day, next day, two day or just standard mail which takes up to shakespeare romeo and juliet, 5-7 business days.
These products are all shipped within the US. The cost of the shipping depends on how soon you want your product to by ruskin, arrive and group can range in price from work essay $5.00 to $17.00 for product ranging in price from $0-$50 USA dollars. A standard package over $50 US dollars is free and is minimal cost for same and next day service. (MAC Cosmetics. Shipping Handling Fee, 2013. www.maccosmetics.com) MAC ‘s distribution process has worked very well for the company over its 29 year existence. Customer service is there #1 concern, and not just selling the products to women, but making sure they. know how to dissertation, confidently wear them, and work wear them well.
The make-up artist at the counter is conflict their most valuable asset, the president of the company will brag. It is by ruskin there one on one approach to the customer that has made it a company that is demanding to be noticed both in and out thesis group, of the essay by ruskin United States. (Estee Lauder News. Company Newsletter, 2013. www.esteelaudernews.com) 1. McKelney, Lauren. www.wordpress.com. April 30, 2012.
MAC Promotion Proposal. 2. Belacel, Amel. ISSUU. MAC Cosmetics Brand Analysis. 3. Viral Marketing. www.viralmarketing.com. On Tata Group? November 9, 2007 4. MKTG 200. Student Edition, 2013. Lamb/Hair/McDaniel. Work? Page 214, 234-235. 5. Michalak, Jody.
Love to word count for essay, Know Makeup (website) Company History MAC Cosmetics. 6. Wikipedia, the free encyclopedia. MAC Cosmetics. 7. Work By Ruskin? Coleman-Lochner, Lauren. (March 21, 2013) Bloomberg Businessweek. A Voice From Essays On The Reading? With MAC, Estee Lauder Enters Emerging Markets 8. Estee Lauder News, 2013. Company Newsletter. www.esteelauder.com. MKTG 200 – Papers 2 3. Professor Rich Lewis. September 24, 2013.
1. MAC is sold in 66 countries across the globe in 1000 outlets. 2. Work By Ruskin? MAC’s founder says “I don’t have the from the attic art of luxury of communicating with my clients, so therefore my salespeople are the link between my philosophy and their customers.” This is the reason why MAC assistants on shop counters are trained to by ruskin, advice customers on makeup techniques, formulas, and thesis on tata latest trends. 3. MAC is present in 53 stores across UK out of work by ruskin which 8 are MAC’s flagship stores proving the fact that the brand pull is essay inventions so strong that consumers. flock these few stores to buy MAC cosmetics giving it a cult like following. So when a mass market brand like L’Oreal or a premium brand like Clinique spend millions on work essay by ruskin, advertising, a brand like MAC does it all by inspiring its stakeholders in non intrusive and subtle ways without spending a fortune on advertising. In 1994 MAC started retail industry’s first charity called MAC AIDS fund. In 1994, Mac was one of the word count for essay introduction first to work essay by ruskin, bring charity to retail sector.
They started MAC AIDS fund for men, women and children affected by AIDS. The same year they launched VIVA GLAM lipstick. MAC decided that every cent of the selling price of the VIVA GLAM lipsticks would go to the M·A·C AIDS Fund. Dissertation? VIVA GLAM’s first lipstick was red in colour which symbolizes the fight against AIDS/HIV throughout the work by ruskin world. With a total of six shades of count introduction Viva Glam lipstick and work essay two shades of Viva Glam ‘lipglass’ now sold worldwide, and through the annual Kids Helping Kids Card Program, M·A·C Cosmetics has provided over $95 million (US) to word count, date for the M·A·C AIDS Fund.
MAC pays for the ads and its production, the packaging, the shipping and of course their lost profit margin out of their own pocket. It worked wonders for essay by ruskin, the MAC brand as the offer was refreshingly straightforward “Every cent of the selling price of MAC Viva Glam Lipstick and group Lipglass is donated to the MAC AIDs Fund to support men, women and children living with HIV and AIDs.” MAC holds Viva Glam gala and fundraiser annually to raise funds for its charity and VIVA GLAM lipstick has become one of the fashion world’s most glamorous and successful fundraising initiatives of all time. From the essay beginning the brand has targeted the premium segments with its pricing. Regular introduction of new trendsetting ranges and colours kept the excitement alive like the latest one is shakespeare romeo ‘Antiquitease’ inspired by antique look. MAC has done some Periodic ads featuring unlikely celebrities such as flamboyant entertainer RuPaul and Elton John. In 1995,RuPaul was signed to a modelling contract for MAC cosmetics, making him the work by ruskin first drag queen supermodel.
M.A.C has never shied away from affiliations with controversial artists. PR strategies used by MAC: Apart from traditional media, MAC regularly gives out new launched products to be reviewed by various beauty websites and beauty blog owners which is one very powerful SEO tool and increase the credibility of the brand. A few examples are as below: Off late MAC has started heavy usage of PPC advertising. Search for cosmetics or any beauty related term on the Internet and you can see MAC all over the Internet. Marketing Tactics used by MAC over dissertation length, time:
1. Halloween: 25 different looks for essay, Halloween—from ghoulish to feminine to shakespeare conflict essay, downright scary— highlighted on its Web site,Maccosmetics.com and at all M.A.C. counters. 2. Work Essay By Ruskin? Back to the attic art of reading, MAC: “Because we share your commitment to work by ruskin, the environment, M·A·C accepts returns of its primary packaging through the dissertation Back to M·A·C Program. By returning six M·A·C primary packaging containers to work essay by ruskin, a M·A·C counter or M·A·C Cosmetics Online, you receive a free M·A·C Lipstick of your choice as our thanks to you. 3. Barbie colours cosmetics line of cosmetics: MAC has teamed up with Mattel to launch a Barbie line. The line is intended for adults, and comes with its own limited-edition MAC-themed Barbie doll. 4. Temporary websites dedicated to a particular new launch like http://www.barbielovesmac.com/
5. BoomBox – BoomBox is London’s first free for all Fashion Club in East London. BoomBox isn’t serious, posey or full of egos; it’s fun, inclusive and unpretentious club unlike fashion which is length meant to be exclusive. The fashion ethos at BoomBox is simple: if all else is work essay by ruskin luxury brands, don’t try and compete – be different, be brilliant on your own terms, create something out of nothing and feed off the sense of community surrounding you. Its frequented by fashion celebrities and is said to have revived London’s fashion scene and London Fashion Week. MAC has tied up with it in smart way by selling MAC and inventions chemist BoomBox co-branded T-Shirts. Team 4: MAC Cosmetics. MDSE 3850 Promotion of Apparel and essay Home Furnishings. Promotion Campaign Report. 1. Executive Summary: This promotion plan for MAC Cosmetics seeks to improve the innovative promotions that MAC is thesis on tata group already implementing and add new ideas to the mix. Work By Ruskin? The overall goal of the plan is to get MAC to its full potential through the use of inventions made chemist fresh advertising and marketing techniques that will fully showcase the company’s many unseen strengths.
The MAC brand has many different opportunities in work essay, its current field and target market and count for essay introduction can be further pushed to its entire capability to achieve a wider range of consumers. Other opportunities for MAC include using new forms of essay by ruskin advertising to from art of reading, reach new cohorts of work essay consumers, showing how MAC can transition from a daytime look to a nighttime look, and, of course, implementing most of MAC’s current promotions in a new and word count introduction improved way. By Ruskin? MAC’s new promotion plan will use its current strengths to its advantage and help redirect the brand so the strengths do not pose any sort of threat to its overall success. Essay Inventions Made Chemist? MAC’s competitors range from drug store makeup brands to higher end, more luxurious brands. Work Essay? MAC’s position relative to its competitors is extremely important to the future of the company and its promotions. MAC has always been seen as a more expensive and group more avant-garde brand. Through this new promotion plan, the work company will be repositioned to reach more consumers that might have been scared to shakespeare romeo and juliet, try MAC products due to its perceived extravagance. Work Essay? The target market for MAC Cosmetics will experience some changes with this new promotion plan as we strive to include a more conservative, mature consumer.
While redirecting this brand, the plan will work to keep the original target market by simply using tactics to appeal to dissertation paper length, a broader range of individuals. The new promotion plan for MAC will focus on print advertising, direct mail, and interactive advertising. The plan is to focus on MAC’s current promotions and further develop these ideas into successful marketing tactics. This new plan will add a new look to work essay by ruskin, the print advertisements that MAC is already executing, send out direct mail-outs to customers to develop more brand loyalty and appeal to new consumers, and introduce the concept of interactive media to on tata group, the brand through smart phone applications, interactive store kiosks, and mall events. 2. Preliminary Research: 2.1 Current Marketing Plan. Frank Toskan founded MAC Cosmetics in 1984 (“The MAC Cosmetic,” 2007). Essay? He was a photographer from Canada who wanted to invent a cosmetic product that was more artistic and achieved the dissertation right look for stage lighting conditions. The cosmetics company is one of the work essay few brands on the market that has colors for all skin tones.
MAC has created over 160 lipsticks and 150 eye shadows to date, while also creating limited edition lines (“The History behind,” 2006). According to Viral Marketing “The Brand personality is cosmopolitan, innovative, irreverent, original, which has almost created a Cult like following. Loved by dissertation, women of all ages, it is one of the few fashion brands which truly transcend age (“The MAC Cosmetic,” 2007).” MAC’s current promotion plan is to portray this personality to their niche market through word of mouth, which Toskan did so by giving the work essay products to top notch make up artist who work with models and actresses. Through this MAC has built a well-known reputation for shakespeare and juliet, itself with using very little advertising. In 1994, once MAC’s name had become better known, MAC started a new charitable promotion plan, the work essay by ruskin MAC AIDS Fund. MAC was on of the first to bring charity to the retail sector.
MAC started this with their new line VIVA LA GLAM, which was a lipstick where every cent went towards men, women and children fighting against AIDS. The first lipstick was the color red, which symbolizes the fight against AIDS throughout the world. MAC has always used celebrity endorsements to advertise for made chemist, VIVA LA GLAM. Currently they have been using Lady GAGA for their advertisement, GAGA is a women who has struggled with her sexuality, hence why she is the perfect person to represent for the MAC AIDS Fund. MAC has provided $128 million dollars towards the work essay by ruskin MAC AIDS Fund, while paying for their own advertisements, packaging, and shipping out of their own pocket (“The History behind,” 2006). MAC also gives back to the community by being environmentally friendly through their BACK 2 MAC campaign. The BACK 2 MAC campaign helps the environment by encouraging consumers to recycle their old cosmetic containers. When a consumer returns six of their old MAC containers they will receive a free lipstick of reading their choice.
The BACK 2 MAC campaign is not only helping out the environment but also giving their customers an opportunity to try out a product that they might not necessarily buy with their own money. MAC is sold in 66 countries around the world and in 1,000 outlets (“The MAC Cosmetic,” 2007). MAC is considered a high quality cosmetic company, selling their products in a higher price range. With their average pricing ranging anywhere from fifteen to twenty dollars for essay, their basic make up, and for a more high end products it can price much more than that. MAC products are often sold in more high-end malls generally having their own stores. If you were to see MAC in a department store it would be a more luxurious department store such as Nordstrom and Dillard’s.
2.2 Target Market Analysis. Since MAC was founded as a product made for an artistic community based off of stage lighting, it is geared towards more of a niche market segment. MAC is a cosmetic company meant for all ages, sexes, and races. Dissertation Paper Length? But when said that it is meant towards all ages, sexes, and races it is not always meant for your average everyday makeup. It is targeted to the more artistic person, who is by ruskin daring to be different and innovative. As stated earlier in the paper, MAC is a more “cult” like culture. Their products are more vibrant and for essay introduction out there, which is meant for a very small segment in work essay, the cosmetics market. MAC doesn’t attract the consumer who is looking for cosmetics that they could wear to word count for essay, work or school, they attract the consumer who is artistic and work essay independent in their life style. While MAC does have it’s charitable organizations through MAC AIDS Fund and BACK 2 MAC to draw in consumers who are willing to give back to their community while also doing something for themselves. MAC doesn’t advertise these organizations well enough to shakespeare romeo conflict, bring in new consumers who are looking for a cosmetic company who gives back, it is more known to their already existing consumers and that is what MAC wants to achieve through this promotional campaign, to broaden MAC’s customers.
They have a strong customer loyalty through their existing consumers, but MAC needs to draw in new consumers. Essay By Ruskin? The way to & theses, do this is to change their promotional strategies and use more than just word-of-mouth. 3. By Ruskin? Situational Analysis. 3.1 Internal Factors. MAC’s current promotion campaign consists of word-of-mouth, magazine ads, online videos, celebrity endorsement and partnerships with other brands. ”M.A.C spends merely $7,795.8M on advertisement each year, which is nothing compared to what other cosmetic companies spend for advertisement, estimated at 2.7 billion yearly (Lootawan, 2010).” MAC relies on word-of-mouth and letting loyal consumers spread their love of MAC Cosmetics. “M.A.C word-of-mouth promotional strategy helps M.A.C cosmetics promote itself by “letting the brand speak for itself. This particular strategy has been used by M.A.C since it was first introduced to famous celebrities, which allowed M.A.C to be exposed to the public and become one of the thesis on tata most famous cosmetic lines today (Lootawan, 2010).” MAC uses celebrity endorsements in almost all of their magazine advertisements. Work? “M.A.C is well-known because of the celebrity exposure it gets from high rank clientele.” Some of MAC’s current celebrity endorsements include Lady Gaga, Boy George, Fergie, Cyndi Lauper and many more (Celebrity Endorsement Ads, 2011).
MAC is very successful at promoting its cosmetics through social media like Facebook, Twitter and romeo Youtube. MAC currently has 2,420,685 Facebook likes and essay 57,535 Twitter followers (Fan Page List, 2011). MAC Cosmetics Youtube channel consists of shakespeare romeo promotional videos featuring celebrities as well as how to videos from MAC makeup artists (MAC Cosmetic’s Channel, 2011). MAC has also had successful promotions through creating product partnerships with other brands such as. Hello Kitty, Barbie, Wonder Woman, Lilly Pulitzer and Alice + Olivia (Temptalia, 2010). SWOT Analysis. MACS current promotion campaign VIVA GLAM has helped MAC AIDS fund raise 202 million dollars since 1994.
MAC promotes its products to work essay by ruskin, all ages, sexes and races by shakespeare conflict, using endorsements from by ruskin different celebrities like Lady Gaga, Boy George, Lil Kim, Pamela Anderson, Johnny Weir, Missy Elliot, Fergie and many more. A Voice From Essays On The Reading? MAC has made itself really well known by using bright flashy advertisements. MAC currently has 2,420,685 Facebook likes and 57,535 Twitter followers (Fan Page List, 2011). Work By Ruskin? WEAKNESSES: MAC only advertises their products online and with magazine ads, not utilizing other forms of promotion. MAC primarily uses extreme makeup in their advertisements, which doesn’t attract a broad audience of people. Paper Length? MAC doesn’t promote their BACK 2 MAC recycling program very well and many consumers don’t know what it is. MAC primarily focuses their promotions on their VIVA GLAM line of products and not all of the work essay other lines they carry. Thesis? OPPORTUNITIES:
By using other forms of advertising, such as direct mail, kiosks and a phone application MAC will reach more consumers. By informing consumers about the BACK 2 MAC program MAC will recycle more used containers and get more consumers shopping in their stores. Work By Ruskin? By showing advertisements with day time makeup looks MAC will attract more conservative women who want to wear their makeup to work and out at night. By using other celebrities like Jennifer Aniston who aren’t as flashy as MACS current endorsers MAC will attract the consumer who wants to wear her makeup a little more modestly. THREATS:
By using multiple advertising outlets MAC could possibly over saturate their consumers with too many advertisements. By showing how MAC makeup can be used for daytime looks consumers may too closely compare MAC to other makeup brands such as Covergirl and Clinique. By providing direct mail coupons customers may only buy MAC products with coupons or during sales promotions and not at full price, which could cost MAC some revenue. While giving free samples and complementary makeovers will draw consumers into MAC they may just take what is free and thesis on tata not make any purchases, loosing revenue for MAC.SWOT Analysis 3.2 External Factors. 188.8.131.52 General Dynamics.
MAC is known for taking a bold approach to cosmetics and not being timid by work, what the standard is to achieve beauty. They were founded as a cosmetic line for word for essay, artists and that still stays true today. Essay By Ruskin? The company’s dynamic reflects a daring side to cosmetics and allows make-up artists to be free of creativity and ideas. MAC has been very successful in essay, producing a quality brand but with such an essay by ruskin, extreme dynamic look it can shy off the everyday consumer. The industry of cosmetics is various in all different types of products while marketing to all different types of consumers. Instead of taking the “matchy-matchy” approach that most cosmetic company’s do with matching the right eye shadow with the appropriate eye color, MAC used a more untraditional route (Harris, 2011). 184.108.40.206 Consumer Behavior. This industry is all about essay made, enhancing and beauty with products that make the skin more even, the eyes more radiant and the lips more luscious.
Although MAC offers products that do what all the traditional make up lines do, MAC takes the by ruskin meaning of dissertation beauty and mixes it with art. They advertise these traditional products in a unique way by favoring the art side by adding words like “studio” and “glamour” into their product names (Harris, 2011). Although MAC has a solid product it is seen that a softer approach would be in their benefit. Essay By Ruskin? A softer approach will draw in the everyday consumer instead of having the inventions made feeling of intimidation. The consumer demands that their cosmetics boost their self-esteem and enhance their feature to help them feel more beautiful.
In the narrow product category of art make up there are changes that could be an advantage to companies with this category. When advertising art make up it creates a very small target market and essay can eliminate possible sales. MAC should try and shy away from a narrow market and essay made market to a more general audience. Instead of using gallant ad campaigns with bold celebrities, maybe consider putting an average consumer in an ad to show the versatility of the products (Irosado, 2009).
As it could appear MAC could use different promotion strategies to achieve this goal of relating more to by ruskin, the traditional consumer. The promotions that they have now are good ideas but not implemented well enough to increase revenue. And Juliet? 3.2.2 Competitors. The competitive situation of MAC is that they are limiting themselves and not using all the resources, opportunities, and work essay by ruskin marketing strategies to the best of the company’s ability. The competitors are higher priced cosmetics that are also known for essay, a great quality and have the approachableness that MAC does not. Because this industry will always be in work essay by ruskin, high demand and new products are coming out daily to fix the new thing that has become ugly, this segment is very competitive. MAC’s competitors are doing well but may be looking for a change as well. 4. Communication Process. 4.1 Promotional Objectives.
The overall goal of the promotion objectives is to increase revenue by expanding the current target market while reaching out to a new consumer with new marketing tactics. The promotional campaign intends to thesis on tata, add marketing by adding social media, interactive advertising, and connecting with the everyday consumer more efficiently. The goal is by the end of essay by ruskin 2012 MAC has executed all the shakespeare essay marketing efforts that are being created with this plan and to see results and feedback by the end of 2012. The plan is to work essay by ruskin, execute brand awareness by the consumer of all the dissertation current promotion strategies like Back 2 MAC, AIDS charity, and the makeovers offered with a product purchase of fifty dollars. To implement this plan MAC must first launch the new print ads and to send out the direct mail created to highlight these promotions. The rest of the promotion mixes will begin and to start measuring results. Results could vary by work by ruskin, seeing if there are more conservative women walking in asking about the products to an increase in consumer wanting to use the back to MAC to their advantage.
We plan to benchmark our performance by comparing sales to previous years and seeing how he consumers respond to word introduction, all the new marketing efforts. It is possible to achieve this in various ways with the promotion plan. Measurements will be taken with how many consumers come in talking about or using the direct mail to their advantage. By benchmarking the progress MAC is able to measure the work by ruskin results in an accurate way and see tangible improvement. 4.2 Positioning Strategy. The MAC Cosmetics brand was founded with the intentions to market towards makeup artists in the fashion, theatre, and romeo and juliet conflict movie industries (“The MAC Cosmetics Story,” 2007). In more recent years MAC has expanded to more of a mass market but still focuses on work essay, the original extravagance and innovative looks that it began with. Count? MAC has generally focused on a price-quality positioning strategy. Work Essay? Through this strategy, MAC keeps product prices in a higher range to highlight the superior quality that the products offered are believed to romeo and juliet conflict, possess (Roy, 2010). Work? Although this positioning strategy seems to work well with MAC’s minimal marketing and advertising strategy that depends widely on word-of-mouth communication, there are some changes that would greatly increase the success that MAC is already experiencing. If MAC were to change their positioning strategy to focus not only on its products’ superior quality through higher prices, but also on the many features and benefits offered, the romeo and juliet conflict company would be much more well-known and successful (Roy, 2010).
With the use of positioning strategies that showcase product features and by ruskin benefits, there will be much less dependence on word-of-mouth since consumers will already be aware of the many wonderful programs and special features that MAC has to inventions made chemist, offer. Through MAC’s many unique aspects and new promotion mix ideas, the company will be much more apparent and appealing in work, the consumer’s mind. Instead of just hearing about how great MAC’s makeup products are, the essay inventions made consumer will actually be able to see, through increased promotion efforts and unique advertising, how MAC truly is superior in by ruskin, product quality, despite its higher prices. The repositioning of MAC will also focus on making the brand appeal to a broader market by showing how the products offered can be worn as day-to-day makeup and do not always have to be seen as extravagant. There are several possible risks involved with changing MAC’s positioning strategy and pushing it to appeal to a more conservative consumer. Paper? As the work essay brand seems to go more mainstream and introduction become more visible to the public eye, it could lose some support from its original market. Work Essay? The new positioning strategy must take precautions as to not lose the from reading original market by keeping MAC’s initial image in tact by simply featuring what MAC already has to offer in new and innovative ways.
This shift of positioning strategy to highlight MAC’s best product features and benefits is the best move for MAC to become more prominent in the cosmetic market and is necessary for the growing success of the brand. Work By Ruskin? The MAC brand has typically been positioned as a more expensive and extremely avant-garde cosmetic company in relation to its competitors. Some of the main competitors that MAC faces include Sephora, Urban Decay, Lancome, and essay inventions made many others that are featured in the positioning map below (“JGlamour Makeup Brushes,” 2009). Currently MAC. seems to fall in essay by ruskin, the higher end as far as pricing goes and extremely avant-garde in style (Belacel, 2010). The consumer has generally seen MAC products as too extravagant and on tata group too innovative when it comes to style, colors, and the overall look the makeup usually is shown to create. With new positioning strategies in work, place, the MAC brand will seek to a voice from the attic on the art of reading, appeal to more conservative consumers by showing the essay market the wide range of romeo and juliet conflict looks MAC makeup products are capable of without losing the original market’s interest. The shift in essay by ruskin, positioning strategies will leave MAC in the same place as far as quality and price go in relation to competitors, but will attempt to cover a wider range of styles to thesis group, appeal to a wider range of consumers while still maintaining the edgy extravagance that captured the loyalty of the work essay original market. Chemist? 4.2.2 Target Market.
MAC will continue to target towards their current market along with expanding into a new market. The target consumer for the new MAC promotion campaign is the essay female of all races within Generation X. The target consumer ages between 30 and 50, born between 1961 and 1981 and estimated at & theses 50 million (Dawson, 2011). These individuals grew up in tough economic times mainly in two-income families. Essay By Ruskin? However, during the Gen Xers childhood, parents started to divorce and the women of the dissertation paper household began to work essay, join the word count introduction workforce. Given that, Gen Xers have put more effort into their families thus succeeding in keeping them together. Seventy-one percent of Gen Xers have children, and have developed a middle ground into how they have chosen to raise them (Dawson, 2011). As parents, this generation is highly involved in their children’s lives and over 80% expect their children to earn at essay by ruskin least a baccalaureate degree (Dawson, 2011). The shift in economy has converted from manufacturing to servicing, which gave Gen Xers the ability to shakespeare romeo, adapt to changes and integrate new technologies.
Technology has been incorporated into their lives since their youth. This generation is comfortable using PDAs, cell phones, e-mail, laptops, Blackberrys and other technology employed in the legal workplace (Kane, n.d.). The Gen Xers have adapted into independent, resourceful and self-sufficient individuals. Work Essay? Gen Xers are a rather educated and sophisticated group, more than 60% of the population has attended higher education institutions (Lyon, 2010). Gen Xers are very determined and enthusiastic about learning new trades and skills. The mentality of this generation is much different than other generations, they work to live rather than live to work. A Voice On The Art Of Reading? These work hard, play hard individuals usually complete tasks within workplace single-handedly. Work By Ruskin? While 60% of the dissertation current workforce is work essay occupied by this generation (Collier, 2009), they are seen as devoting long hours to their careers and continuing their education. Within Gen X’s population, 86% is employed, 70% devote 40 or more hours to work each week and those with a doctorate or professional degree devote 50 or more hours;
79% of the Gen X women work (Dawson, 2011). Gen Xers posses an entrepreneurial spirit with immense leadership qualities. Already in CEO or management positions within financial services, communications and retail, these individuals are seen to a voice essays on the, be in the best position within the job market and will inherit Baby Boomers positions once they retire. Work? In 2005, the a voice from the attic on the wealthy members of Gen X have an essay, average income of $213,000 and spend 18% or more than Baby Boomers (20%) on luxury goods (American Express, 2005). & Theses? Studies have shown that 70% of women won’t leave the house without applying something to by ruskin, her face, with 68% saying cosmetics make them feel more confident (Douglas, 2010). Word For Essay? Generation X spends an work essay by ruskin, average of $3,235 a year on fragrance, cosmetics and beauty products, 60% more than Baby Boomers (American Express, 2005).
Pam Danziger, founder of Unity Marketing explains, “Marketers tend to think of the Gen Xers as a cohort too small to shakespeare romeo, even worry about, but this research shows that Gen X may be small in size but they are mighty in luxury buying power,” therefore “ they are in a more youthful life stage and are accumulating a lot of big material goods… influence is destined to grow even more in the luxury market in the future…” (American Express, 2005). Moreover, the affluent Gen Xers earn an average of $152,500 per year (American Express, 2005). Work Essay? In general, Gen Xers are luxury consumers and indulge in luxury experiences, yet the wealthy spends about three times more than the affluent. Differences amongst the purchase behavior of the two social classes are that the affluent like to find deals on luxury goods with the sense of being a smart shopper. & Theses? Luxury goods entice Gen Xers, they take value and quality into consideration first rather than the latest trend. They’re extremely skeptical when it comes to advertisements, flashiness does not attract them into making a purchase. Work Essay By Ruskin? As the Internet emerged into the Gen Xers society they became the first mass-media generation. The Nielson Company has averaged in 2009 that global consumers spent more than 5.5 hours on social networking sites with Generation X being one of the main users (Porterfield, 2010). Adults ages 30 and over are involved in social networking at 40% much less when compared to 72% of young adults.
Gen X moms favor Facebook while 44% use social media for word-of-mouth recommendations on on tata, brands and products along with 73% feel they find trustworthy information about work essay by ruskin, products and shakespeare romeo conflict essay services through niche online communities (Porterfield, 2010). Essay By Ruskin? Two-thirds of Facebook moms are a fan of one company or brand therefore Facebook is a fertile ground for marketers to engage mothers and drive sales (Porterfield, 2010). Traditional media referencing TV, Internet, Newspapers, Magazines and cell phones reach 46% to 86% of Generation X (Continuum Crew, 2010 ). Moms within Gen X use the Internet for purchases, online research, sales sites and essay inventions made Groupon, which are amongst the most popular methods used to find great deals. Over half of these tech-savvy consumers own a Smartphone in which generates research and shopping. Application downloads on Smartphones have increased, users spend roughly 80 minutes per by ruskin day using apps and download at minimum nine new apps per month (AdMob, 2010). Length? 4.3 Brand Communication. To communicate efficiently to by ruskin, Gen Xers, MAC will implement tactics such as word-of-mouth and social media.
MAC will deliver messages that are relevant to the new target market with straightforward information. MAC will sign new celebrity endorsers that have a positive sophisticated image. It is found that on on tata, average, ads that contain a celebrity endorser produced 9.4 % higher consumer readership than ads without a celebrity endorser (Dolliver, 2011). As all technological spaces are emerging, MAC will be repositioning its brand via interactive advertisements and Smartphone apps. MAC will cater to Gen Xers values of work genuine promotional strategies. 5. Promotion Mix.
MAC’s new promotional campaign will use three forms of advertising mediums including print, direct and interactive. MAC’s new promotions will include expanding MAC’s current use of print media by introducing new magazine advertisements. Inventions Made? MAC will also begin to use direct media by sending out many different direct mail-outs to essay, customers. Lastly MAC will introduce the use of interactive media to its promotional campaign thru many different new advertising methods. MAC will introduce a new MAC Cosmetics phone application as well as a new MAC in store kiosk. MAC will also host interactive mall events to further promote their make-up to all consumers. MAC’s new promotion campaign will include interactive media with a new phone application that will make the MAC make-up world more accessible to all consumers who own Smartphone’s.
With this application consumers will be able to dissertation, purchase MAC products from the palm of their hand. The MAC Cosmetics application will be full of information for consumers to learn more about MAC Cosmetics as well as make purchases. The MAC phone application will feature icons for the MAC shop, new products, today’s obsession, what’s hot now, and a shopping cart. The icon for work essay by ruskin, today’s obsession will have daily tips from MAC makeup artists on what product they are obsessing over essay made chemist, that day and videos about new ways to wear MAC makeup. The what’s hot now icon on MAC’s phone application will feature all of MAC’s best selling products with a comment box for consumers to write what they think of MAC products. With this feature consumers will be able to explore MAC products and see what their fellow consumers like. The what’s new icon on the MAC phone application will present all of MAC’s new products as well as any new MAC collaborations with other brands, like Barbie and Hello Kitty. The MAC shop icon will be where consumers can search through all of the work essay by ruskin MAC products by type of cosmetic and color. Once a consumer has chosen what makeup products they want to shakespeare essay, purchase they will then go to work essay by ruskin, the shopping cart icon where consumers can purchase their products to be shipped to their home with a credit or debit card. The MAC application will be updated daily with today’s obsession and anytime something new is thesis on tata being introduced. Work By Ruskin? By creating a MAC phone application consumers will be able to shop whenever and wherever they are.
MAC’s new promotion campaign will also include interactive media thru a new MAC kiosk located in MAC stand-alone stores as well as in MAC locations in inventions made, department stores. Work Essay By Ruskin? These touch screen kiosks will allow consumers to be interactive with MAC cosmetics and explore all that MAC has to offer. With the paper MAC interactive kiosk consumers will be able to walk into a MAC store and go up to the kiosk and browse through the MAC merchandise. When a consumer finds a product they like they will be able to click on it and be taken to essay by ruskin, a page that will contain a full explanation about that MAC product, photos of different ways that specific product can be used, and other complementary products that go along with the selected product. If a consumer decides that they want to purchase that product they can click on the buy now button and the kiosk will inform the consumers if that product is available in from the attic essays art of reading, the store. If that product is not available in the store the work consumer can purchase the product right from the kiosk with complementary shipping from MAC. If a consumer wants to receive more information about a product or test the product all they will do is thesis click the button located right next to the buy now button, labeled try me.
By clicking this a MAC consultant will be notified and be able to assist the consumer with information about the specific MAC product or try the essay product on the consumer. If it is a product that is unavailable in that specific MAC store the consultant will be able to show the consumer different products that are similar to dissertation & theses, the one the consumer has selected. The MAC kiosk will be a great tool to use especially when the store is very busy. A consumer will not feel like they are just standing around waiting for a MAC employee, but they will be able to by ruskin, do research on these kiosks and thesis find exactly what they are wanting. These kiosks will always be kept up to date with current and new MAC products. Work Essay By Ruskin? By having MAC kiosks more consumers will be able to art of, be informed about work essay, all MAC has to offer. Dissertation & Theses? In this new promotional campaign MAC Cosmetics will also expand on their use of interactive media by hosting many different mall events throughout the United States. These events will feature complimentary makeovers, free samples, live models and a chance for customers to take pictures as a model. These events will be a great way for customers to be able to learn more about MAC cosmetics in a fun and interactive way. MAC will send out emails to all existing customers in the area informing them about the work essay mall event. MAC will also have signs on a voice the attic essays, mall entrance doors as well as fun music playing to draw in curious people walking by.
At these mall events MAC will give free makeovers to any consumer who would like to participate. After they receive their makeovers customers will be able to participate in a MAC photo shoot, where the customers will take a picture and work by ruskin receive an online code where they can upload their picture for free. Word Count For Essay Introduction? Make consultants will also be passing out small free samples to consumers so they can get a feel of work how great MAC make-up is. MAC will also have hired models walking around and dissertation length posing with various makeovers to show consumers everything they can do with MAC make-up. By hosting these mall events MAC will introduce their make-up to new consumers and MAC will get more customers in the store and work by ruskin increase revenue. These mall events will be hosted throughout the year at major malls such as North Park Center in group, Dallas, Texas, Mall of America in Bloomington, Minnesota, Glendale Galleria in Glendale, California and many more. About six to eight MAC mall events will be held throughout the year in high traffic malls where MAC stores are located. Work? By having MAC mall events MAC will be able to introduce their products to a new consumer in a fun and inventions chemist interactive way.
MAC’s new promotion campaign will also consist of direct media with many new direct mail-outs to send to MAC customers. These direct mail-outs will be sent to all loyal MAC customers who already have a history of purchasing MAC products. This concept will directly let MAC choose who receives which offers, and pinpoint the work essay customers within different groups desired to target. Dissertation Paper? Similar to work essay by ruskin, the direct mail-outs Victoria’s Secret sends out to their loyal angel card holders, the MAC direct mail-outs will give consumers the opportunity to learn about all MAC promotions and sales, as well as any new products. The Attic On The Art Of Reading? The MAC direct mail-outs will showcase the latest trends and colors for the season and potentially drive the customer to shop online or in the store. MAC will send out direct mail-outs presenting the newest MAC products. For example the by ruskin new color tones for winter 2011 or Summer 2012. These direct mail-outs will feature the new color palettes for each season. These direct mail-outs will also feature new products introduced to MAC cosmetics; for for essay, example a new type of foundation or powder launched at MAC. Another direct mail-out MAC will send to its customers is one about the BACK 2 MAC program.
Every six months MAC will mail out work essay, information about the BACK 2 MAC program to remind customers to save up their old containers to be recycled and a voice the attic essays on the bring them back to the MAC store to receive a free MAC lipstick. Essay? A third direct mail-out to be sent out to MAC customers is one to dissertation, advertise MAC sales or promotions. Special offers such as free shipping on orders over a hundred dollars, free cosmetic bag with purchase of a new product, and coupons such as ten dollars off of fifty dollars or twenty dollars off of one hundred dollars. All of these direct mail-outs will drive the consumers in to the MAC store to make purchases or to by ruskin, shop online. The last form of dissertation & theses direct mail-outs to work essay, be sent to MAC customers is one informing consumers about MAC’s support for AIDS, the & theses MAC AIDS Fund.
This advertisement will inform consumers about how MAC donates the entire selling price from its VIVA GLAM lipstick and lipglass to the MAC AIDS Fund. This will reach a wide audience of people and increase sales of the MAC VIVA GLAM lipstick and lipglass that benefit AIDS patients. By informing consumers about the great organizations that MAC benefits more consumers will want to work, purchase MAC products. MAC will send out one direct mail-out each month to loyal customers. MAC will alternate what they are advertising for each month between new products, BACK 2 MAC, sales and promotions, and the MAC AIDS Fund so that consumers are not overwhelmed with just one advertisement. MAC currently uses print advertisements in their promotion campaign, so for paper length, this new promotion campaign MAC will expand and work essay by ruskin add new print advertisements. MAC will have print advertisements in magazines such as Vogue, Elle, Cosmopolitan and dissertation paper Glamour. This promotion campaign proposes a new use of print advertisements called “I am a MAC Girl.” The “I am a MAC Girl” campaign is a way to work essay by ruskin, promote MAC cosmetics to the fun flirty girl who wants to wear her make-up a little more natural and light. Thesis On Tata? These advertisements will feature fresh-faced celebrities such as Jennifer Aniston, Rachel Bilson and Reese Witherspoon. By using celebrities wearing their make-up a little lighter and more natural MAC will be able to attract a different consumer then they currently attract. MAC currently uses celebrities like Fergie and Lady Gaga wearing heavy make-up that the everyday girl cannot always pull off.
This campaign will keep the MAC’s current advertisements while adding the “I am a MAC Girl” advertisements to essay, attract all different consumers. MAC will switch off each month running their VIVA GLAM advertisements they currently have with the “I am a MAC Girl” advertisements so that the same magazine does not always have the romeo and juliet same advertisements. Work? For example in April Vogue magazine will have a VIVA GLAM advertisement featuring Lady Gaga and then in May Vogue magazine will have an “I am a MAC Girl” advertisement featuring Jennifer Aniston. By using these new print advertisements MAC will be able to target the more conservative women who wants to wear MAC make-up to work while also keeping the advertisements to the extravagant woman. Dissertation Length? As a whole MAC’s use of interactive, print, and direct advertisements will all tie in together and work very cohesively. Work Essay? Consumers will be able to dissertation, see MAC advertisements in multiples places including in stores, in magazines, on line, and in essay by ruskin, their mail. This way MAC consumers will not be overwhelmed with only seeing one form of advertising over and over again. Group? All of the MAC advertisements in all four medias will have a cohesive look using black and work essay by ruskin white with accents of dissertation paper color. By keeping the essay colors consistent through all advertisements the advertisements will be visually appealing to essays on the, all consumers. The MAC promotional campaign will work together to bring in new customers and generate more revenue. 6. Evaluation.
MAC has made a brand image for itself mainly using word-of-mouth, so by bringing in new types of advertising and a new target market the only place it can go is up. This new promotion campaign isn’t changing the product in anyway it is just bringing in a different promotional strategy to reach out to a different target market, which is why it wont hurt the by ruskin company. We are doing so through informational advertising, with promoting BACK 2 MAC and inventions the MAC AIDS Fund more, which also advertises the company with an emotional appeal to those who support these charities. Work Essay? Another way we are using emotional appeal is by bringing in paper, celebrities to the promotion so that MAC consumers can relate to essay, these celebrities or strive to be like them through using MAC products. Once the new promotional strategy has launched these advertisements MAC will test whether or not they are successful through different posttests. MAC will use focus groups and consumer jury to get the best results and feedback. MAC can actually talk to customers or potential customers in the target market and get the true reaction of what they got out of the new promotional strategies.
By receiving feedback from count for essay introduction MAC’s new target market MAC will be able to adjust any advertisements if needed to get the best results possible. 7. By Ruskin? Reference List. Belacel, A. (2010). Paper Length? Brand analysis MAC cosmetics. Issuu, 2, 9-10. Retrieved from http://issuu.com/amelbelacel/docs/brand_report_mac Bobbi Brown pretty powerful 2.0. Essay By Ruskin? (n.d.).
Retrieved fromhttp://www.bobbibrowncosmetics.com/index.tmpl Product index of celebrity endorsements. (2011). Celebrity Endorsement Ads. Retrieved from dissertation length http://www.celebrityendorsementads.com/celebrity-endorsements/product-index/. Collier, E. (2009, July 7). Workplace Warfare: baby boomers, gen X and gen Y. Retrieved November 2011 from Career Faqs:http://www.careerfaqs.com.au/news/news-and-views/workplace-warfare-baby-boomers-gen-x-and-gen-y/ Continuum Crew. (2010 , February 15). Work Essay? Boomers’ media habits shift online, Turn Social. Retrieved November 2011 from Marketing Profs:http://www.marketingprofs.com/charts/2010/3409/boomers-media-habits-shift-online-turn-social Coty, Inc. (n.d.).
Retrieved from http://www.answers.com/topic/coty-inc Dawson, A. (2011, October 26). Studies say generation X is balanced and happy. Retrieved November 2011 from shakespeare CNN Living:http://articles.cnn.com/2011-10-26/living/living_gen-x-satisfied_1_marketing-study-half-of-gen-xers-divorce-rates?_s=PM:LIVING Dolliver, M. (2011, March 10). Do celebrity endorsements help or hurt a brand?Retrieved November 2011 from Yahoo! Advertising:http://www.yadvertisingblog.com/blog/2011/03/10/do-celebrity-endorsements-help-or-hurt-a-brand/ Douglas, J. (2010, March 15). Women spend $13,000 each on work essay by ruskin, makeup. Dissertation? Retrieved November 2011 from Shine Fashion:http://shine.yahoo.com/event/satc/women-spend-13-000-each-on-makeup-1147372 Estee Lauder companies. (n.d.). Retrieved fromhttp://www.answers.com/topic/the-estee-lauder-co Fan Page List, (2011). MAC cosmetics on work by ruskin, Facebook and Twitter. Retrieved fromhttp://fanpagelist.com/user/maccosmetics Harris, G. From Art Of? (2011, July 24). The beauty beat.
Retrieved fromhttp://www.nbcmiami.com/the-scene/shopping/The-Beauty-Beat-724-730-125725463.html Irosado, L. (2009, August). Marketing channels and supple change. Retrieved from http://irosado.blogspot.com/ JGlamour makeup brushes compete with MAC Cosmetics Sephora. (2009, June 03). Retrieved from http://www.prlog.org/10248724-jglamour-makeup-brushes-compete-with-mac-cosmetics-sephora.html Kane, S. (n.d.). Generation X. Work By Ruskin? Retrieved November 2011. from About.com Legal Careers: http://legalcareers.about.com/od/practicetips/a/GenerationX.htm Lancome Paris. (2011). Dissertation? Retrieved from work http://www.lancome-usa.com/ L’Oreal. A Voice From The Attic On The Reading? (n.d.). Retrieved from http://www.loreal.com/dispatch.aspx? Lyon, E. (2010, March 1).
Examining generation X: stats, demographics, segments, predictions. Retrieved November 2011 from Sparxoo:http://sparxoo.com/2010/03/01/examining-generation-x-stats-demographics-segments-predictions/ MAC Cosmetics. (n.d.). Retrieved fromhttp://www.maccosmetics.com/index.tmpl MAC Cosmetic’s Channel. (2011). MAC COSMETICS. By Ruskin? Retrieved fromhttp://www.youtube.com/user/MACcosmeticsblend=1ob=4. M.A.C (Make-up Art Cosmetics). (2010, November 14). Retrieved fromhttp://lootawan081.blogspot.com/ May 2010 mobile Metrics report.
Retrieved (2010, May 30). AdMob. November 2011 from AdMob: http://metrics.admob.com/ Porterfield, A. (2010, March 5). Social media differences among teens, Boomers and from art of reading Moms: New Study Findings. Retrieved November 2011 from work essay by ruskin Social Media Examiner: http://www.socialmediaexaminer.com/social-media-differences-among-teens-boomers-and-moms-new-study-findings/ Procter Gamble. (n.d.). On Tata Group? Retrieved fromhttp://www.answers.com/topic/procter-gamble Revlon. By Ruskin? (2011). Retrieved from http://www.revlon.com/#/2. Rimmel London. Essay Inventions Made Chemist? (2011). Work By Ruskin? Retrieved fromhttp://www.rimmellondon.com/us/en/landingpage/ Roy, P. (2010).
Positioning strategy- different types of product positioning.Marketing and PR, Retrieved from http://paulami-roy.suite101.com/marketing-strategy-different-positioning-tactics-a247638 Sephora. (2011). From On The Reading? Retrieved from http://www.sephora.com/ Temptalia. (2010, January 5). Your guide to upcoming MAC cosmetics collections. By Ruskin? Retrieved from http://www.temptalia.com/your-guide-to-upcoming-mac-cosmetics-collections-2011. The History behind MAC cosmetics. Paper Length? (2006, April 6). Retrieved fromhttp://www.sokissandmakeup.com/the-history-behind-mac-cosmetics-inc/ The MAC cosmetics story. Work? (2007, November 09). Retrieved. from http://marketingdope.blogspot.com/2007/11/mac-cosmetics-story.html The second annual american express platinum luxury survey. Dissertation? (2005, June). American Express.
Retrieved November 2011 from Odyssey Media Group:http://www.odysseymediagroup.com/nan/Editorial-Corporate-Cards.asp?ReportID=109663 Top 10 cosmetic brands. (2011, August 13). Retrieved fromhttp://www.lifestylo.com/2011/08/top-10-cosmetic-brands/ Urban Decay: beauty with an edge. (2010, November 4). 2011). Retrieved from essay by ruskin http://www.urbandecay.com/Lootawan, R. Authors: Hope Hamilton, Danielle Veeder, Emily Diemer, Chelsea Byrd, Lauren McKelvey. “All ages, all races, all sexes, all MAC. ” This is the motto for one of the fastest growing cosmetics brands in the world, MAC Cosmetics. For MAC artists and MAC customers this motto is not just words it is truly what drives the essay inventions made culture of this amazing Company. At any given time you will find grandmothers shopping for color with their granddaughters.
Isn’t that an amazing and work wonderful concept? As you read further you will see why MAC is on tata group such a unique entity in this well established industry. Every aspect of what makes MAC Cosmetics; “MAC” being its products, its location, its brand image, or its phenomenal price point is unique and goes against most typical marketing methods. As we all know product, price, promotion and place are the four fundamental variables that either make or break a company. MAC is currently the only Estee Lauder brand that is by ruskin not only introduction meeting; but exceeding its annual sales goals. Last year MAC surpassed its sales plan by 121 million dollars!
Which is unheard of in this industry. Lets start at the beginning Makeup Art Cosmetics or MAC was created in 1984 by two men, Frank Tosken a seasoned makeup artist and photographer and Frank Angelo a successful business owner of a chain of by ruskin salons in Toronto. They found that cosmetic products that were readily available were not performing well under the made intense environment of studio lights perspiration, had little color payoff, and were constantly having to be touched up. So, they created cosmetic formulas and colors that met these specific needs. By using them backstage on models and performers news traveled fast about how wonderful these products were. At first MAC was only available to the professional makeup artist. Work Essay? Then by shakespeare romeo conflict essay, word of mouth news spread to work by ruskin, the general population and was maid available to the public and in 1994 was purchased by word count for essay, the Estee Lauder who owns many of the world’s favorite cosmetic brands.
MAC continues to be a testimonial brand of work essay cosmetics, meaning that there is no paid advertising for MAC to promote their traditional product. They rely solely on testimonials and word of dissertation paper mouth advertisement. However you may find that MAC does advertise only one product and that is Viva Glam Lipstick. Essay? This is the attic essays their charity lipstick and it comes in five different shades. 100% of the retail selling price of work by ruskin this lipstick goes to men, women and essay children living with or affected by HIV and AIDS. This money goes to living needs and not research. MAC feels that there are enough efforts going towards finding a cure for AIDS, and that providing money to help make someone’s life a little easier makes more of an impact right now. Currently the spokes person for Viva Glam V lipstick is Pamela Anderson, and she has helped us raise 4 million dollars for our MAC Aids Fund. As a whole, Viva Glam Lipstick has raised over 50 million dollars since its inception in 1984 and work essay by ruskin this number continues to climb.
MAC’s image is Sleek, Clean, and straight forward. This is expressed through their packaging, product names, store image and how their employees dress. MAC is romeo essay all about allowing the work by ruskin product to be the focus, so product packaging is simple and made of plastic or glass. There are no fancy shapes or embellishments. This allows for a reasonable price point and for the product not to be overshadowed by opulent, gaudy packaging.
MAC products also have straight forward, logical names. For example eye shadow is called eye shadow, and a voice from art of their matte foundation is called Studio Matte Finish Foundation, instead of something creative like “eye transformer” or “Perfect Finish Foundation” which many other cosmetic companies use to sensationalize a product. Often you will find that MAC products have the word “Studio” or “Pro”, before the actual name of the product. Such as Studio Fix Powder Foundation or Pro Eye makeup Remover. This is done so that customers know that this product was actually used by makeup professionals and used in a studio setting. Work Essay By Ruskin? This means, that these products do what they are meant to do quickly, and essay chemist are easy to use and create a great finish that lasts. MAC wants its customers to be able to work, experience their product line in a clean straightforward way. So, MAC locations are minimalistic, with Marble counter tops, wood or poured cement floors, and open tester units. When a customer walks into a MAC location they tend to be a bit overwhelmed, because all of the products are open and chemist available to experience. That is why the MAC makeup artist is so important.
All MAC employees are professional makeup artists. This means that before they are hired they must have experience doing makeup in a retail environment, or doing makeup as a profession. Essay? That is shakespeare conflict essay why customers have such high expectations when they come to a MAC counter. They know that the experience they’ll have is work essay by ruskin going to be at a higher level since they are dealing with professionals. The MAC image is also communicated to the customer by the way artist dresses for work. A 100% Black attire is required. There is no particular dress code other than that. A Voice From The Attic On The Art Of Reading? An artist is free to express their personal clothing style, as long as it is black. This allows an by ruskin, Artist to express and display their creativity and individuality, and in turn allows the customer to word, get a feel for work essay, the style of a makeup artist. Each customer is drawn to conflict, the MAC artist that represents a look that they can identify with, which is the start of a great customer experience. MAC artists are also required to work essay, wear a minimum of fifteen MAC and only MAC products on their faces every day.
This is great advertisement, and shows the versatility of the products. MAC is considered to from, be a luxury cosmetic line. Channel, Lancome, and Christian Dior are also luxury brands. Their prices tend to be quite high, for example the average price of a foundation from any of these lines is around $40.00. The most expensive MAC foundation is $27.50. They keep their prices down so that everyone can afford the experience of wearing professional products.
The straight forward packaging and the lack of mass advertising and free gift promotions allows them to keep their prices reasonable. So you may ask, why doesn’t MAC have a free gift? Every three to four weeks MAC launches a new color collection. Through research MAC has found that its employees its customer base are drawn to the newest, trendiest products, so by launching new color collections frequently they inspire and excite their employees and appeals to of what their customer base really wants. By doing this MAC has a steady flow of income throughout the year. Most of the other companies have severely high revenue during “free gift” promotions and work by ruskin low revenue points when they don’t. They have trained their customer to thesis, buy when they’re getting something for free, which can be a tough cycle to break.
They constantly have to work by ruskin, come up with bigger and better “gifts” to keep their customers coming back. To reward their customer for being loyal to the brand, and to shakespeare and juliet conflict essay, also help the environment, MAC has a consciousness program called “Back to MAC”. Whenever a customer brings back six empty containers of product, they can choose a gift lipstick of their choice, excluding Viva Glam since that is the charity lipstick. This incentive program allows customers to try a MAC lipstick if they never have, as well as try a new color that they may not usually pick without the essay commitment. To touch the & theses professional customer MAC offers a 40% discount to professional makeup artists, and a 30% discount to industry professionals such as models, actors, new anchors, nail techs, and hair stylists. Remember, MAC was created by industry professionals and so they always keep that customer in mind. Although so many customers are willing to purchase MAC product without a free gift, the work company realizes that a promotional and dissertation paper length limited life items will excite their customer in a similar way. So, seasonally MAC creates limited life products in work essay by ruskin, their color collections, and essay inventions made these products are only available until they sell out. Making such exclusive products creates a sense of urgency in the customer so that they buy now instead of later. One of their most popular promotions happens during the holiday season. MAC launches a series of holiday brush, eye shadow, lipstick, and lip gloss sets at an incredibly low price.
For example a holiday brush set has six travel size brushes in a cute little bag for around $42.00. If you were to purchase these same six brushes at regular price it would cost you around $245.00 and you don’t get a bag. These holiday sets are shipped in limited quantities, and are one of the by ruskin most anticipated launches of the year. Receiving these collections on time is very important to & theses, each location, and work since the MAC customer thrives on & theses, trends timing is everything. MAC was founded in Toronto, and our Factory and Distribution center is there as well.
Through meticulous planning each MAC location receives new color launches about 2 ? 3 weeks before the product goes on counter. MAC doesn’t believe in having a lot of back stock sitting in stock rooms. So it is the responsibility of the Retail Manager to send a stock concerns list to work by ruskin, the buyer of the store and the company ships them more product based on their most popular items and what is put on & theses, the list.
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Action Verbs #8212; By Skills Categories. Remember to use the below action verbs when describing your skills and accomplishments in a resume or cover letterby doing so, you’ll maximize the essay by ruskin, effectiveness of a voice from essays art of your accomplishments, and make potential employers take notice. By Ruskin? Be sure to dissertation & theses, use these action verbs in their proper context. Always use action verbs in the past tense when describing responsibilities and work accomplishments from a previous job . Here’s an example: Trained, led, and supervised a team of content editors in the development, structuring, and writing of online classes. Always use action verbs in the present tense when describing responsibilities and accomplishments in a current job . Remember, these are leads words that start off each descriptive bullet point of dissertation a job description. Train, lead, and supervise a team of content editors in work by ruskin the development, structuring, and writing of thesis on tata group online classes.
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Department of Mathematical Sciences, Unit Catalogue 2003/04. Aims: This course is designed to cater for work essay, first year students with widely different backgrounds in school and college mathematics. Dissertation & Theses? It will treat elementary matters of advanced arithmetic, such as summation formulae for progressions and will deal with matters at a certain level of work essay by ruskin abstraction. This will include the principle of mathematical induction and some of its applications. Complex numbers will be introduced from first principles and developed to a level where special functions of a complex variable can be discussed at an elementary level. Objectives: Students will become proficient in the use of mathematical induction. Also they will have practice in real and complex arithmetic and be familiar with abstract ideas of primes, rationals, integers etc, and their algebraic properties. Calculations using classical circular and hyperbolic trigonometric functions and the complex roots of unity, and their uses, will also become familiar with practice.
Natural numbers, integers, rationals and reals. Highest common factor. Lowest common multiple. Prime numbers, statement of prime decomposition theorem, Euclid's Algorithm. Proofs by induction. Elementary formulae. Polynomials and their manipulation. Finite and infinite APs, GPs.
Binomial polynomials for and juliet conflict essay, positive integer powers and by ruskin, binomial expansions for non-integer powers of a+ b . Finite sums over multiple indices and changing the order of summation. Algebraic and geometric treatment of complex numbers, Argand diagrams, complex roots of unity. Trigonometric, log, exponential and hyperbolic functions of real and complex arguments. Gaussian integers. Trigonometric identities. Polynomial and transcendental equations. MA10002: Functions, differentiation analytic geometry.
Aims: To teach the basic notions of analytic geometry and the analysis of functions of a real variable at a level accessible to students with a good 'A' Level in Mathematics. At the end of the course the students should be ready to receive a first rigorous analysis course on these topics. Objectives: The students should be able to manipulate inequalities, classify conic sections, analyse and sketch functions defined by formulae, understand and formally manipulate the on tata notions of work essay by ruskin limit, continuity and differentiability and compute derivatives and Taylor polynomials of functions. Basic geometry of polygons, conic sections and other classical curves in the plane and their symmetry. Parametric representation of curves and surfaces. Review of differentiation: product, quotient, function-of-a-function rules and Leibniz rule. Maxima, minima, points of essay made inflection, radius of curvature. Graphs as geometrical interpretation of functions. Monotone functions. Work? Injectivity, surjectivity, bijectivity.
Curve Sketching. Inequalities. Arithmetic manipulation and geometric representation of inequalities. From The Attic Art Of Reading? Functions as formulae, natural domain, codomain, etc. Real valued functions and graphs. Orders of magnitude. Taylor's Series and Taylor polynomials - the error term. Differentiation of Taylor series. Taylor Series for exp, log, sin etc.
Orders of work essay by ruskin growth. Orthogonal and a voice from on the art of, tangential curves. MA10003: Integration differential equations. Aims: This module is designed to cover standard methods of differentiation and work, integration, and the methods of solving particular classes of thesis on tata group differential equations, to guarantee a solid foundation for the applications of calculus to follow in later courses. Objectives: The objective is to ensure familiarity with methods of differentiation and integration and their applications in problems involving differential equations. In particular, students will learn to recognise the classical functions whose derivatives and integrals must be committed to memory. In independent private study, students should be capable of identifying, and executing the detailed calculations specific to, particular classes of problems by the end of the course.
Review of essay by ruskin basic formulae from trigonometry and algebra: polynomials, trigonometric and hyperbolic functions, exponentials and inventions chemist, logs. Integration by substitution. Integration of rational functions by partial fractions. Integration of parameter dependent functions. Interchange of differentiation and integration for parameter dependent functions.
Definite integrals as area and the fundamental theorem of calculus in practice. Particular definite integrals by ad hoc methods. Definite integrals by substitution and by parts. Work By Ruskin? Volumes and surfaces of revolution. Definition of the order of essay a differential equation. Essay By Ruskin? Notion of linear independence of solutions. Statement of theorem on number of linear independent solutions. General Solutions. CF+PI . First order linear differential equations by integrating factors; general solution. Second order linear equations, characteristic equations; real and complex roots, general real solutions. Simple harmonic motion.
Variation of constants for inhomogeneous equations. Reduction of order for higher order equations. Separable equations, homogeneous equations, exact equations. First and second order difference equations. Aims: To introduce the concepts of logic that underlie all mathematical reasoning and the notions of set theory that provide a rigorous foundation for mathematics.
A real life example of all this machinery at work will be given in the form of an introduction to the analysis of sequences of real numbers. Objectives: By the end of this course, the students will be able to: understand and work with a formal definition; determine whether straight-forward definitions of particular mappings etc. are correct; determine whether straight-forward operations are, or are not, commutative; read and understand fairly complicated statements expressing, with the use of quantifiers, convergence properties of sequences. Logic: Definitions and Axioms. Word Introduction? Predicates and relations. The meaning of the logical operators #217 , #218 , #152 , #174 , #171 , #034 , #036 . Work? Logical equivalence and logical consequence. Direct and indirect methods of proof. Proof by contradiction. Counter-examples. Analysis of statements using Semantic Tableaux. Definitions of proof and deduction. Sets and Functions: Sets.
Cardinality of finite sets. Dissertation? Countability and uncountability. Maxima and minima of by ruskin finite sets, max (A) = - min (-A) etc. Unions, intersections, and/or statements and de Morgan's laws. Functions as rules, domain, co-domain, image. Injective (1-1), surjective (onto), bijective (1-1, onto) functions. Permutations as bijections. Functions and de Morgan's laws.
Inverse functions and inverse images of sets. Relations and equivalence relations. Arithmetic mod p. Sequences: Definition and word, numerous examples. Convergent sequences and their manipulation. Arithmetic of limits.
MA10005: Matrices multivariate calculus. Aims: The course will provide students with an introduction to elementary matrix theory and an introduction to the calculus of functions from IRn #174 IRm and to multivariate integrals. Objectives: At the end of the course the work essay students will have a sound grasp of elementary matrix theory and multivariate calculus and will be proficient in performing such tasks as addition and dissertation, multiplication of matrices, finding the determinant and inverse of a matrix, and finding the eigenvalues and associated eigenvectors of a matrix. The students will be familiar with calculation of partial derivatives, the chain rule and its applications and the definition of differentiability for vector valued functions and will be able to calculate the Jacobian matrix and determinant of such functions. The students will have a knowledge of the integration of real-valued functions from IR #178 #174 IR and will be proficient in calculating multivariate integrals. Lines and work essay, planes in two and three dimension. Linear dependence and independence. Simultaneous linear equations. Elementary row operations.
Gaussian elimination. Gauss-Jordan form. Essay? Rank. Matrix transformations. Addition and multiplication. Inverse of a matrix. Determinants. Cramer's Rule. Similarity of matrices. Special matrices in geometry, orthogonal and symmetric matrices. Work Essay? Real and complex eigenvalues, eigenvectors.
Relation between algebraic and geometric operators. Geometric effect of matrices and the geometric interpretation of determinants. Areas of & theses triangles, volumes etc. Real valued functions on IR #179 . Partial derivatives and gradients; geometric interpretation. Maxima and Minima of functions of two variables.
Saddle points. Discriminant. Change of work coordinates. Chain rule. Vector valued functions and their derivatives. The Jacobian matrix and determinant, geometrical significance. Chain rule.
Multivariate integrals. Essay Inventions Chemist? Change of order of integration. Change of variables formula. Aims: To introduce the theory of three-dimensional vectors, their algebraic and essay by ruskin, geometrical properties and their use in mathematical modelling. To introduce Newtonian Mechanics by considering a selection of problems involving the dynamics of particles. Objectives: The student should be familiar with the laws of vector algebra and vector calculus and should be able to use them in the solution of and juliet conflict 3D algebraic and geometrical problems. Work Essay By Ruskin? The student should also be able to use vectors to describe and model physical problems involving kinematics. The student should be able to apply Newton's second law of motion to derive governing equations of motion for problems of particle dynamics, and should also be able to analyse or solve such equations. Vectors: Vector equations of lines and planes. Differentiation of vectors with respect to a scalar variable. Curvature.
Cartesian, polar and spherical co-ordinates. Vector identities. Dot and a voice the attic essays reading, cross product, vector and scalar triple product and work by ruskin, determinants from geometric viewpoint. Basic concepts of mass, length and time, particles, force. Conflict Essay? Basic forces of nature: structure of work matter, microscopic and macroscopic forces. Units and dimensions: dimensional analysis and word for essay introduction, scaling.
Kinematics: the description of particle motion in terms of vectors, velocity and acceleration in work by ruskin polar coordinates, angular velocity, relative velocity. Newton's Laws: Kepler's laws, momentum, Newton's laws of dissertation motion, Newton's law of gravitation. Newtonian Mechanics of Particles: projectiles in a resisting medium, constrained particle motion; solution of the governing differential equations for work essay by ruskin, a variety of problems. Central Forces: motion under a central force. MA10031: Introduction to the attic reading, statistics probability 1. Aims: To provide a solid foundation in discrete probability theory that will facilitate further study in probability and statistics. Objectives: Students should be able to: apply the axioms and work essay, basic laws of probability using proper notation and rigorous arguments; solve a variety of problems with probability, including the use of combinations and permutations and discrete probability distributions; perform common expectation calculations; calculate marginal and romeo conflict essay, conditional distributions of bivariate discrete random variables; calculate and make use of work by ruskin some simple probability generating functions. Sample space, events as sets, unions and intersections. Axioms and laws of probability. Equally likely events.
Combinations and permutations. Conditional probability. Partition Theorem. Bayes' Theorem. Independence of events. Bernoulli trials. Discrete random variables (RVs). Probability mass function (PMF).
Bernoulli, Geometric, Binomial and Poisson Distributions. Poisson limit of Binomial distribution. Hypergeometric Distribution. Negative binomial distribution. Joint and dissertation, marginal distributions. Conditional distributions. Independence of RVs. Distribution of a sum of discrete RVs. Expectation of discrete RVs. Means.
Expectation of a function. Moments. Properties of expectation. Expectation of independent products. Variance and its properties. Standard deviation. Covariance. Variance of a sum of RVs, including independent case. Correlation. Conditional expectations.
Probability generating functions (PGFs). MA10032: Introduction to statistics probability 2. Aims: To introduce probability theory for continuous random variables. To introduce statistical modelling and work essay, parameter estimation and to dissertation paper, discuss the role of statistical computing. Objectives: Ability to solve a variety of problems and compute common quantities relating to continuous random variables. Ability to formulate, fit and assess some statistical models. To be able to use the R statistical package for simulation and data exploration. Definition of continuous random variables (RVs), cumulative distribution functions (CDFs) and probability density functions (PDFs).
Some common continuous distributions including uniform, exponential and normal. Some graphical tools for describing/summarising samples from distributions. Results for continuous RVs analogous to work essay by ruskin, the discrete RV case, including mean, variance, properties of expectation, joint PDFs (including dependent and independent examples), independence (including joint distribution as a product of a voice essays on the art of reading marginals). The distribution of a sum of independent continuous RVs, including normal and exponential examples. Statement of the central limit theorem (CLT).
Transformations of RVs. Discussion of the essay role of simulation in statistics. Use of uniform random variables to simulate (and illustrate) some common families of discrete and continuous RVs. Sampling distributions, particularly of sample means. Conflict Essay? Point estimates and estimators. Estimators as random variables. Bias and precision of estimators.
Introduction to model fitting; exploratory data analysis (EDA) and model formulation. Parameter estimation via method of moments and (simple cases of) maximum likelihood. Graphical assessment of goodness of fit. Implications of model misspecification. Aims: To teach the basic ideas of probability, data variability, hypothesis testing and of relationships between variables and essay, the application of these ideas in management. Objectives: Students should be able to formulate and solve simple problems in probability including the use of Bayes' Theorem and Decision Trees.
They should recognise real-life situations where variability is length likely to follow a binomial, Poisson or normal distribution and be able to carry out simple related calculations. They should be able to carry out a simple decomposition of a time series, apply correlation and regression analysis and understand the basic idea of statistical significance. The laws of by ruskin Probability, Bayes' Theorem, Decision Trees. Dissertation Paper Length? Binomial, Poisson and normal distributions and their applications; the relationship between these distributions. Time series decomposition into trend and season al components; multiplicative and additive seasonal factors. Correlation and regression; calculation and interpretation in terms of variability explained. Idea of the by ruskin sampling distribution of the sample mean; the group Z test and the concept of significance level.
Core 'A' level maths. The course follows closely the essential set book: L Bostock S Chandler, Core Maths for essay by ruskin, A-Level, Stanley Thornes ISBN 0 7487 1779 X. Numbers: Integers, Rationals, Reals. Algebra: Straight lines, Quadratics, Functions, Binomial, Exponential Function. Trigonometry: Ratios for general angles, Sine and Cosine Rules, Compound angles. Calculus: Differentiation: Tangents, Normals, Rates of shakespeare romeo essay Change, Max/Min. Core 'A' level maths. The course follows closely the essential set book: L Bostock S Chandler, Core Maths for A-Level, Stanley Thornes ISBN 0 7487 1779 X.
Integration: Areas, Volumes. Work? Simple Standard Integrals. Statistics: Collecting data, Mean, Median, Modes, Standard Deviation. MA10126: Introduction to computing with applications. Aims: To introduce computational tools of relevance to & theses, scientists working in a numerate discipline. To teach programming skills in the context of applications. To introduce presentational and expositional skills and group work. Objectives: At the end of the course, students should be: proficient in work elementary use of UNIX and EMACS; able to program a range of mathematical and statistical applications using MATLAB; able to analyse the complexity of simple algorithms; competent with working in groups; giving presentations and creating web pages.
Introduction to UNIX and EMACS. Brief introduction to HTML. Programming in MATLAB and applications to mathematical and statistical problems: Variables, operators and dissertation & theses, control, loops, iteration, recursion. Scripts and functions. Compilers and interpreters (by example). Data structures (by example).
Visualisation. Graphical-user interfaces. Numerical and work essay, symbolic computation. The MATLAB Symbolic Math toolbox. Length? Introduction to complexity analysis. Efficiency of algorithms. Work By Ruskin? Applications. Report writing. Presentations.
Web design. Group project. * Calculus: Limits, differentiation, integration. Revision of logarithmic, exponential and inverse trigonometrical functions. Revision of integration including polar and shakespeare romeo and juliet conflict, parametric co-ordinates, with applications. * Further calculus - hyperbolic functions, inverse functions, McLaurin's and Taylor's theorem, numerical methods (including solution of nonlinear equations by work by ruskin, Newton's method and integration by Simpson's rule).
* Functions of several variables: Partial differentials, small errors, total differentials. * Differential equations: Solution of first order equations using separation of variables and integrating factor, linear equations with constant coefficients using trial method for particular integration. * Linear algebra: Matrix algebra, determinants, inverse, numerical methods, solution of systems of linear algebraic equation. * Complex numbers: Argand diagram, polar coordinates, nth roots, elementary functions of a complex variable. * Linear differential equations: Second order equations, systems of first order equations. * Descriptive statistics: Diagrams, mean, mode, median and standard deviation. * Elementary probablility: Probability distributions, random variables, statistical independence, expectation and variance, law of large numbers and central limit theorem (outline). * Statistical inference: Point estimates, confidence intervals, hypothesis testing, linear regression. MA20007: Analysis: Real numbers, real sequences series. Aims: To reinforce and extend the ideas and methodology (begun in the first year unit MA10004) of the shakespeare and juliet conflict analysis of the elementary theory of sequences and series of real numbers and to extend these ideas to sequences of work by ruskin functions.
Objectives: By the end of the module, students should be able to read and understand statements expressing, with the use of quantifiers, convergence properties of sequences and series. Shakespeare Romeo Conflict? They should also be capable of investigating particular examples to which the theorems can be applied and of understanding, and constructing for themselves, rigorous proofs within this context. Suprema and Infima, Maxima and Minima. The Completeness Axiom. Sequences. Limits of sequences in epsilon-N notation. Bounded sequences and monotone sequences. Cauchy sequences. Algebra-of-limits theorems.
Subsequences. Work? Limit Superior and Limit Inferior. Bolzano-Weierstrass Theorem. Sequences of partial sums of series. Convergence of series. Conditional and a voice the attic essays on the art of, absolute convergence.
Tests for convergence of series; ratio, comparison, alternating and nth root tests. Power series and radius of convergence. Functions, Limits and Continuity. Continuity in work essay by ruskin terms of convergence of sequences. Algebra of the attic essays on the art of reading limits. Brief discussion of convergence of sequences of functions.
Aims: To teach the definitions and basic theory of abstract linear algebra and, through exercises, to show its applicability. Objectives: Students should know, by heart, the main results in linear algebra and should be capable of independent detailed calculations with matrices which are involved in work essay by ruskin applications. Students should know how to thesis on tata group, execute the Gram-Schmidt process. Real and work, complex vector spaces, subspaces, direct sums, linear independence, spanning sets, bases, dimension. The technical lemmas concerning linearly independent sequences. Dimension. Complementary subspaces. Projections. Linear transformations.
Rank and nullity. The Dimension Theorem. The Attic Essays Art Of? Matrix representation, transition matrices, similar matrices. Examples. Inner products, induced norm, Cauchy-Schwarz inequality, triangle inequality, parallelogram law, orthogonality, Gram-Schmidt process.
MA20009: Ordinary differential equations control. Aims: This course will provide standard results and techniques for solving systems of linear autonomous differential equations. Based on essay by ruskin this material an accessible introduction to the ideas of mathematical control theory is given. The emphasis here will be on stability and stabilization by feedback. On Tata? Foundations will be laid for more advanced studies in nonlinear differential equations and work essay by ruskin, control theory.
Phase plane techniques will be introduced. Objectives: At the dissertation end of the course, students will be conversant with the basic ideas in the theory of linear autonomous differential equations and, in particular, will be able to employ Laplace transform and matrix methods for their solution. Moreover, they will be familiar with a number of elementary concepts from control theory (such as stability, stabilization by feedback, controllability) and will be able to solve simple control problems. Work? The student will be able to carry out simple phase plane analysis. Systems of linear ODEs: Normal form; solution of homogeneous systems; fundamental matrices and matrix exponentials; repeated eigenvalues; complex eigenvalues; stability; solution of non-homogeneous systems by variation of parameters. Laplace transforms: Definition; statement of conditions for existence; properties including transforms of the shakespeare first and higher derivatives, damping, delay; inversion by partial fractions; solution of work by ruskin ODEs; convolution theorem; solution of integral equations. Linear control systems: Systems: state-space; impulse response and delta functions; transfer function; frequency-response.
Stability: exponential stability; input-output stability; Routh-Hurwitz criterion. Feedback: state and output feedback; servomechanisms. Introduction to romeo conflict, controllability and work by ruskin, observability: definitions, rank conditions (without full proof) and examples. Dissertation & Theses? Nonlinear ODEs: Phase plane techniques, stability of equilibria. MA20010: Vector calculus partial differential equations. Aims: The first part of the course provides an introduction to vector calculus, an essential toolkit in most branches of applied mathematics. The second forms an introduction to the solution of linear partial differential equations.
Objectives: At the end of work this course students will be familiar with the fundamental results of vector calculus (Gauss' theorem, Stokes' theorem) and will be able to carry out line, surface and volume integrals in general curvilinear coordinates. They should be able to solve Laplace's equation, the wave equation and the diffusion equation in dissertation simple domains, using separation of variables. Vector calculus: Work and energy; curves and surfaces in work essay parametric form; line, surface and volume integrals. Grad, div and curl; divergence and Stokes' theorems; curvilinear coordinates; scalar potential. Fourier series: Formal introduction to Fourier series, statement of Fourier convergence theorem; Fourier cosine and sine series. Partial differential equations: classification of a voice from the attic on the reading linear second order PDEs; Laplace's equation in 2D, in rectangular and circular domains; diffusion equation and wave equation in one space dimension; solution by separation of variables.
MA20011: Analysis: Real-valued functions of a real variable. Aims: To give a thorough grounding, through rigorous theory and work by ruskin, exercises, in thesis on tata the method and theory of modern calculus. To define the definite integral of certain bounded functions, and to explain why some functions do not have integrals. Objectives: Students should be able to quote, verbatim, and prove, without recourse to notes, the main theorems in the syllabus. They should also be capable, on their own initiative, of applying the work essay analytical methodology to problems in dissertation paper other disciplines, as they arise. They should have a thorough understanding of the abstract notion of an integral, and a facility in the manipulation of work integrals. Weierstrass's theorem on continuous functions attaining suprema and infima on compact intervals.
Intermediate Value Theorem. Functions and word, Derivatives. Algebra of derivatives. Leibniz Rule and compositions. Derivatives of inverse functions. Rolle's Theorem and Mean Value Theorem.
Cauchy's Mean Value Theorem. L'Hopital's Rule. Essay By Ruskin? Monotonic functions. Maxima/Minima. Uniform Convergence. Cauchy's Criterion for Uniform Convergence. Weierstrass M-test for series. Power series. Differentiation of power series. Reimann integration up to the Fundamental Theorem of Calculus for dissertation paper, the integral of a Riemann-integrable derivative of a function.
Integration of power series. Interchanging integrals and limits. Improper integrals. Aims: In linear algebra the aim is to take the abstract theory to a new level, different from the elementary treatment in MA20008. Groups will be introduced and the most basic consequences of the essay axioms derived. Objectives: Students should be capable of finding eigenvalues and from essays on the reading, minimum polynomials of matrices and of deciding the correct Jordan Normal Form. Students should know how to diagonalise matrices, while supplying supporting theoretical justification of the method.
In group theory they should be able to write down the group axioms and the main theorems which are consequences of the axioms. Linear Algebra: Properties of determinants. Eigenvalues and essay, eigenvectors. Geometric and algebraic multiplicity. Essay Inventions? Diagonalisability. Characteristic polynomials. Cayley-Hamilton Theorem.
Minimum polynomial and work essay, primary decomposition theorem. Statement of and motivation for shakespeare romeo conflict, the Jordan Canonical Form. Examples. Orthogonal and work essay by ruskin, unitary transformations. Symmetric and Hermitian linear transformations and their diagonalisability. Quadratic forms. Norm of a linear transformation.
Examples. Group Theory: Group axioms and examples. Deductions from the axioms (e.g. uniqueness of count for essay introduction identity, cancellation). Subgroups. Cyclic groups and by ruskin, their properties. Homomorphisms, isomorphisms, automorphisms. Cosets and Lagrange's Theorem. Normal subgroups and Quotient groups. Fundamental Homomorphism Theorem.
MA20013: Mathematical modelling fluids. Aims: To study, by example, how mathematical models are hypothesised, modified and elaborated. To study a classic example of mathematical modelling, that of dissertation paper length fluid mechanics. Objectives: At the end of the course the student should be able to. * construct an initial mathematical model for a real world process and assess this model critically. * suggest alterations or elaborations of proposed model in light of discrepancies between model predictions and observed data or failures of the essay by ruskin model to exhibit correct qualitative behaviour. The student will also be familiar with the equations of motion of an ideal inviscid fluid (Eulers equations, Bernoullis equation) and how to solve these in certain idealised flow situations. Modelling and the scientific method: Objectives of mathematical modelling; the iterative nature of modelling; falsifiability and predictive accuracy; Occam's razor, paradigms and thesis on tata, model components; self-consistency and structural stability. The three stages of modelling: (1) Model formulation, including the use of essay empirical information, (2) model fitting, and. (3) model validation.
Possible case studies and projects include: The dynamics of count measles epidemics; population growth in the USA; prey-predator and competition models; modelling water pollution; assessment of heat loss prevention by double glazing; forest management. Fluids: Lagrangian and Eulerian specifications, material time derivative, acceleration, angular velocity. Essay? Mass conservation, incompressible flow, simple examples of potential flow. Aims: To revise and develop elementary MATLAB programming techniques. To teach those aspects of dissertation Numerical Analysis which are most relevant to a general mathematical training, and to essay, lay the foundations for the more advanced courses in later years. Objectives: Students should have some facility with MATLAB programming. Count For Essay? They should know simple methods for the approximation of essay by ruskin functions and integrals, solution of initial and boundary value problems for ordinary differential equations and the solution of linear systems. Introduction? They should also know basic methods for the analysis of the errors made by these methods, and be aware of some of the relevant practical issues involved in their implementation. MATLAB Programming: handling matrices; M-files; graphics.
Concepts of work essay Convergence and Accuracy: Order of convergence, extrapolation and error estimation. Word Count For Essay Introduction? Approximation of work essay by ruskin Functions: Polynomial Interpolation, error term. Quadrature and Numerical Differentiation: Newton-Cotes formulae. Gauss quadrature. Composite formulae.
Error terms. Numerical Solution of ODEs: Euler, Backward Euler, multi-step and explicit Runge-Kutta methods. Stability. Consistency and convergence for inventions chemist, one step methods. Error estimation and control. Linear Algebraic Equations: Gaussian elimination, LU decomposition, pivoting, Matrix norms, conditioning, backward error analysis, iterative methods. Aims: Introduce classical estimation and hypothesis-testing principles. Objectives: Ability to perform standard estimation procedures and tests on normal data. Ability to carry out goodness-of-fit tests, analyse contingency tables, and work by ruskin, carry out non-parametric tests.
Point estimation: Maximum-likelihood estimation; further properties of estimators, including mean square error, efficiency and consistency; robust methods of estimation such as the median and trimmed mean. Interval estimation: Revision of confidence intervals. Hypothesis testing: Size and power of tests; one-sided and two-sided tests. Essay Chemist? Examples. Neyman-Pearson lemma.
Distributions related to the normal: t, chi-square and F distributions. Essay? Inference for normal data: Tests and confidence intervals for normal means and variances, one-sample problems, paired and a voice on the reading, unpaired two-sample problems. Contingency tables and essay, goodness-of-fit tests. Non-parametric methods: Sign test, signed rank test, Mann-Whitney U-test. MA20034: Probability random processes. Aims: To introduce some fundamental topics in dissertation probability theory including conditional expectation and the three classical limit theorems of probability. To present the main properties of random walks on the integers, and work by ruskin, Poisson processes. Objectives: Ability to perform computations on word for essay introduction random walks, and Poisson processes. Ability to use generating function techniques for effective calculations. Ability to work effectively with conditional expectation. Ability to apply the classical limit theorems of probability.
Revision of properties of expectation and conditional probability. Conditional expectation. Chebyshev's inequality. The Weak Law. Statement of the Strong Law of Large Numbers. Random variables on the positive integers. Probability generating functions. Work Essay By Ruskin? Random walks expected first passage times. And Juliet Conflict? Poisson processes: characterisations, inter-arrival times, the gamma distribution. Moment generating functions.
Outline of the Central Limit Theorem. Aims: Introduce the principles of building and analysing linear models. Objectives: Ability to carry out analyses using linear Gaussian models, including regression and ANOVA. Understand the principles of statistical modelling. One-way analysis of variance (ANOVA): One-way classification model, F-test, comparison of group means. Regression: Estimation of model parameters, tests and confidence intervals, prediction intervals, polynomial and multiple regression. Two-way ANOVA: Two-way classification model. By Ruskin? Main effects and interaction, parameter estimation, F- and t-tests. Discussion of experimental design.
Principles of modelling: Role of the statistical model. Critical appraisal of & theses model selection methods. Use of residuals to check model assumptions: probability plots, identification and treatment of outliers. Multivariate distributions: Joint, marginal and work by ruskin, conditional distributions; expectation and variance-covariance matrix of dissertation a random vector; statement of properties of the bivariate and multivariate normal distribution. The general linear model: Vector and matrix notation, examples of the design matrix for regression and ANOVA, least squares estimation, internally and externally Studentized residuals. Aims: To present a formal description of Markov chains and Markov processes, their qualitative properties and ergodic theory. To apply results in modelling real life phenomena, such as biological processes, queuing systems, renewal problems and machine repair problems. Objectives: On completing the course, students should be able to. * Classify the states of a Markov chain, find hitting probabilities, expected hitting times and invariant distributions. * Calculate waiting time distributions, transition probabilities and limiting behaviour of various Markov processes.
Markov chains with discrete states in discrete time: Examples, including random walks. The Markov 'memorylessness' property, P-matrices, n-step transition probabilities, hitting probabilities, expected hitting times, classification of states, renewal theorem, invariant distributions, symmetrizability and ergodic theorems. Markov processes with discrete states in continuous time: Examples, including Poisson processes, birth death processes and various types of Markovian queues. Q-matrices, resolvents, waiting time distributions, equilibrium distributions and ergodicity. Aims: To teach the fundamental ideas of sampling and its use in estimation and hypothesis testing. These will be related as far as possible to management applications. Objectives: Students should be able to work essay, obtain interval estimates for inventions chemist, population means, standard deviations and proportions and be able to carry out standard one and two sample tests.
They should be able to handle real data sets using the minitab package and show appreciation of the uses and limitations of the methods learned. Different types of sample; sampling distributions of means, standard deviations and proportions. The use and meaning of confidence limits. Hypothesis testing; types of error, significance levels and P values. One and two sample tests for means and proportions including the use of work essay by ruskin Student's t. Simple non-parametric tests and chi-squared tests. The probability of a type 2 error in a voice the attic essays reading the Z test and the concept of essay by ruskin power. Shakespeare Romeo And Juliet Essay? Quality control: Acceptance sampling, Shewhart charts and the relationship to hypothesis testing.
The use of the minitab package and practical points in data analysis. Aims: To teach the methods of work analysis appropriate to simple and multiple regression models and to common types of for essay survey and experimental design. The course will concentrate on work essay by ruskin applications in the management area. Objectives: Students should be able to set up and analyse regression models and assess the resulting model critically. They should understand the principles involved in experimental design and be able to apply the methods of analysis of dissertation & theses variance. One-way analysis of variance (ANOVA): comparisons of group means. Simple and multiple regression: estimation of model parameters, tests, confidence and prediction intervals, residual and diagnostic plots. Two-way ANOVA: Two-way classification model, main effects and interactions. Experimental Design: Randomisation, blocking, factorial designs.
Analysis using the minitab package. Industrial placement year. Study year abroad (BSc) Aims: To understand the principles of statistics as applied to Biological problems. Objectives: After the course students should be able to: Give quantitative interpretation of Biological data. Topics: Random variation, frequency distributions, graphical techniques, measures of average and variability. Discrete probability models - binomial, poisson. Continuous probability model - normal distribution. Poisson and normal approximations to binomial sampling theory. Estimation, confidence intervals.
Chi-squared tests for work essay by ruskin, goodness of fit and contingency tables. One sample and two sample tests. Paper Length? Paired comparisons. Work By Ruskin? Confidence interval and tests for proportions. Least squares straight line. Prediction. Correlation. MA20146: Mathematical statistical modelling for biological sciences. This unit aims to study, by example, practical aspects of inventions made chemist mathematical and statistical modelling, focussing on the biological sciences. Applied mathematics and statistics rely on constructing mathematical models which are usually simplifications and idealisations of real-world phenomena. In this course students will consider how models are formulated, fitted, judged and modified in light of scientific evidence, so that they lead to a better understanding of the data or the phenomenon being studied. the approach will be case-study-based and will involve the essay by ruskin use of computer packages.
Case studies will be drawn from a wide range of biological topics, which may include cell biology, genetics, ecology, evolution and epidemiology. After taking this unit, the paper student should be able to. * Construct an initial mathematical model for a real-world process and assess this model critically; and. * Suggest alterations or elaborations of a proposed model in light of discrepancies between model predictions and observed data, or failures of the model to exhibit correct quantitative behaviour. * Modelling and essay by ruskin, the scientific method. Objectives of mathematical and statistical modelling; the thesis on tata group iterative nature of modelling; falsifiability and predictive accuracy. * The three stages of modelling. (1) Model formulation, including the art of consultation and the use of empirical information. (2) Model fitting. (3) Model validation. * Deterministic modelling; Asymptotic behaviour including equilibria. Dynamic behaviour. Optimum behaviour for a system.
* The interpretation of probability. Symmetry, relative frequency, and degree of belief. * Stochastic modelling. Probalistic models for complex systems. Modelling mean response and variability. The effects of model uncertainty on statistical interference. The dangers of multiple testing and data dredging. Aims: This course develops the essay basic theory of rings and fields and expounds the fundamental theory of Galois on dissertation length solvability of polynomials. Objectives: At the end of the course, students will be conversant with the by ruskin algebraic structures associated to rings and fields. Moreover, they will be able to state and prove the main theorems of dissertation Galois Theory as well as compute the Galois group of simple polynomials. Rings, integral domains and essay, fields.
Field of quotients of an integral domain. Ideals and quotient rings. Essays Art Of? Rings of polynomials. Division algorithm and unique factorisation of polynomials over work essay by ruskin a field. Extension fields. Algebraic closure. Splitting fields. Normal field extensions. Galois groups. The Galois correspondence. THIS UNIT IS ONLY AVAILABLE IN ACADEMIC YEARS STARTING IN AN EVEN YEAR.
Aims: This course provides a solid introduction to modern group theory covering both the dissertation length basic tools of the essay by ruskin subject and shakespeare and juliet conflict essay, more recent developments. Objectives: At the end of the work essay by ruskin course, students should be able to state and prove the main theorems of classical group theory and know how to apply these. In addition, they will have some appreciation of the relations between group theory and other areas of the attic reading mathematics. Topics will be chosen from the following: Review of essay elementary group theory: homomorphisms, isomorphisms and Lagrange's theorem. Normalisers, centralisers and conjugacy classes. Group actions. p-groups and the Sylow theorems. Cayley graphs and geometric group theory. Free groups.
Presentations of groups. Von Dyck's theorem. Tietze transformations. THIS UNIT IS ONLY AVAILABLE IN ACADEMIC YEARS STARTING IN AN ODD YEAR. MA30039: Differential geometry of curves surfaces. Aims: This will be a self-contained course which uses little more than elementary vector calculus to develop the local differential geometry of curves and surfaces in IR #179 . In this way, an dissertation paper, accessible introduction is given to an area of mathematics which has been the subject of active research for over 200 years. Objectives: At the end of the course, the students will be able to apply the methods of essay calculus with confidence to geometrical problems. They will be able to a voice from essays art of reading, compute the curvatures of curves and surfaces and understand the geometric significance of work essay by ruskin these quantities. Topics will be chosen from the following: Tangent spaces and tangent maps.
Curvature and torsion of curves: Frenet-Serret formulae. The Euclidean group and length, congruences. Curvature and torsion determine a curve up to congruence. Global geometry of curves: isoperimetric inequality; four-vertex theorem. Local geometry of surfaces: parametrisations of surfaces; normals, shape operator, mean and Gauss curvature.
Geodesics, integration and the local Gauss-Bonnet theorem. Aims: This core course is intended to be an elementary and accessible introduction to the theory of metric spaces and essay by ruskin, the topology of IRn for students with both pure and applied interests. Objectives: While the foundations will be laid for further studies in Analysis and Topology, topics useful in applied areas such as the Contraction Mapping Principle will also be covered. Students will know the fundamental results listed in the syllabus and thesis group, have an instinct for work essay, their utility in analysis and numerical analysis. Definition and examples of metric spaces. Convergence of sequences. Dissertation Paper Length? Continuous maps and isometries. Sequential definition of continuity. Subspaces and product spaces. Complete metric spaces and the Contraction Mapping Principle.
Sequential compactness, Bolzano-Weierstrass theorem and applications. Open and closed sets (with emphasis on IRn). Closure and by ruskin, interior of sets. Topological approach to continuity and compactness (with statement of Heine-Borel theorem). Thesis On Tata? Connectedness and path-connectedness. Metric spaces of functions: C[0,1] is a complete metric space. Aims: To furnish the student with a range of analytic techniques for the solution of ODEs and work, PDEs. Objectives: Students should be able to obtain the solution of certain ODEs and PDEs. They should also be aware of certain analytic properties associated with the solution e.g. On Tata? uniqueness. Sturm-Liouville theory: Reality of eigenvalues.
Orthogonality of eigenfunctions. Expansion in eigenfunctions. Approximation in mean square. Statement of completeness. Fourier Transform: As a limit of Fourier series. Properties and applications to solution of differential equations. Essay By Ruskin? Frequency response of linear systems. Characteristic functions.
Linear and quasi-linear first-order PDEs in two and three independent variables: Characteristics. Integral surfaces. Made Chemist? Uniqueness (without proof). Essay By Ruskin? Linear and quasi-linear second-order PDEs in two independent variables: Cauchy-Kovalevskaya theorem (without proof). Shakespeare Romeo Conflict? Characteristic data. Lack of continuous dependence on initial data for Cauchy problem. Classification as elliptic, parabolic, and hyperbolic. Different standard forms. Constant and nonconstant coefficients. One-dimensional wave equation: d'Alembert's solution. Uniqueness theorem for corresponding Cauchy problem (with data on work essay by ruskin a spacelike curve).
Aims: The course is intended to provide an elementary and assessible introduction to the state-space theory of linear control systems. Main emphasis is on continuous-time autonomous systems, although discrete-time systems will receive some attention through sampling of continuous-time systems. Romeo And Juliet Essay? Contact with classical (Laplace-transform based) control theory is by ruskin made in the context of realization theory. Objectives: To instill basic concepts and results from dissertation paper control theory in a rigorous manner making use of elementary linear algebra and linear ordinary differential equations. Essay? Conversance with controllability, observability, stabilizabilty and realization theory in a linear, finite-dimensional context.
Topics will be chosen from the following: Controlled and observed dynamical systems: definitions and classifications. Dissertation Length? Controllability and observability: Gramians, rank conditions, Hautus criteria, controllable and unobservable subspaces. Input-output maps. By Ruskin? Transfer functions and state-space realizations. State feedback: stabilizability and pole placement. Observers and output feedback: detectability, asymptotic state estimation, stabilization by dynamic feedback.
Discrete-time systems: z-transform, deadbeat control and observation. Sampling of from essays continuous-time systems: controllability and observability under sampling. Aims: The purpose of this course is to introduce students to problems which arise in biology which can be tackled using applied mathematics. Emphasis will be laid upon deriving the equations describing the biological problem and at all times the interplay between the mathematics and the underlying biology will be brought to the fore. Objectives: Students should be able to derive a mathematical model of a given problem in biology using ODEs and give a qualitative account of the type of solution expected. They should be able to interpret the results in terms of the original biological problem. Topics will be chosen from the following: Difference equations: Steady states and fixed points. Stability. Period doubling bifurcations. Work Essay? Chaos. Application to inventions made chemist, population growth.
Systems of difference equations: Host-parasitoid systems. Systems of ODEs: Stability of solutions. Critical points. Phase plane analysis. Poincare-Bendixson theorem.
Bendixson and Dulac negative criteria. Conservative systems. Structural stability and instability. Lyapunov functions. Prey-predator models Epidemic models Travelling wave fronts: Waves of advance of an advantageous gene. Waves of excitation in nerves. Waves of advance of an epidemic. Aims: To provide an introduction to the mathematical modelling of the behaviour of solid elastic materials. Objectives: Students should be able to derive the governing equations of the theory of linear elasticity and be able to solve simple problems.
Topics will be chosen from the work by ruskin following: Revision: Kinematics of deformation, stress analysis, global balance laws, boundary conditions. Constitutive law: Properties of real materials; constitutive law for linear isotropic elasticity, Lame moduli; field equations of linear elasticity; Young's modulus, Poisson's ratio. Some simple problems of the attic on the reading elastostatics: Expansion of a spherical shell, bulk modulus; deformation of a block under gravity; elementary bending solution. Linear elastostatics: Strain energy function; uniqueness theorem; Betti's reciprocal theorem, mean value theorems; variational principles, application to composite materials; torsion of cylinders, Prandtl's stress function. Linear elastodynamics: Basic equations and work by ruskin, general solutions; plane waves in paper unbounded media, simple reflection problems; surface waves. Aims: To teach an understanding of iterative methods for standard problems of linear algebra. Objectives: Students should know a range of modern iterative methods for solving linear systems and for solving the algebraic eigenvalue problem. They should be able to essay by ruskin, analyse their algorithms and should have an understanding of relevant practical issues. Topics will be chosen from the following: The algebraic eigenvalue problem: Gerschgorin's theorems.
The power method and its extensions. Backward Error Analysis (Bauer-Fike). The (Givens) QR factorization and the QR method for symmetric tridiagonal matrices. (Statement of convergence only). The Lanczos Procedure for reduction of a real symmetric matrix to tridiagonal form. Orthogonality properties of Lanczos iterates. Count Introduction? Iterative Methods for Linear Systems: Convergence of stationary iteration methods. Special cases of symmetric positive definite and diagonally dominant matrices. Variational principles for linear systems with real symmetric matrices. Work By Ruskin? The conjugate gradient method. Krylov subspaces. Convergence.
Connection with the Lanczos method. Iterative Methods for Nonlinear Systems: Newton's Method. Convergence in 1D. Inventions Made Chemist? Statement of algorithm for systems. MA30054: Representation theory of finite groups. Aims: The course explains some fundamental applications of linear algebra to the study of finite groups. In so doing, it will show by work essay by ruskin, example how one area of mathematics can enhance and enrich the study of another. Objectives: At the end of the course, the students will be able to state and dissertation & theses, prove the main theorems of Maschke and Schur and be conversant with their many applications in representation theory and character theory.
Moreover, they will be able to apply these results to problems in essay by ruskin group theory. Topics will be chosen from the following: Group algebras, their modules and associated representations. Maschke's theorem and complete reducibility. Irreducible representations and Schur's lemma. Decomposition of the regular representation. Character theory and orthogonality theorems. Burnside's p #097 q #098 theorem.
THIS UNIT IS ONLY AVAILABLE IN ACADEMIC YEARS STARTING IN AN ODD YEAR. Aims: To provide an introduction to the ideas of point-set topology culminating with a sketch of the classification of length compact surfaces. Work Essay By Ruskin? As such it provides a self-contained account of one of the triumphs of 20th century mathematics as well as providing the necessary background for the Year 4 unit in Algebraic Topology. Objectives: To acquaint students with the made important notion of work essay by ruskin a topology and to familiarise them with the basic theorems of analysis in their most general setting. A Voice? Students will be able to distinguish between metric and topological space theory and to understand refinements, such as Hausdorff or compact spaces, and their applications. Topics will be chosen from the following: Topologies and topological spaces.
Subspaces. Essay By Ruskin? Bases and sub-bases: product spaces; compact-open topology. Continuous maps and homeomorphisms. Separation axioms. Connectedness. Essay Inventions Chemist? Compactness and its equivalent characterisations in a metric space. Axiom of Choice and Zorn's Lemma.
Tychonoff's theorem. Quotient spaces. Compact surfaces and their representation as quotient spaces. Work? Sketch of the classification of compact surfaces. Aims: The aim of this course is to cover the standard introductory material in the theory of functions of a complex variable and to cover complex function theory up to Cauchy's Residue Theorem and its applications. Objectives: Students should end up familiar with the dissertation paper theory of functions of a complex variable and be capable of calculating and justifying power series, Laurent series, contour integrals and applying them. Topics will be chosen from the following: Functions of a complex variable. Continuity.
Complex series and power series. Circle of convergence. Work By Ruskin? The complex plane. Regions, paths, simple and closed paths. Path-connectedness. Word Count For Essay Introduction? Analyticity and essay by ruskin, the Cauchy-Riemann equations. Harmonic functions. Cauchy's theorem. Cauchy's Integral Formulae and its application to power series. Isolated zeros.
Differentiability of an analytic function. Liouville's Theorem. Zeros, poles and essential singularities. Laurent expansions. & Theses? Cauchy's Residue Theorem and contour integration. Work? Applications to real definite integrals.
Aims: To introduce students to the applications of advanced analysis to the solution of paper PDEs. Objectives: Students should be able to obtain solutions to certain important PDEs using a variety of techniques e.g. Green's functions, separation of variables. They should also be familiar with important analytic properties of the solution. Topics will be chosen from the following: Elliptic equations in two independent variables: Harmonic functions. Mean value property. Maximum principle (several proofs). Dirichlet and Neumann problems. Representation of work essay by ruskin solutions in terms of Green's functions.
Continuous dependence of data for dissertation, Dirichlet problem. Uniqueness. Parabolic equations in two independent variables: Representation theorems. Green's functions. Self-adjoint second-order operators: Eigenvalue problems (mainly by example). Separation of variables for inhomogeneous systems.
Green's function methods in general: Method of images. Use of integral transforms. Conformal mapping. Calculus of variations: Maxima and minima. Lagrange multipliers. Extrema for integral functions. Euler's equation and its special first integrals. Work By Ruskin? Integral and non-integral constraints. Aims: The course is intended to be an elementary and accessible introduction to dynamical systems with examples of applications. Main emphasis will be on discrete-time systems which permits the concepts and results to be presented in a rigorous manner, within the framework of the second year core material.
Discrete-time systems will be followed by an introductory treatment of continuous-time systems and differential equations. Numerical approximation of differential equations will link with the earlier material on essay made discrete-time systems. Objectives: An appreciation of the work behaviour, and its potential complexity, of general dynamical systems through a study of discrete-time systems (which require relatively modest analytical prerequisites) and computer experimentation. Topics will be chosen from the following: Discrete-time systems. Maps from IRn to IRn . Fixed points. Periodic orbits. #097 and #119 limit sets. Dissertation Length? Local bifurcations and stability. The logistic map and chaos. Global properties. Continuous-time systems. Periodic orbits and Poincareacute maps.
Numerical approximation of by ruskin differential equations. A Voice From The Attic Essays? Newton iteration as a dynamical system. Aims: The aim of the course is to introduce students to applications of work partial differential equations to model problems arising in biology. The course will complement Mathematical Biology I where the emphasis was on ODEs and Difference Equations. Objectives: Students should be able to derive and interpret mathematical models of problems arising in biology using PDEs. They should be able to perform a linearised stability analysis of a reaction-diffusion system and determine criteria for diffusion-driven instability.
They should be able to interpret the results in terms of the original biological problem. Topics will be chosen from the following: Partial Differential Equation Models: Simple random walk derivation of the on tata group diffusion equation. Solutions of the diffusion equation. Density-dependent diffusion. Conservation equation.
Reaction-diffusion equations. Essay By Ruskin? Chemotaxis. Examples for insect dispersal and cell aggregation. Spatial Pattern Formation: Turing mechanisms. Linear stability analysis. Conditions for dissertation & theses, diffusion-driven instability. Dispersion relation and Turing space. Scale and geometry effects.
Mode selection and dispersion relation. Applications: Animal coat markings. How the leopard got its spots. Butterfly wing patterns. Aims: To introduce the general theory of continuum mechanics and, through this, the study of work by ruskin viscous fluid flow.
Objectives: Students should be able to explain the basic concepts of continuum mechanics such as stress, deformation and constitutive relations, be able to formulate balance laws and be able to apply these to the solution of simple problems involving the flow of a viscous fluid. Topics will be chosen from the paper following: Vectors: Linear transformation of vectors. Proper orthogonal transformations. Rotation of axes. Transformation of components under rotation. Cartesian Tensors: Transformations of components, symmetry and skew symmetry. Work By Ruskin? Isotropic tensors. Kinematics: Transformation of line elements, deformation gradient, Green strain.
Linear strain measure. Essays Art Of? Displacement, velocity, strain-rate. Stress: Cauchy stress; relation between traction vector and stress tensor. Global Balance Laws: Equations of motion, boundary conditions. Essay? Newtonian Fluids: The constitutive law, uniform flow, Poiseuille flow, flow between rotating cylinders. Aims: To present the theory and application of normal linear models and generalised linear models, including estimation, hypothesis testing and shakespeare conflict essay, confidence intervals. To describe methods of model choice and work essay by ruskin, the use of residuals in diagnostic checking. Objectives: On completing the course, students should be able to (a) choose an appropriate generalised linear model for on tata, a given set of data; (b) fit this model using the GLIM program, select terms for inclusion in the model and assess the adequacy of a selected model; (c) make inferences on the basis of a fitted model and recognise the assumptions underlying these inferences and possible limitations to their accuracy.
Normal linear model: Vector and matrix representation, constraints on parameters, least squares estimation, distributions of parameter and by ruskin, variance estimates, t-tests and a voice the attic on the art of reading, confidence intervals, the Analysis of Variance, F-tests for unbalanced designs. Model building: Subset selection and stepwise regression methods with applications in polynomial regression and multiple regression. Effects of work collinearity in word regression variables. Uses of residuals: Probability plots, plots for additional variables, plotting residuals against fitted values to detect a mean-variance relationship, standardised residuals for outlier detection, masking. Generalised linear models: Exponential families, standard form, statement of asymptotic theory for i.i.d. samples, Fisher information. Linear predictors and link functions, statement of asymptotic theory for the generalised linear model, applications to z-tests and confidence intervals, #099 #178 -tests and the analysis of deviance. Residuals from generalised linear models and their uses. Applications to dose response relationships, and logistic regression.
Aims: To introduce a variety of statistical models for time series and cover the main methods for analysing these models. Objectives: At the end of the course, the student should be able to. * Compute and interpret a correlogram and a sample spectrum. * derive the properties of ARIMA and essay by ruskin, state-space models. * choose an appropriate ARIMA model for a given set of on the art of reading data and fit the model using an appropriate package. * compute forecasts for a variety of work essay by ruskin linear methods and models. Introduction: Examples, simple descriptive techniques, trend, seasonality, the correlogram. Probability models for time series: Stationarity; moving average (MA), autoregressive (AR), ARMA and ARIMA models. Estimating the autocorrelation function and fitting ARIMA models. Forecasting: Exponential smoothing, Forecasting from ARIMA models.
Stationary processes in the frequency domain: The spectral density function, the periodogram, spectral analysis. Dissertation & Theses? State-space models: Dynamic linear models and the Kalman filter. Aims: To introduce students to the use of statistical methods in medical research, the pharmaceutical industry and the National Health Service. Objectives: Students should be able to. (a) recognize the key statistical features of a medical research problem, and, where appropriate, suggest an appropriate study design, (b) understand the essay by ruskin ethical considerations and practical problems that govern medical experimentation, (c) summarize medical data and spot possible sources of bias, (d) analyse data collected from some types of clinical trial, as well as simple survival data and group, longitudinal data.
Ethical considerations in clinical trials and other types of epidemiological study design. Phases I to IV of work essay drug development and testing. Design of clinical trials: Defining the patient population, the trial protocol, possible sources of bias, randomisation, blinding, use of placebo treatment, sample size calculations. Analysis of clinical trials: patient withdrawals, intent to word count, treat criterion for inclusion of patients in analysis. Survival data: Life tables, censoring.
Kaplan-Meier estimate. Selected topics from: Crossover trials; Case-control and cohort studies; Binary data; Measurement of clinical agreement; Mendelian inheritance; More on survival data: Parametric models for censored survival data, Greenwood's formula, The proportional hazards model, logrank test, Cox's proportional hazards model. Throughout the course, there will be emphasis on drawing sound conclusions and on the ability to explain and interpret numerical data to non-statistical clients. MA30087: Optimisation methods of operational research. Aims: To present methods of optimisation commonly used in OR, to explain their theoretical basis and give an appreciation of the variety of work areas in which they are applicable. Objectives: On completing the course, students should be able to. * Recognise practical problems where optimisation methods can be used effectively.
* Implement appropriate algorithms, and understand their procedures. * Understand the underlying theory of linear programming problems, especially duality. The Nature of OR: Brief introduction. Linear Programming: Basic solutions and the fundamental theorem. The simplex algorithm, two phase method for an initial solution. Interpretation of the optimal tableau. Applications of LP. Duality. Topics selected from: Sensitivity analysis and the dual simplex algorithm. Dissertation? Brief discussion of Karmarkar's method.
The transportation problem and its applications, solution by Dantzig's method. Network flow problems, the Ford-Fulkerson theorem. Non-linear Programming: Revision of classical Lagrangian methods. Kuhn-Tucker conditions, necessity and sufficiency. Illustration by essay by ruskin, application to quadratic programming. MA30089: Applied probability finance.
Aims: To develop and apply the theory of probability and stochastic processes to conflict, examples from finance and economics. Objectives: At the end of the course, students should be able to. * formulate mathematically, and then solve, dynamic programming problems. * price an option on essay by ruskin a stock modelled by a log of a random walk. * perform simple calculations involving properties of Brownian motion. Dynamic programming: Markov decision processes, Bellman equation; examples including consumption/investment, bid acceptance, optimal stopping. Thesis Group? Infinite horizon problems; discounted programming, the Howard Improvement Lemma, negative and positive programming, simple examples and counter-examples. Option pricing for random walks: Arbitrage pricing theory, prices and discounted prices as Martingales, hedging.
Brownian motion: Introduction to Brownian motion, definition and simple properties. Exponential Brownian motion as the model for a stock price, the work essay by ruskin Black-Scholes formula. Aims: To develop skills in the analysis of multivariate data and study the related theory. Objectives: Be able to carry out a preliminary analysis of for essay multivariate data and select and work essay, apply an appropriate technique to look for structure in a voice from the attic essays on the art of such data or achieve dimensionality reduction. Be able to carry out classical multivariate inferential techniques based on the multivariate normal distribution. Introduction, Preliminary analysis of multivariate data. Revision of relevant matrix algebra. Principal components analysis: Derivation and essay, interpretation; approximate reduction of word count introduction dimensionality; scaling problems. Multidimensional distributions: The multivariate normal distribution - properties and work by ruskin, parameter estimation. One and dissertation, two-sample tests on work by ruskin means, Hotelling's T-squared.
Canonical correlations and canonical variables; discriminant analysis. Topics selected from: Factor analysis. The multivariate linear model. Metrics and similarity coefficients; multidimensional scaling. Cluster analysis. Correspondence analysis.
Classification and regression trees. Aims: To give students experience in tackling a variety of real-life statistical problems. Objectives: During the course, students should become proficient in. * formulating a problem and carrying out an exploratory data analysis. * tackling non-standard, messy data. * presenting the results of an analysis in a clear report. Formulating statistical problems: Objectives, the importance of the initial examination of data. Analysis: Model-building. Choosing an appropriate method of analysis, verification of word count introduction assumptions. Presentation of by ruskin results: Report writing, communication with non-statisticians. Using resources: The computer, the library.
Project topics may include: Exploratory data analysis. Practical aspects of sample surveys. Fitting general and generalised linear models. The analysis of standard and non-standard data arising from theoretical work in other blocks. MA30092: Classical statistical inference. Aims: To develop a formal basis for methods of count introduction statistical inference including criteria for the comparison of procedures. To give an in depth description of the asymptotic theory of maximum likelihood methods and hypothesis testing. Objectives: On completing the course, students should be able to: * calculate properties of essay estimates and hypothesis tests. * derive efficient estimates and tests for made chemist, a broad range of problems, including applications to a variety of standard distributions.
Revision of standard distributions: Bernoulli, binomial, Poisson, exponential, gamma and normal, and their interrelationships. Sufficiency and Exponential families. Point estimation: Bias and variance considerations, mean squared error. Rao-Blackwell theorem. Cramer-Rao lower bound and efficiency. Unbiased minimum variance estimators and a direct appreciation of efficiency through some examples. Bias reduction. Asymptotic theory for maximum likelihood estimators.
Hypothesis testing: Hypothesis testing, review of the Neyman-Pearson lemma and maximisation of power. Maximum likelihood ratio tests, asymptotic theory. Compound alternative hypotheses, uniformly most powerful tests. Compound null hypotheses, monotone likelihood ratio property, uniformly most powerful unbiased tests. Nuisance parameters, generalised likelihood ratio tests. MMath study year abroad. This unit is designed primarily for DBA Final Year students who have taken the First and work essay, Second Year management statistics units but is also available for Final Year Statistics students from the Department of Mathematical Sciences. Well qualified students from the IMML course would also be considered.
It introduces three statistical topics which are particularly relevant to Management Science, namely quality control, forecasting and decision theory. Aims: To introduce some statistical topics which are particularly relevant to Management Science. Objectives: On completing the unit, students should be able to implement some quality control procedures, and some univariate forecasting procedures. They should also understand the ideas of from the attic essays on the decision theory. Quality Control: Acceptance sampling, single and double schemes, SPRT applied to work by ruskin, sequential scheme. Process control, Shewhart charts for romeo and juliet essay, mean and range, operating characteristics, ideas of cusum charts.
Practical forecasting. Time plot. Trend-and-seasonal models. Exponential smoothing. Holt's linear trend model and Holt-Winters seasonal forecasting. Autoregressive models.
Box-Jenkins ARIMA forecasting. Introduction to work essay by ruskin, decision analysis for discrete events: Revision of Bayes' Theorem, admissability, Bayes' decisions, minimax. Decision trees, expected value of perfect information. Utility, subjective probability and its measurement. MA30125: Markov processes applications. Aims: To study further Markov processes in both discrete and continuous time. Group? To apply results in areas such genetics, biological processes, networks of work queues, telecommunication networks, electrical networks, resource management, random walks and a voice on the art of, elsewhere. Objectives: On completing the course, students should be able to. * Formulate appropriate Markovian models for a variety of work essay by ruskin real life problems and apply suitable theoretical results to obtain solutions.
* Classify a variety of birth-death processes as explosive or non-explosive. * Find the Q-matrix of a time-reversed chain and make effective use of time reversal. Topics covering both discrete and continuous time Markov chains will be chosen from: Genetics, the Wright-Fisher and Moran models. Epidemics. Telecommunication models, blocking probabilities of Erlang and Engset. Models of interference in & theses communication networks, the ALOHA model. Series of M/M/s queues. Essay By Ruskin? Open and closed migration processes. Explosions.
Birth-death processes. Branching processes. Resource management. A Voice From Reading? Electrical networks. Random walks, reflecting random walks as queuing models in one or more dimensions. The strong Markov property. The Poisson process in time and space. Work By Ruskin? Other applications. Aims: To satisfy as many of the objectives as possible as set out in the individual project proposal.
Objectives: To produce the deliverables identified in & theses the individual project proposal. Defined in the individual project proposal. MA30170: Numerical solution of PDEs I. Aims: To teach numerical methods for elliptic and parabolic partial differential equations via the finite element method based on variational principles. Objectives: At the end of the course students should be able to derive and implement the finite element method for work essay by ruskin, a range of standard elliptic and parabolic partial differential equations in one and dissertation, several space dimensions. They should also be able to derive and work by ruskin, use elementary error estimates for these methods.
* Variational and weak form of elliptic PDEs. Romeo And Juliet Conflict Essay? Natural, essential and mixed boundary conditions. Linear and quadratic finite element approximation in one and several space dimensions. An introduction to convergence theory. * System assembly and solution, isoparametric mapping, quadrature, adaptivity.
* Applications to PDEs arising in applications. * Brief introduction to time dependent problems. Aims: The aim is to work by ruskin, explore pure mathematics from a problem-solving point of paper length view. In addition to conventional lectures, we aim to encourage students to work on solving problems in small groups, and to give presentations of solutions in workshops. Objectives: At the end of the course, students should be proficient in formulating and testing conjectures, and will have a wide experience of different proof techniques. The topics will be drawn from cardinality, combinatorial questions, the essay by ruskin foundations of measure, proof techniques in algebra, analysis, geometry and topology. Aims: This is an dissertation, advanced pure mathematics course providing an introduction to classical algebraic geometry via plane curves. It will show some of the essay by ruskin links with other branches of mathematics. Objectives: At the end of the course students should be able to shakespeare conflict, use homogeneous coordinates in projective space and to distinguish singular points of plane curves.
They should be able to demonstrate an understanding of the difference between rational and nonrational curves, know examples of both, and work essay, be able to describe some special features of plane cubic curves. To be chosen from: Affine and projective space. Paper? Polynomial rings and homogeneous polynomials. Ideals in the context of polynomial rings,the Nullstellensatz. Plane curves; degree; Bezout's theorem. Work? Singular points of plane curves. Rational maps and morphisms; isomorphism and birationality. Curves of thesis low degree (up to 3). Genus. Elliptic curves; the group law, nonrationality, the j invariant. Weierstrass p function.
Quadric surfaces; curves of quadrics. Duals. THIS UNIT IS ONLY AVAILABLE IN ACADEMIC YEARS STARTING IN AN EVEN YEAR. Aims: The course will provide a solid introduction to one of the Big Machines of modern mathematics which is also a major topic of work essay current research. In particular, this course provides the necessary prerequisites for post-graduate study of Algebraic Topology.
Objectives: At the end of the course, the students will be conversant with the basic ideas of homotopy theory and, in particular, will be able to compute the fundamental group of several topological spaces. Topics will be chosen from the following: Paths, homotopy and the fundamental group. Homotopy of maps; homotopy equivalence and deformation retracts. Computation of the fundamental group and applications: Fundamental Theorem of Algebra; Brouwer Fixed Point Theorem. Covering spaces. Path-lifting and homotopy lifting properties. Deck translations and the fundamental group. Universal covers. Loop spaces and their topology. Inductive definition of higher homotopy groups.
Long exact sequence in homotopy for fibrations. MA40042: Measure theory integration. Aims: The purpose of this course is to lay the basic technical foundations and thesis on tata, establish the main principles which underpin the work classical notions of area, volume and the related idea of an integral. Objectives: The objective is to familiarise students with measure as a tool in analysis, functional analysis and probability theory. Students will be able to quote and apply the main inequalities in the subject, and to understand their significance in a wide range of contexts. Students will obtain a full understanding of the Lebesgue Integral. Topics will be chosen from the following: Measurability for sets: algebras, #115 -algebras, #112 -systems, d-systems; Dynkin's Lemma; Borel #115 -algebras. Measure in the abstract: additive and #115 -additive set functions; monotone-convergence properties; Uniqueness Lemma; statement of Caratheodory's Theorem and discussion of the #108 -set concept used in its proof; full proof on handout. Lebesgue measure on IRn: existence; inner and outer regularity. Dissertation & Theses? Measurable functions.
Sums, products, composition, lim sups, etc; The Monotone-Class Theorem. Probability. Sample space, events, random variables. Essay By Ruskin? Independence; rigorous statement of the Strong Law for coin tossing. Integration.
Integral of a non-negative functions as sup of the integrals of simple non-negative functions dominated by shakespeare romeo conflict, it. Monotone-Convergence Theorem; 'Additivity'; Fatou's Lemma; integral of 'signed' function; definition of Lp and of L p; linearity; Dominated-Convergence Theorem - with mention that it is not the `right' result. Product measures: definition; uniqueness; existence; Fubini's Theorem. Absolutely continuous measures: the idea; effect on integrals. Statement of the Radon-Nikodm Theorem. Inequalities: Jensen, Holder, Minkowski.
Completeness of Lp. Aims: To introduce and study abstract spaces and general ideas in analysis, to apply them to examples, to lay the foundations for the Year 4 unit in Functional analysis and to motivate the Lebesgue integral. Objectives: By the end of the unit, students should be able to state and prove the principal theorems relating to essay, uniform continuity and uniform convergence for real functions on metric spaces, compactness in spaces of continuous functions, and elementary Hilbert space theory, and to apply these notions and the theorems to simple examples. Topics will be chosen from:Uniform continuity and uniform limits of continuous functions on [0,1]. Abstract Stone-Weierstrass Theorem. Uniform approximation of continuous functions. Polynomial and from the attic on the art of reading, trigonometric polynomial approximation, separability of C[0,1]. Total Boundedness. Diagonalisation. Ascoli-Arzelagrave Theorem.
Complete metric spaces. Baire Category Theorem. Nowhere differentiable function. Picard's theorem for x = f(x,t). Work Essay? Metric completion M of a metric space M. Real inner product spaces. Hilbert spaces.
Cauchy-Schwarz inequality, parallelogram identity. Examples: l #178 , L #178 [0,1] := C[0,1]. Separability of L #178 . Orthogonality, Gram-Schmidt process. Bessel's inequality, Pythagoras' Theorem. Projections and dissertation paper length, subspaces. Orthogonal complements. Riesz Representation Theorem. Complete orthonormal sets in separable Hilbert spaces. Completeness of trigonometric polynomials in work essay L #178 [0,1].
Fourier Series. Aims: A treatment of the qualitative/geometric theory of dynamical systems to a level that will make accessible an area of mathematics (and allied disciplines) that is highly active and rapidly expanding. Objectives: Conversance with concepts, results and techniques fundamental to dissertation & theses, the study of qualitative behaviour of dynamical systems. An ability to work by ruskin, investigate stability of equilibria and periodic orbits. A basic understanding and appreciation of group bifurcation and chaotic behaviour.
Topics will be chosen from the following: Stability of equilibria. Lyapunov functions. Work By Ruskin? Invariance principle. Periodic orbits. Poincareacute maps. Hyperbolic equilibria and dissertation paper, orbits. Essay? Stable and unstable manifolds. Word Introduction? Nonhyperbolic equilibria and orbits. Centre manifolds. Bifurcation from a simple eigenvalue. Introductory treatment of chaotic behaviour.
Horseshoe maps. Symbolic dynamics. MA40048: Analytical geometric theory of differential equations. Aims: To give a unified presention of systems of ordinary differential equations that have a Hamiltonian or Lagrangian structure. Geomtrical and analytical insights will be used to by ruskin, prove qualitative properties of solutions. These ideas have generated many developments in modern pure mathematics, such as sympletic geometry and ergodic theory, besides being applicable to the equations of classical mechanics, and count introduction, motivating much of by ruskin modern physics. Objectives: Students will be able to state and prove general theorems for Lagrangian and romeo conflict, Hamiltonian systems.
Based on these theoretical results and essay by ruskin, key motivating examples they will identify general qualitative properties of group solutions of these systems. Lagrangian and Hamiltonian systems, phase space, phase flow, variational principles and Euler-Lagrange equations, Hamilton's Principle of least action, Legendre transform, Liouville's Theorem, Poincare recurrence theorem, Noether's Theorem. MA40050: Nonlinear equations bifurcations. Aims: To extend the real analysis of work implicitly defined functions into the numerical analysis of iterative methods for computing such functions and to teach an awareness of practical issues involved in applying such methods. Objectives: The students should be able to solve a variety of nonlinear equations in many variables and should be able to assess the performance of their solution methods using appropriate mathematical analysis. Topics will be chosen from the following: Solution methods for nonlinear equations: Newtons method for systems. Quasi-Newton Methods.
Eigenvalue problems. Theoretical Tools: Local Convergence of Newton's Method. A Voice From The Attic? Implicit Function Theorem. Work Essay? Bifcurcation from the trivial solution. Applications: Exothermic reaction and buckling problems. Continuous and discrete models. Analysis of word for essay introduction parameter-dependent two-point boundary value problems using the shooting method.
Practical use of the shooting method. The Lyapunov-Schmidt Reduction. Application to analysis of discretised boundary value problems. Work Essay? Computation of solution paths for systems of nonlinear algebraic equations. Pseudo-arclength continuation. Dissertation & Theses? Homotopy methods. Computation of turning points. Bordered systems and their solution.
Exploitation of symmetry. Essay By Ruskin? Hopf bifurcation. The Attic Essays Reading? Numerical Methods for Optimization: Newton's method for unconstrained minimisation, Quasi-Newton methods. Aims: To introduce the theory of infinite-dimensional normed vector spaces, the linear mappings between them, and spectral theory. Objectives: By the end of the unit, the students should be able to work, state and prove the principal theorems relating to Banach spaces, bounded linear operators, compact linear operators, and spectral theory of compact self-adjoint linear operators, and apply these notions and theorems to simple examples.
Topics will be chosen from the following: Normed vector spaces and their metric structure. Banach spaces. Young, Minkowski and Holder inequalities. Examples - IRn, C[0,1], l p, Hilbert spaces. Dissertation? Riesz Lemma and finite-dimensional subspaces. The space B(X,Y) of bounded linear operators is a Banach space when Y is complete. Work? Dual spaces and essay made chemist, second duals.
Uniform Boundedness Theorem. Work Essay? Open Mapping Theorem. Closed Graph Theorem. Projections onto closed subspaces. Dissertation Paper? Invertible operators form an open set. Work? Power series expansion for (I-T)- #185 . Compact operators on a voice the attic Banach spaces. Spectrum of an operator - compactness of spectrum. Operators on Hilbert space and their adjoints. Spectral theory of self-adjoint compact operators.
Zorn's Lemma. Hahn-Banach Theorem. Canonical embedding of X in X* * is isometric, reflexivity. Simple applications to weak topologies. Aims: To stimulate through theory and especially examples, an interest and appreciation of the power of this elegant method in analysis and probability. Applications of the theory are at the heart of this course. Objectives: By the end of the course, students should be familiar with the main results and techniques of discrete time martingale theory. They will have seen applications of essay martingales in proving some important results from classical probability theory, and they should be able to recognise and apply martingales in solving a variety of more elementary problems. Topics will be chosen from the following: Review of fundamental concepts. Conditional expectation. Martingales, stopping times, Optional-Stopping Theorem.
The Convergence Theorem. L #178 -bounded martingales, the random-signs problem. Dissertation Length? Angle-brackets process, Leacutevy's Borel-Cantelli Lemma. Uniform integrability. UI martingales, the work essay Downward Theorem, the Strong Law, the Submartingale Inequality. Likelihood ratio, Kakutani's theorem. MA40061: Nonlinear optimal control theory. Aims: Four concepts underpin control theory: controllability, observability, stabilizability and optimality. Of these, the first two essentially form the focus of the thesis Year 3/4 course on linear control theory. In this course, the latter notions of stabilizability and optimality are developed. Together, the courses on linear control theory and nonlinear optimal control provide a firm foundation for participating in theoretical and practical developments in work by ruskin an active and count for essay introduction, expanding discipline.
Objectives: To present concepts and work by ruskin, results pertaining to robustness, stabilization and dissertation length, optimization of (nonlinear) finite-dimensional control systems in a rigorous manner. Emphasis is work essay by ruskin placed on for essay optimization, leading to conversance with both the work essay by ruskin Bellman-Hamilton-Jacobi approach and the maximum principle of Pontryagin, together with their application. Topics will be chosen from the following: Controlled dynamical systems: nonlinear systems and linearization. Shakespeare Essay? Stability and robustness. Stabilization by feedback. Lyapunov-based design methods. Stability radii. Work Essay By Ruskin? Small-gain theorem. Optimal control.
Value function. The Bellman-Hamilton-Jacobi equation. Verification theorem. Quadratic-cost control problem for linear systems. Riccati equations. The Pontryagin maximum principle and transversality conditions (a dynamic programming derivation of a restricted version and statement of the general result with applications). Proof of the maximum principle for the linear time-optimal control problem.
MA40062: Ordinary differential equations. Aims: To provide an accessible but rigorous treatment of initial-value problems for nonlinear systems of ordinary differential equations. Foundations will be laid for advanced studies in dynamical systems and control. The material is also useful in mathematical biology and numerical analysis. Objectives: Conversance with existence theory for essay inventions made, the initial-value problem, locally Lipschitz righthand sides and uniqueness, flow, continuous dependence on initial conditions and parameters, limit sets. Topics will be chosen from the following: Motivating examples from diverse areas. Existence of solutions for the initial-value problem. Work Essay? Uniqueness.
Maximal intervals of existence. Dependence on initial conditions and parameters. Flow. Global existence and dynamical systems. Limit sets and attractors. Aims: To satisfy as many of the objectives as possible as set out in the individual project proposal.
Objectives: To produce the deliverables identified in the individual project proposal. Defined in the individual project proposal. MA40171: Numerical solution of PDEs II. Aims: To teach an understanding of linear stability theory and its application to ODEs and evolutionary PDEs. Objectives: The students should be able to analyse the stability and convergence of a range of & theses numerical methods and assess the by ruskin practical performance of these methods through computer experiments. Solution of initial value problems for ODEs by Linear Multistep methods: local accuracy, order conditions; formulation as a one-step method; stability and convergence. Introduction to physically relevant PDEs. Well-posed problems.
Truncation error; consistency, stability, convergence and dissertation & theses, the Lax Equivalence Theorem; techniques for finding the stability properties of particular numerical methods. Numerical methods for parabolic and work by ruskin, hyperbolic PDEs. MA40189: Topics in Bayesian statistics. Aims: To introduce students to the ideas and techniques that underpin the theory and practice of the Bayesian approach to statistics. Objectives: Students should be able to formulate the thesis on tata Bayesian treatment and analysis of many familiar statistical problems. Bayesian methods provide an alternative approach to data analysis, which has the ability to incorporate prior knowledge about a parameter of interest into the statistical model. The prior knowledge takes the form of a prior (to sampling) distribution on the parameter space, which is updated to essay, a posterior distribution via Bayes' Theorem, using the data. Summaries about the group parameter are described using the posterior distribution.
The Bayesian Paradigm; decision theory; utility theory; exchangeability; Representation Theorem; prior, posterior and predictive distributions; conjugate priors. Tools to undertake a Bayesian statistical analysis will also be introduced. Simulation based methods such as Markov Chain Monte Carlo and importance sampling for use when analytical methods fail. Aims: The course is intended to provide an elementary and assessible introduction to the state-space theory of linear control systems. Essay By Ruskin? Main emphasis is on continuous-time autonomous systems, although discrete-time systems will receive some attention through sampling of continuous-time systems. Contact with classical (Laplace-transform based) control theory is made in the context of shakespeare romeo conflict essay realization theory.
Objectives: To instill basic concepts and essay, results from dissertation & theses control theory in a rigorous manner making use of elementary linear algebra and essay by ruskin, linear ordinary differential equations. Word For Essay Introduction? Conversance with controllability, observability, stabilizabilty and realization theory in essay a linear, finite-dimensional context. Content: Topics will be chosen from the following: Controlled and observed dynamical systems: definitions and classifications. Controllability and observability: Gramians, rank conditions, Hautus criteria, controllable and unobservable subspaces. Input-output maps. Essay Made Chemist? Transfer functions and state-space realizations. State feedback: stabilizability and work, pole placement. Observers and group, output feedback: detectability, asymptotic state estimation, stabilization by dynamic feedback.
Discrete-time systems: z-transform, deadbeat control and observation. Sampling of continuous-time systems: controllability and work essay by ruskin, observability under sampling. Aims: To introduce students to the applications of advanced analysis to the solution of PDEs. Objectives: Students should be able to obtain solutions to certain important PDEs using a variety of techniques e.g. Green's functions, separation of variables. Thesis Group? They should also be familiar with important analytic properties of the solution.
Content: Topics will be chosen from the following: Elliptic equations in two independent variables: Harmonic functions. By Ruskin? Mean value property. Maximum principle (several proofs). Dirichlet and Neumann problems. Representation of solutions in romeo essay terms of work Green's functions. Continuous dependence of data for Dirichlet problem. Uniqueness. Parabolic equations in two independent variables: Representation theorems. Green's functions. Self-adjoint second-order operators: Eigenvalue problems (mainly by example).
Separation of variables for word for essay, inhomogeneous systems. Green's function methods in general: Method of images. Work Essay By Ruskin? Use of integral transforms. Conformal mapping. Calculus of variations: Maxima and minima. Lagrange multipliers. Extrema for integral functions. Euler's equation and its special first integrals. Integral and non-integral constraints.
Aims: The aim of the course is to introduce students to made, applications of work essay by ruskin partial differential equations to model problems arising in biology. The course will complement Mathematical Biology I where the essay inventions emphasis was on ODEs and Difference Equations. Objectives: Students should be able to derive and interpret mathematical models of work essay by ruskin problems arising in biology using PDEs. They should be able to perform a linearised stability analysis of a reaction-diffusion system and and juliet conflict, determine criteria for work essay, diffusion-driven instability. They should be able to interpret the word results in terms of the original biological problem. Content: Topics will be chosen from the following: Partial Differential Equation Models: Simple random walk derivation of the diffusion equation. Solutions of the diffusion equation.
Density-dependent diffusion. Conservation equation. Reaction-diffusion equations. Chemotaxis. Examples for insect dispersal and cell aggregation. Spatial Pattern Formation: Turing mechanisms. Linear stability analysis.
Conditions for diffusion-driven instability. Dispersion relation and Turing space. Scale and geometry effects. Mode selection and dispersion relation. Applications: Animal coat markings.
How the leopard got its spots. Butterfly wing patterns. Aims: To introduce the general theory of continuum mechanics and, through this, the study of viscous fluid flow. Objectives: Students should be able to essay, explain the and juliet basic concepts of essay continuum mechanics such as stress, deformation and constitutive relations, be able to formulate balance laws and be able to apply these to the solution of simple problems involving the dissertation & theses flow of a viscous fluid. Content: Topics will be chosen from the following: Vectors: Linear transformation of vectors. Proper orthogonal transformations. Rotation of axes. Transformation of essay components under rotation. Cartesian Tensors: Transformations of components, symmetry and skew symmetry.
Isotropic tensors. Kinematics: Transformation of line elements, deformation gradient, Green strain. Linear strain measure. Displacement, velocity, strain-rate. Stress: Cauchy stress; relation between traction vector and stress tensor. Global Balance Laws: Equations of motion, boundary conditions. Newtonian Fluids: The constitutive law, uniform flow, Poiseuille flow, flow between rotating cylinders. Aims: To present the theory and application of count for essay normal linear models and generalised linear models, including estimation, hypothesis testing and confidence intervals.
To describe methods of model choice and the use of residuals in diagnostic checking. To facilitate an in-depth understanding of the topic. Objectives: On completing the course, students should be able to. (a) choose an appropriate generalised linear model for a given set of data; (b) fit this model using the GLIM program, select terms for inclusion in work the model and assess the adequacy of a selected model; (c) make inferences on the basis of a fitted model and recognise the assumptions underlying these inferences and possible limitations to their accuracy; (d) demonstrate an in-depth understanding of the topic. Content: Normal linear model: Vector and matrix representation, constraints on parameters, least squares estimation, distributions of parameter and variance estimates, t-tests and confidence intervals, the a voice essays Analysis of Variance, F-tests for unbalanced designs.
Model building: Subset selection and stepwise regression methods with applications in polynomial regression and work, multiple regression. Effects of collinearity in regression variables. Uses of residuals: Probability plots, plots for additional variables, plotting residuals against word introduction fitted values to detect a mean-variance relationship, standardised residuals for outlier detection, masking. Generalised linear models: Exponential families, standard form, statement of asymptotic theory for i.i.d. samples, Fisher information. Linear predictors and essay, link functions, statement of paper length asymptotic theory for work by ruskin, the generalised linear model, applications to paper length, z-tests and essay by ruskin, confidence intervals, #099 #178 -tests and the analysis of deviance. Dissertation? Residuals from generalised linear models and their uses. Applications to dose response relationships, and logistic regression. Aims: To introduce a variety of statistical models for time series and cover the main methods for analysing these models.
To facilitate an in-depth understanding of the topic. Objectives: At the end of the course, the student should be able to: * Compute and interpret a correlogram and a sample spectrum; * derive the properties of ARIMA and state-space models; * choose an appropriate ARIMA model for a given set of work by ruskin data and fit the model using an appropriate package; * compute forecasts for a variety of linear methods and models; * demonstrate an in-depth understanding of the topic. Content: Introduction: Examples, simple descriptive techniques, trend, seasonality, the correlogram. Probability models for time series: Stationarity; moving average (MA), autoregressive (AR), ARMA and ARIMA models. Estimating the autocorrelation function and fitting ARIMA models. Forecasting: Exponential smoothing, Forecasting from ARIMA models. Stationary processes in the frequency domain: The spectral density function, the periodogram, spectral analysis. State-space models: Dynamic linear models and the Kalman filter. MA50089: Applied probability finance. Aims: To develop and apply the theory of probability and stochastic processes to examples from finance and economics.
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To facilitate an in-depth understanding of the topic. Objectives: Be able to carry out a preliminary analysis of multivariate data and select and apply an appropriate technique to essay inventions made chemist, look for work essay, structure in such data or achieve dimensionality reduction. Be able to carry out classical multivariate inferential techniques based on the multivariate normal distribution. Be able to demonstrate an paper length, in-depth understanding of the by ruskin topic. Content: Introduction, Preliminary analysis of multivariate data. Revision of relevant matrix algebra. Principal components analysis: Derivation and interpretation; approximate reduction of dimensionality; scaling problems. Multidimensional distributions: The multivariate normal distribution - properties and parameter estimation.
One and two-sample tests on means, Hotelling's T-squared. Canonical correlations and canonical variables; discriminant analysis. Topics selected from: Factor analysis. The multivariate linear model. Metrics and similarity coefficients; multidimensional scaling. Cluster analysis. Correspondence analysis. Dissertation? Classification and regression trees.
MA50092: Classical statistical inference. Aims: To develop a formal basis for work by ruskin, methods of statistical inference including criteria for the comparison of procedures. Word Count Introduction? To give an in depth description of the asymptotic theory of maximum likelihood methods. To facilitate an in-depth understanding of the topic. Objectives: On completing the course, students should be able to: * calculate properties of estimates and hypothesis tests; * derive efficient estimates and tests for a broad range of problems, including applications to a variety of essay standard distributions; * demonstrate an in-depth understanding of the topic. Revision of standard distributions: Bernoulli, binomial, Poisson, exponential, gamma and word introduction, normal, and their interrelationships. Sufficiency and Exponential families. Point estimation: Bias and variance considerations, mean squared error. Rao-Blackwell theorem. Cramer-Rao lower bound and efficiency.
Unbiased minimum variance estimators and a direct appreciation of efficiency through some examples. Bias reduction. Asymptotic theory for maximum likelihood estimators. Hypothesis testing: Hypothesis testing, review of the Neyman-Pearson lemma and maximisation of work power. Maximum likelihood ratio tests, asymptotic theory. Compound alternative hypotheses, uniformly most powerful tests. Compound null hypotheses, monotone likelihood ratio property, uniformly most powerful unbiased tests. Nuisance parameters, generalised likelihood ratio tests. MA50125: Markov processes applications. Aims: To study further Markov processes in both discrete and continuous time.
To apply results in areas such genetics, biological processes, networks of queues, telecommunication networks, electrical networks, resource management, random walks and word, elsewhere. Work Essay? To facilitate an in-depth understanding of the topic. Objectives: On completing the length course, students should be able to: * Formulate appropriate Markovian models for a variety of real life problems and apply suitable theoretical results to obtain solutions; * Classify a variety of birth-death processes as explosive or non-explosive; * Find the Q-matrix of a time-reversed chain and make effective use of time reversal; * Demonstrate an in-depth understanding of the topic. Content: Topics covering both discrete and continuous time Markov chains will be chosen from: Genetics, the Wright-Fisher and Moran models. Epidemics.
Telecommunication models, blocking probabilities of Erlang and Engset. Models of interference in communication networks, the ALOHA model. Series of M/M/s queues. Open and closed migration processes. Explosions. Work By Ruskin? Birth-death processes. Branching processes.
Resource management. Electrical networks. Random walks, reflecting random walks as queuing models in one or more dimensions. The strong Markov property. The Poisson process in time and space. Other applications. MA50170: Numerical solution of PDEs I.
Aims: To teach numerical methods for elliptic and parabolic partial differential equations via the finite element method based on variational principles. Objectives: At the end of the thesis group course students should be able to derive and implement the finite element method for a range of standard elliptic and parabolic partial differential equations in one and several space dimensions. By Ruskin? They should also be able to inventions made chemist, derive and essay, use elementary error estimates for these methods. Variational and weak form of elliptic PDEs. Natural, essential and mixed boundary conditions. Linear and quadratic finite element approximation in count for essay introduction one and several space dimensions. Work By Ruskin? An introduction to convergence theory. System assembly and solution, isoparametric mapping, quadrature, adaptivity. Applications to PDEs arising in applications. Parabolic problems: methods of lines, and simple timestepping procedures. Shakespeare? Stability and convergence.
MA50174: Theory methods 1b-differential equations: computation and applications. Content: Introduction to Maple and Matlab and work essay by ruskin, their facilities: basic matrix manipulation, eigenvalue calculation, FFT analysis, special functions, solution of simultaneous linear and nonlinear equations, simple optimization. Basic graphics, data handling, use of toolboxes. Problem formulation and solution using Matlab. Numerical methods for solving ordinary differential equations: Matlab codes and student written codes.
Convergence and Stability. Shooting methods, finite difference methods and spectral methods (using FFT). Sample case studies chosen from: the two body problem, the three body problem, combustion, nonlinear control theory, the Lorenz equations, power electronics, Sturm-Liouville theory, eigenvalues, and orthogonal basis expansions. Finite Difference Methods for classical PDEs: the wave equation, the heat equation, Laplace's equation. MA50175: Theory methods 2 - topics in differential equations. Aims: To describe the theory and phenomena associated with hyperbolic conservation laws, typical examples from applications areas, and their numerical approximation; and to introduce students to the literature on the subject.
Objectives: At the end of the course, students should be able to recognise the importance of inventions chemist conservation principles and be familiar with phenomena such as shocks and rarefaction waves; and they should be able to choose appropriate numerical methods for work essay, their approximation, analyse their behaviour, and implement them through Matlab programs. Content: Scalar conservation laws in and juliet essay 1D: examples, characteristics, shock formation, viscosity solutions, weak solutions, need for an entropy condition, total variation, existence and uniqueness of solutions.Design of conservative numerical methods for hyperbolic systems: interface fluxes, Roe's first order scheme, Lax-Wendroff methods, finite volume methods, TVD schemes and the Harten theorem, Engquist-Osher method. The Riemann problem: shocks and the Hugoniot locus, isothermal flow and by ruskin, the shallow water equations, the Godunov method, Euler equations of compressible fluid flow. System wave equation in 2D. R.J. LeVeque, Numerical Methods for Conservation Laws (2nd Edition), Birkhuser, 1992. K.W. Morton D.F. Mayers, Numerical Solution of Partial Differential Equations, CUP, 1994.R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, CUP, 2002. MA50176: Methods applications 1: case studies in mathematical modelling and industrial mathematics.
Content: Applications of the theory and shakespeare conflict, techniques learnt in the prerequisites to solve real problems drawn from from the industrial collaborators and/or from the industrially related research work of the key staff involved. Instruction and practical experience of a set of problem solving methods and techniques, such as methods for simplifying a problem, scalings, perturbation methods, asymptotic methods, construction of similarity solutions. Comparison of mathematical models with experimental data. By Ruskin? Development and refinement of mathematical models. Case studies will be taken from micro-wave cooking, Stefan problems, moulding glass, contamination in pipe networks, electrostatic filtering, DC-DC conversion, tests for elasticity. Students will work in teams under the pressure of project deadlines. They will attend lectures given by external industrialists describing the application of mathematics in an industrial context.
They will write reports and give presentations on romeo conflict the case studies making appropriate use of computer methods, graphics and work essay by ruskin, communication skills. MA50177: Methods and applications 2: scientific computing. Content: Units, complexity, analysis of algorithms, benchmarks. Floating point arithmetic. Programming in Fortran90: Makefiles, compiling, timing, profiling. Data structures, full and sparse matrices. Libraries: BLAS, LAPACK, NAG Library. Visualisation. Dissertation? Handling modules in other languages such as C, C++. Software on essay the Web: Netlib, GAMS.
Parallel Computation: Vectorisation, SIMD, MIMD, MPI. Performance indicators. Case studies illustrating the lectures will be chosen from the topics:Finite element implementation, iterative methods, preconditioning; Adaptive refinement; The algebraic eigenvalue problem (ARPACK); Stiff systems and the NAG library; Nonlinear 2-point boundary value problems and bifurcation (AUTO); Optimisation; Wavelets and data compression. Content: Topics will be chosen from the from on the art of following: The algebraic eigenvalue problem: Gerschgorin's theorems. The power method and its extensions. Backward Error Analysis (Bauer-Fike). The (Givens) QR factorization and the QR method for symmetric tridiagonal matrices. (Statement of convergence only). The Lanczos Procedure for reduction of a real symmetric matrix to tridiagonal form.
Orthogonality properties of Lanczos iterates. Iterative Methods for Linear Systems: Convergence of stationary iteration methods. Special cases of symmetric positive definite and diagonally dominant matrices. Variational principles for linear systems with real symmetric matrices. The conjugate gradient method. Work By Ruskin? Krylov subspaces. Convergence. Connection with the reading Lanczos method. Iterative Methods for Nonlinear Systems: Newton's Method. Convergence in 1D. Statement of algorithm for systems.
Content: Topics will be chosen from the work following: Difference equations: Steady states and fixed points. Stability. Period doubling bifurcations. Chaos. Application to population growth. Systems of difference equations: Host-parasitoid systems.Systems of ODEs: Stability of solutions. Critical points. Phase plane analysis. Poincari-Bendixson theorem. Bendixson and Dulac negative criteria. Conservative systems.
Structural stability and instability. Lyapunov functions. Travelling wave fronts: Waves of advance of an advantageous gene. Waves of shakespeare and juliet conflict essay excitation in nerves. Work Essay By Ruskin? Waves of essay made chemist advance of an epidemic. Content: Topics will be chosen from the following: Revision: Kinematics of deformation, stress analysis, global balance laws, boundary conditions. Constitutive law: Properties of real materials; constitutive law for linear isotropic elasticity, Lami moduli; field equations of linear elasticity; Young's modulus, Poisson's ratio. Some simple problems of elastostatics: Expansion of a spherical shell, bulk modulus; deformation of a block under gravity; elementary bending solution. Linear elastostatics: Strain energy function; uniqueness theorem; Betti's reciprocal theorem, mean value theorems; variational principles, application to composite materials; torsion of cylinders, Prandtl's stress function. Linear elastodynamics: Basic equations and general solutions; plane waves in unbounded media, simple reflection problems; surface waves.
MA50181: Theory methods 1a - differential equations: theory methods. Content: Sturm-Liouville theory: Reality of eigenvalues. Orthogonality of essay eigenfunctions. Expansion in eigenfunctions. Approximation in a voice from the attic art of mean square. Statement of completeness. Fourier Transform: As a limit of Fourier series.
Properties and applications to solution of differential equations. Frequency response of linear systems. Characteristic functions. Linear and quasi-linear first-order PDEs in two and work by ruskin, three independent variables: Characteristics. Integral surfaces. Uniqueness (without proof).
Linear and quasi-linear second-order PDEs in two independent variables: Cauchy-Kovalevskaya theorem (without proof). Characteristic data. Lack of continuous dependence on initial data for Cauchy problem. Classification as elliptic, parabolic, and hyperbolic. Different standard forms. Constant and nonconstant coefficients. One-dimensional wave equation: d'Alembert's solution. Uniqueness theorem for thesis, corresponding Cauchy problem (with data on a spacelike curve). Content: Definition and examples of metric spaces.
Convergence of sequences. Essay? Continuous maps and isometries. Sequential definition of a voice on the reading continuity. Subspaces and product spaces. Work Essay By Ruskin? Complete metric spaces and thesis on tata group, the Contraction Mapping Principle. Sequential compactness, Bolzano-Weierstrass theorem and work essay, applications. Open and closed sets. Closure and romeo and juliet, interior of sets. Topological approach to continuity and compactness (with statement of Heine-Borel theorem). Equivalence of Compactness and sequential compactness in metric spaces.
Connectedness and path-connectedness. Metric spaces of functions: C[0,1] is a complete metric space. MA50183: Specialist reading course. * advanced knowledge in the chosen field. * evidence of independent learning. * an ability to read critically and master an essay by ruskin, advanced topic in mathematics/ statistics/probability. Content: Defined in the individual course specification. MA50183: Specialist reading course.
advanced knowledge in the chosen field. evidence of independent learning. an ability to dissertation, read critically and master an advanced topic in mathematics/statistics/probability. Content: Defined in the individual course specification. MA50185: Representation theory of finite groups.
Content: Topics will be chosen from the following: Group algebras, their modules and associated representations. Maschke's theorem and complete reducibility. Irreducible representations and Schur's lemma. Decomposition of the regular representation. Character theory and orthogonality theorems. Burnside's p #097 q #098 theorem. Content: Topics will be chosen from the following: Functions of a complex variable. Continuity.
Complex series and power series. Circle of convergence. The complex plane. Regions, paths, simple and closed paths. Path-connectedness. Analyticity and the Cauchy-Riemann equations. Work Essay By Ruskin? Harmonic functions. Cauchy's theorem. Cauchy's Integral Formula and inventions made chemist, its application to power series. Isolated zeros. Differentiability of an analytic function.
Liouville's Theorem. Zeros, poles and essential singularities. Laurent expansions. Cauchy's Residue Theorem and contour integration. Applications to real definite integrals. On completion of the course, the student should be able to demonstrate:- * Advanced knowledge in the chosen field.
* Evidence of independent learning. * An ability to initiate mathematical/statistical research. * An ability to read critically and master an advanced topic in mathematics/ statistics/probability to work by ruskin, the extent of being able to expound it in a coherent, well-argued dissertation. * Competence in a document preparation language to the extent of being able to typeset a dissertation with substantial mathematical/statistical content. Content: Defined in the individual project specification. MA50190: Advanced mathematical methods. Objectives: Students should learn a set of mathematical techniques in a variety of areas and be able to apply them to either solve a problem or to construct an accurate approximation to the solution. They should demonstrate an understanding of both the theory and the range of applications (including the limitations) of all the & theses techniques studied.
Content: Transforms and Distributions: Fourier Transforms, Convolutions (6 lectures, plus directed reading on complex analysis and calculus of residues). Asymptotic expansions: Laplace's method, method of steepest descent, matched asymptotic expansions, singular perturbations, multiple scales and averaging, WKB. (12 lectures, plus directed reading on applications in continuum mechanics). Dimensional analysis: scaling laws, reduction of PDEs and ODEs, similarity solutions. (6 lectures, plus directed reading on symmetry group methods). References: L. Dresner, Similarity Solutions of essay by ruskin Nonlinear PDEs , Pitman, 1983; JP Keener, Principles of Applied Mathematics, Addison Wesley, 1988; P. Olver, Symmetry Methods for romeo and juliet conflict, PDEs, Springer; E.J. Hinch, Perturbation Methods, CUP. Objectives: At the end of the course students should be able to use homogeneous coordinates in projective space and to distinguish singular points of plane curves.
They should be able to demonstrate an understanding of the difference between rational and nonrational curves, know examples of both, and be able to describe some special features of work plane cubic curves. Content: To be chosen from: Affine and projective space. Polynomial rings andhomogeneous polynomials. Ideals in the context of essay polynomial rings,the Nullstellensatz. Work? Plane curves; degree; Bezout's theorem. Singular points of plane curves. Rational maps and morphisms; isomorphism and on tata, birationality. Curves of work essay low degree (up to 3). Genus. Elliptic curves; the group law, nonrationality, the j invariant. Weierstrass p function.
Quadric surfaces; curves of quadrics. Duals. MA50194: Advanced statistics for use in health contexts 2. * To equip students with the skills to use and interpret advanced multivariate statistics; * To provide an appreciation of the applications of advanced multivariate analysis in health and medicine. Learning Outcomes: On completion of this unit, students will: * Learn and understand how and why selected advanced multivariate analyses are computed; * Practice conducting, interpreting and reporting analyses. * To learn independently; * To critically evaluate and & theses, assess research and evidence as well as a variety of other information; * To utilise problem solving skills.
* Advanced information technology and computing technology (e.g. SPSS); * Independent working skills; * Advanced numeracy skills. Content: Introduction to STATA, power and sample size, multidimensional scaling, logistic regression, meta-analysis, structural equation modelling. Student Records Examinations Office, University of Bath, Bath BA2 7AY. Tel: +44 (0) 1225 384352 Fax: +44 (0) 1225 386366.
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