# Brainstorming college application essays

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Tips for Brainstorming College Application Essay Topics |…

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Frequently Asked Questions About the Criterion ® Online Writing Evaluation Service.
Students get a response to their writing while it is fresh in their minds. They find out immediately how their work compares to a standard and **essays**, what they should do to improve it. **Maps Project**! The Criterion service also provides an application environment for writing and revision that students can use independently, 24 hours a day. This environment, coupled with the opportunity for instant feedback, provides the directed writing practice that is so beneficial for students.
How many topics are available?
The Criterion Online Writing Evaluation Topics Library for K12 Education includes writing prompts for grades 412.

Criterion essay topics are constructed to elicit writing in various modes that include persuasive, expository, descriptive and narrative. **Online Writing**! All prompts are grade level appropriate in vocabulary and **brainstorming college application essays**, appeal to student interests. Each topic may be scored on either a 6-point or 4-point scale and the associated rubrics are shown with each prompt.
Currently, there are 61 College Level I topics appropriate for first-year writing courses, practice and placement; 64 College Level II topics appropriate for second-year writing courses and practice; 10 College Preparatory topics; 14 GRE ® test topics; and 35 TOEFL ® test topics.
Instructors can also create and assign their own writing prompts for a student assignment. Because instructors can create their own topics, the **research**, topic library is endless.
The Criterion service library of topics contains assignments representing the **college**, following writing genres: persuasive, informative, narrative, expository, issue and argumentative.

Where do Criterion service topics come from? Criterion topics come from a number of sources, including ETS testing programs such as the Praxis ® tests, the GRE and TOEFL tests, and client programs such as NAEP® and the English Placement Test designed for California State University. Criterion topics have been developed based on representative samples that are mode-specific and that utilize 6-point holistic scales based on widely accepted writing standards. How does the Criterion service handle an unusual writing style? The Criterion service looks for specific features of syntax, organization and vocabulary. If the essay under consideration is not sufficiently similar to those in its database of already-scored essays, the Criterion service posts a warning, called an Advisory, saying that it is report unable to give an accurate score. Advisories usually result from essays that are too brief or those in which the vocabulary is college essays unusual or the content is on city life vs. country life off-topic.

Will the use of the Criterion service stifle creative writing among students?
Not necessarily. **Brainstorming College Application**! The Criterion service is on globalization designed to be used for evaluating writing done under testing conditions situations in which even the most creative writers concentrate on playing it safe with straightforward and competent writing.
Will the Criterion service catch cheating or plagiarism?
No. **College Application Essays**! The Criterion service simply evaluates the **papers mining**, essay.

It is up to the institution to ensure that students are working independently and submitting their own work.
Instructors can opt to display a writer's sample for some topics on **application** the Create Assignment screen. Students can then view the **segovia thesis**, samples and refer to them while they write their own essays. The sample essays are in a read-only format and cannot be copied and pasted into another document.
What information does the Criterion service report to **college application essays**, educators?
Educators have easy and secure access to **maps**, each student's portfolio of essays, diagnostic reports and scores, as well as summary information on the performance of entire classes.
What information does the Criterion service report to students?
Typically, students get diagnostic feedback, as well as a holistic evaluation, each time they submit an essay.

However, educators can block students from seeing their scores and may choose to do so if they use the **application essays**, Criterion service for **vs. country** benchmarking.
Can instructors limit student feedback?
Yes. Instructors can elect to report all, some or none of the **brainstorming**, feedback analysis. **Google Essay**! When creating an assignment, instructors turn the score analysis feature on **application** or off, as well as select which diagnostic feedback to report.
Can instructors limit access to assignments?
Yes. Instructors can limit access when selecting assignment options. **Essay Writing**! For example, the date and time an assignment is available are selected by instructors during setup. **Application Essays**! They can also limit how many times a student can write and revise an assignment.
Can instructors impose time limits on assignments?

Yes. Many assignments available from the **online essay writing for dummies**, Criterion service library of topics have time limits associated with them. When creating the assignment, instructors select whether to impose a time limit, or they can turn off the **brainstorming application**, time limit function to allow unlimited writing and revision time.
How is the Criterion service feedback different from the Microsoft Word® Spelling and Grammar tool?
The Microsoft Word Spelling and Grammar tool can provide writers with a quick analysis of **writing** common errors. However, the Criterion service, as an instructional tool used to improve writing, targets more precise feedback. Research shows that the spelling error detection and correction module in *application*, the Criterion service has better precision than the spelling error detection and correction module used in MS Word. **Essay**! We continually strive through research and **college essays**, user input to improve the **essay**, precision of all our feedback categories.
What is the Writer's Handbook?
The Writer's Handbook is an intuitive online tool that a student can access while reviewing diagnostic feedback.

It explains every error or feature reported by defining it and providing examples of correct and incorrect use. There are five Writer's Handbook versions available: Elementary, Middle Schools, Descriptive, High School/College and ELL. There are also four bilingual versions available: Spanish/English, Simplified Chinese/English, Japanese/English and **application essays**, Korean/English.
Using the Criterion Service for **phd thesis on globalization** Remediation, Placement and Assessment.
In the vast majority of **brainstorming college application essays** cases, ETS researchers generally found either exact or adjacent agreement (within one point) between the Criterion service scores and those of a trained essay reader or instructor.

Both used the **on data mining**, same scoring guidelines and **college**, scoring system.
How can the Criterion service be used for writing remediation and in basic skills writing classes?
Instructors assign the Criterion service standard topics or use their own topics to give students opportunities for additional writing practice. The Criterion service topics library contains a group of writing assignments called College Level Preparatory. These topics are graded against a lower level scoring rubric and can be assigned to gradually move incoming freshmen up to the first-year writing level. Instructors may assign topics to encourage students to focus on essential problem areas that will improve their writing. The immediate feedback features of the **essay writing for dummies**, Criterion service provide additional motivation for students to write and revise their essays when writing on their own.
How are the Criterion scores used for placement?

Students may be assigned to classes on the basis of **college application** their scores on a Criterion service-scored essay or the combination of a Criterion service score and other indicators. The electronic score should not be the **project**, sole basis for a placement decision. It is essays best to combine a Criterion score with the score of a human reader in the same way that institutions combine scores from two different human readers. If the **bless me ultima book report**, two scores differ by more than one point, a different reader should also evaluate the essay.
How is the Criterion service used for assessment purposes?
Some institutions use the Criterion service scores for **application** exit testing combining a Criterion service score with the score from a reader in *project*, the same way they combine scores from two different readers. If the **brainstorming college**, two scores differ by more than one point, a different reader also evaluates the essay. Some institutions use the Criterion service for benchmark testing, assigning the Criterion service-scored essays at *papers on data mining*, specified points during an academic term.

How can the Criterion service be used in a writing lab?
When the Criterion service is used in *brainstorming college application*, a writing lab, tutors and writing mentors have access to topics, feedback and **google maps essay**, student portfolios. They also have a way to communicate with instructors about student progress. Use of the **brainstorming college application essays**, Criterion service in a writing lab facilitates writing across the curriculum when students use the lab to check in-progress writing for all of their classes. Providing access to **phd thesis on globalization**, an open-ended instructor's topic allows students to write an essays essay about segovia thesis, any subject assigned by any instructor. The interactive features of the Criterion service promote communication between classroom learning and writing lab support.
How do students feel about being scored by a machine?
Most of today's students have had experience with instant feedback in computer programs and are comfortable with the **essays**, idea of computerized scoring.
Can the **book report**, Criterion service score essays on other topics?

Yes. Using the Scored Instructor Topic feature, teachers can create their own topics that are parallel to **essays**, the Criterion service library prompts, and the students' essays will receive Criterion scores upon completion. A link in the Criterion service provides step-by-step instructions on **online essay for dummies** how to create either a persuasive or expository topic that can be scored.
A Criterion score is an overall score (usually on a 4- or 6-point scale) that is given to an essay. The Criterion service scoring compares a student's writing to thousands of essays written and evaluated by writing instructors.
The essays used to build the scoring models have been scored by trained readers and **brainstorming application essays**, were written by students under timed-testing conditions. The writers had no opportunity to revise, use a spell-checker or reflect on what they had written. So when students write on the Criterion service topics in *bless book*, a regular class, working under more relaxed conditions, instructors and students should recognize that students' scores may not precisely compare to those of the **brainstorming college application essays**, samples.
The Criterion score is a holistic score based on **essay on city life life** the traits of word choice, convention and fluency/organization. **College Application**! The Criterion score also takes content relevance into account by analyzing the degree of similarity between prompt-specific vocabulary and that of the response.
Does Criterion provide trait scores?

Yes. The trait scores are shown as Developing, Proficient and Advanced. These are based on a normative range, where the **segovia thesis**, majority (60 percent) of student scores falls. **Essays**! Responses scoring within this range are considered proficient at this grade level. Responses scoring below this range are considered developing these traits at this grade level. **Online Essay**! Responses scoring above this range are considered advanced at this grade level.
How does the **essays**, Criterion service come up with its scores?

The Criterion service is based on a technology called e-rater ® that was developed by Educational Testing Service. The e-rater scoring engine compares the **bless me ultima book report**, new essay to samples of essays previously scored by readers, looking for similarities in sentence structure, organization and vocabulary. **Brainstorming College Application Essays**! Essays earning high scores are those with characteristics most similar to the high-scoring essays in the sample group; essays earning low scores share characteristics with low-scoring essays in the sample group.
What is the technology used in the e-rater scoring?
The e-rater scoring engine is an application of Natural Language Processing (NLP), a field of computer technology that uses computational methods to analyze characteristics of text. Researchers have been using NLP for the past 50 years to translate text from segovia thesis, one language to another and to summarize text. **College Application Essays**! Internet search engines currently use NLP to retrieve information.

The e-rater scoring engine uses NLP to identify the features of the faculty-scored essays in its sample collection and **research on data mining**, store them with their associated weights in a database. **Brainstorming**! When e-rater evaluates a new essay, it compares its features to **segovia thesis**, those in the database in order to assign a score.
Because the **college essays**, e-rater scoring engine is not doing any actual reading, the validity of its scoring depends on the scoring of the sample essays from which the e-rater database is created.
Can students trick the **segovia thesis**, Criterion service?
Yes. Since the **brainstorming college essays**, e-rater engine cannot really understand English, it can be fooled by on globalization an illogical, but well-written, argument. Educators can discourage students from deliberately trying to fool the Criterion service by announcing that a random sample of essays will be read by independent readers. The Criterion service will also display an Advisory along with the e-rater score when an essay displays certain characteristics that warrant attention compared to other essays scored against the same topic.
Must students be connected to the Internet to use the Criterion service?
Students can initially compose their essays offline, using any word-processing application.

However, they will ultimately need an Internet connection to be able to cut and paste their essays into application, the Criterion essay submission box so their work can be scored and **me ultima book report essay**, analyzed. For assignments that are timed, essays should be composed online only to ensure accountability by all students and to **application essays**, accurately reflect their writing skills in this environment.
Can I import student identifiers from my data management system?
Yes. The Criterion service has import capabilities for administrators at *segovia thesis*, several levels. A Criterion Administrator can easily import by using templates provided in the system.

Details are provided in both the **brainstorming college essays**, HELP text and the Criterion® User Manual and Administrator Supplement .
Can I save my data?
Yes. The Criterion service has export features that easily allow users to create export files, and **on data**, an archive portfolios feature that can be used to create export files in a comma-delimited format (.csv) that can be opened by most text editors and spreadsheet programs. **College Application Essays**! Detailed instructions for both features are provided in the Criterion ® User Manual and Administrator Supplement .
Understanding the Analysis of Organization and Development in Student Essays.
There is now broad acceptance of automated essay scoring technology for large-scale assessment and classroom instruction. Instructors and educational researchers encourage the **me ultima book essay**, development of improved essay evaluation applications that not only generate a numerical rating for an essay, but also analyze grammar, usage, mechanics and discourse structure. In terms of classroom instruction, the goal is to **college application essays**, develop applications that give students more opportunity to practice writing on their own with automated feedback that helps them revise their work, and ultimately improve their writing skills. This technology is a helpful supplement to **phd thesis on globalization**, traditional teacher instruction. Specifically, it is more effective for students to **brainstorming college essays**, receive feedback that refers explicitly to their own writing rather than just general feedback.

The Criterion service's capability to analyze organizational elements serves as a critical complement to other tools in the application that provide feedback related to grammar, usage, mechanics and style features in student essays.
Which organizational elements are analyzed?
This cutting edge, first-of-its-kind technology employs machine learning to identify organizational elements in *papers on data*, student essays, including introductory or background material, thesis statements, main ideas, supporting ideas and conclusions. The system makes decisions that exemplify how educators perform this task. **Brainstorming Application Essays**! For instance, when grading students' essays, educators provide comments on the discourse structure. Instructors may indicate that there is no thesis statement, or that the main idea has insufficient support. This kind of feedback from an instructor helps students reflect on the discourse structure of their writing.

How did the system learn how to do the analysis?
Trained readers annotate large samples of student essay responses with essay-based organizational elements. The annotation schema reflects the organizational structure of essay-writing genres, such as persuasive writing, which are highly structured. The increased use of **segovia thesis** automated essay-scoring technology allows for the collection of a large corpus of students' essay responses that we use for annotation purposes.
How can this analysis help students?
As students become more sophisticated writers, they start to think about the **brainstorming college**, organizational structure in their writing. The Criterion service application offers students feedback about this aspect of **phd thesis on globalization** their writing. Students who use the tool can see a comprehensive analysis of the existing organizational elements in their essays. For instance, if a student writes an essays essay, and the system feedback indicates that the essay has no conclusion, then the **segovia thesis**, student can begin to work on this new organizational element.

This kind of automated feedback is an initial step in students' improvement of the organization and development of their essays. This kind of feedback also resembles traditional feedback that a student might receive from a professor.
Understanding Organization and Development Feedback.
The algorithm developed to automatically identify essay-based organizational elements is based on samples of teacher-annotated essay data. Two readers were trained to annotate essay data with appropriate organizational labels.
What is the agreement rate between two readers on the labeling task?
Two readers are in general agreement on all labeling tasks.
What is the agreement rate between the **brainstorming essays**, system and the reader?
The trained reader's assessment is in general agreement with the system. **Bless Report Essay**! In the vast majority of cases, ETS researchers generally found either exact or adjacent agreement (within one point) between the Criterion service scores and those of a trained essay reader.

Both used the **college application**, same scoring guidelines and scoring system.
Does the system label each individual sentence with a corresponding organizational label?
Yes. Sometimes multiple sentences are associated with a single organizational element, and the entire block of text is highlighted and appears to be assigned a single label. **Essay Life Life**! In fact, each sentence is labeled individually.
Does the system label according to sentence position only?
No. Many features, including word usage, rhetorical strategy information, possible sequence of organizational elements and syntactic information are used to determine the final organizational label.
The Criterion service is available 24 hours a day and only requires an Internet connection and a web browser, and is PC and Mac®compatible. It can also be used on **brainstorming college** the iPad®, but an external keyboard is recommended.
For a complete description of **papers mining** minimum and recommended standards and network configuration suggestions, please refer to the System Requirements Sheet.

Where can I find additional information about the Criterion service and the e-rater technology?
The research papers on the ETS website are sources of more information about the **brainstorming college**, Criterion service and its underlying technology.
If you are interested in ordering, have questions about pricing or would like to speak to **segovia thesis**, a Criterion Specialist, contact us today.
Tests and Products.
ETS Measuring the Power of **brainstorming college essays** Learning. ®
Copyright 2017 by Educational Testing Service.

All rights reserved.
All trademarks are property of **bless book** their respective owners.

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Accepted by Cornell pdf 2 Кб

Plagiarism - What it is and how to avoid it.
Plagiarism is *application essays*, a serious academic offence. Each year a number of bless me ultima report essay cases of plagiarism are brought to the attention of the Dean of college application Arts and the President#146;s Office. Depending on **online essay writing for dummies** the severity of the offence, students found guilty of plagiarism may lose credit for the assignment in question, be awarded a mark of zero in the course, or face suspension from the University.
Most simply, plagiarism is intellectual theft. Any use of another author#146;s research, ideas, or language without proper attribution may be considered plagiarism. Because such definitions include many shades of accidental or intentional plagiarism, these need to be described more fully.
This is the most obvious case: a student submits, as his or her own work, an *brainstorming college application* essay that has been written by someone else. Usually the original source is *essay on city life life*, a published journal article or book chapter. The use of brainstorming college application unpublished work, including the work of research papers another student , is just as serious.
In such cases, plagiarism cannot be avoided by paraphrasing the original or acknowledging its use in *application essays* footnotes . The work is the property of another author and **google essay project** should not be used.

See Example #1. A student may also lift portions of another text and use them in his or her own work. For example, a student might add her or his own conclusions or introduction to an essay. Or a student might scatter his or her own comments through a text taken substantially from another source. These practices are unacceptable. Even with some attribution, the bulk of the work has been done by another. See Example #1.

In many cases, a student will lift ideas, phrases, sentences, and paragraphs from a variety of sources and stitch them together into **brainstorming college essays** an essay. These situations often seem difficult to *essay on city life*, assess. Most essays, after all, are attempts to bring together a range of sources and arguments. But the line between plagiarism and original work is not difficult to draw. See Example #2.
Lazy plagiarism crops up in many student essays, and is usually the result of sloppy note-taking or research shortcuts. Examples include:
inadvertent use of another#146;s language, usually when the student fails to distinguish between direct quotes and general observations when taking notes. In such cases, the presence of a footnote does not excuse the use of college application essays another#146;s language without quotation marks. use of footnotes or material quoted in other sources as if they were the results of your research. sloppy or inadequate footnoting which leaves out sources or page references.
Although it may not be the student#146;s intention to deceive, it is often difficult for instructors to distinguish between purposeful and accidental plagiarism.

See Example #3.
This is not intended to discourage students from **bless report essay** pursuing specific interests. If you want to use a previously completed essay as a starting point for new research, you should receive the instructor#146;s approval and provide her or him with a copy of the brainstorming, original essay. If you want to use substantially similar essays to *google maps essay project*, satisfy the requirements of two related courses, you should get approval from all the instructors concerned.
It is not hard to draw the distinction between original and thoroughly plagiarized work.

But the grey areas between these extremes are more vexing. Students should avoid any hint of dishonesty by maintaining good research habits and paying attention to a few basic rules of writing and documentation.
Most written assignments begin with the collection of research notes -- a combination of ideas or quotes from other sources, and **essays** the student#146;s own ideas. Whether you keep notes on index cards, in a loose-leaf binder, or on old envelopes in a desk drawer, it is important to record and organize them in such a way that vital information is *maps*, not lost.
Keep careful and **essays** complete track of sources. Accurately copy the author, title, and **phd thesis on globalization** other information about the source publication, including the number(s) of the page(s) from which notes or quotes were taken.
Distinguish carefully between your ideas and the ideas of others. This is a simple question of intellectual honesty. If you use another#146;s conclusions, acknowledge them.

If you come to the same conclusions as another on **college essays** your own, you should still acknowledge the agreement.
Distinguish carefully between your own words and those of others. If necessary, highlight or use coloured index cards for directly quoted material.
As you begin to tie your ideas together in written form, consider the following:
Begin by organizing your essay in an original manner . Avoid mimicking the pattern or order of argument used by others. Remember: this is your humble contribution to a debate or a body of research; it is not (in most case) an attempt to summarize or paraphrase the work of others.

As you weave the ideas and language of others into your work, make clear choices about the use of quoted material . In other words, either quote directly, or state the idea(s) in your own language. Do not mess around with close paraphrases or purely cosmetic changes. See Example #4.
Read the first draft carefully . Is the distinction between your work and **segovia thesis** the work of others clear and unambiguous? You might even take an early draft and highlight all those passages that summarize, paraphrase, or quote other sources. Is there enough of your own work left in the essay?
Many cases of plagiarism occur in *brainstorming college application essays* the documentation rather than the body of the on globalization, essay. You should have a clear idea of the variety of purposes a footnote (or endnote) may serve, and **brainstorming college application essays** many different ways you can acknowledge the work of others. For specific cases See Example #5 . Also note the following:

Always record your source of the research papers on data, information; never use or rely on another author#146;s footnotes.
The footnote should allow the reader to find or check the college application, material being cited. Provide exact page numbers for direct quotes, and a range of page numbers for more general points.
If you included more than one source or reference in a footnote, the relevance or order of the various sources should be clear to the reader.
Once your essay is complete, consider each portion that is drawn from another source, and ask yourself the following:
Is the idea or argument expressed entirely my own?
Is the general language or choice of online words (including even phrases or rough paraphrases) my own?
If either answer is no, the work must be credited to the original author. **College**. And if the answer to the second question is no, the project, passage should either be quoted directly or rewritten in the student#146;s own words and credited directly.
Complete or Near-Complete Plagiarism.
Despite minor changes to the text, the passages are substantially unchanged.

In the first case, the plagiarist also lifts the footnote from the original. Note that the use of even very brief passages (such as the wings of aspiration) constitutes plagiarism. Use of such passages throughout an essay would constitute complete plagiarism ; use of such passages occasionally would constitute near-complete plagiarism . **Brainstorming College Application Essays**. [This example is drawn from a longer discussion regarding plagiarism in the graduate school essays of Martin Luther King Jr. Students interested in a well-illustrated discussion of student plagiarism, might want to consult this: Becoming Martin Luther King -- Plagiarism and Originality: A Round Table, Journal of American History (June 1991, pp. 11-123. **Essay Life**. The example used below is on p. **Brainstorming College Application Essays**. 25.]
The second case illustrates a more typical instance of student plagiarism.

Even the footnote to *on globalization*, the original does not excuse the substantial use of the original#146;s language.
It is Eros, not Agape, that loves in proportion to the value of its object. By the pursuit of value in its object, Platonic love is let up and away from the brainstorming essays, world, on wings of aspiration, beyond all transient things and persons to the realm of the Ideas. Agape, as described in the Gospels and Epistles, is spontaneous and **essay for dummies** #145;uncaused#146;, indifferent to human merit, and **application essays** creates value in those upon whom it is bestowed out of pure generosity. It flows down from God into this transient, sinful world; those whom it touches become conscious of their own utter unworthiness; they are impelled to forgive and love their enemies. because the God of essay writing grace imparts worth to them by the act of brainstorming loving them.* [footnote* is to Anders Nygren, Agape and **segovia thesis** Eros . (New York, 1932), pp. 52-56]
As Nygren set out to *brainstorming college application*, contrast these two Greek words he finds that Eros loves in proportion to the value of the object.

By the pursuit of value in its objects. Platonic love is let up and away from the online writing for dummies, world, on wings of aspiration, beyond all transient things and persons to the realm of the Ideas. Agape as described in *brainstorming application* the Gospels and Epistles, is spontaneous and uncaused, indifferent to human merit, and creates value in those upon whom it is bestowed out of pure generosity. It flows down from God into the transient, sinful world; those whom it touches become conscious of their own utter unworthiness; they are impelled to forgive and love their enemies, because the God of Grace imparts worth to them by the act of loving them.*
[Footnote* is to Nygren, Agape and Eros , pp. **Bless Me Ultima**. 52-56]
The strike officially began on May 29, and on June 1 the manufacturers met publicly to *brainstorming college application essays*, plan their resistance. Their strategies were carried out on two fronts. They pressured the proprietors into holding out *essay on city life*, indefinitely by refusing to send new collars and cuffs to *college essays*, any laundry.

Also the manufacturers attempted to undermine directly the segovia thesis, union#146;s efforts to weather the college essays, strike. They tried to *segovia thesis*, create a negative image of the union through the press, which they virtually controlled. **College Application**. They prevented a few collar manufacturers in other cities from patronizing the unions#146; cooperative laundry even though it claimed it could provide the same services for 25 percent less. Under these circumstances, the collar ironers#146; tactics were much less useful.
The strike began on May 29, and on June 1 the manufacturers met publicly to plan their response. **Writing**. They had two strategies. **Brainstorming College Essays**. They pressured the proprietors into **book report** holding out indefinitely by refusing to send new collars and **brainstorming application** cuffs to *segovia thesis*, any laundry, and they attempted to undermine directly the union#146;s efforts to weather the strike. They also tried to create a negative image of the union through the newspapers, which they virtually controlled. They prevented a few collar manufacturers in other cities from using the unions#146; cooperative laundry even though it could provide the same services for 25 percent less.

Under these circumstances, the collar ironers#146; tactics were much less useful. 1. 1. Carole Turbin, And We are Nothing But Women: Irish Working Women in Troy, pp. 225-26 in Women of America . Edited by Mary Beth Norton (Boston: Houghton Mifflin, 1979). Here two sources are combined to create a new passage. As it stands, the college application, passage is clearly plagiarized. If a footnote were added acknowledging the sources, the substantial use of the language of the original passage would still open the student to charges of plagiarism. An example of an honest and acceptable use of the information derived from these sources is provided at the bottom of the page. Note that the acceptable version uses the facts of the online essay for dummies, original sources, but organizes and expresses them in the student#146;s own language.

Despite the strong public opposition, the Reagan administration continued to install so many North American men, supplies, and facilities in Honduras that one expert called it the USS Honduras , a [stationary] aircraft carrier or sorts. (Walter LaFeber, Inevitable Revolutions (New York, 1989), 309.)
By December 1981, American agents--some CIA, some U.S. Special Forces--were working through Argentine intermediaries to set up contra safe houses, training centres, and base camps along the brainstorming college application, Nicaraguan-Honduran border. **Research**. (Peter Kornbluh, Nicaragua, in Michael Klare (ed), Low Intensity Warfare (New York, 1983), 139.)
Despite strong public opposition, by December 1981 the Reagan Administration was working through Argentine intermediaries to *brainstorming college application*, install contra safe houses, training centres, and base camps in Honduras. One expert called Honduras the USS Honduras , a stationary aircraft carrier or sorts.
In the early 1980s, the Reagan Administration made increasing use of Honduras as a base for segovia thesis the contra war. The Administration set up a number of military and training facilities--some American, some contra, and some housing Argentine mercenaries--along the border between Nicaragua and Honduras. The country, as one observer noted, was little more than a [stationary] aircraft carrier, which he described as the USS Honduras . 2.
2. **Brainstorming College Application Essays**. See Walter Lafeber, Inevitable Revolutions (New York, 1989), p. 307-310 (quote p. **Segovia Thesis**. 309); and Peter Kornbluh, Nicaragua, in Michael Klare (ed), Low Intensity Warfare (New York, 1983), 139.

In this example, the student may have made a sincere effort to write an *college essays* original passage, but sloppy research and documentation raise the possibility of research on data mining plagiarism. Note the characteristic errors: confusion of original and student#146;s language, quotation marks in the wrong place, improper or incomplete footnotes.
Despite the college application essays, strong public opposition, the Reagan administration continued to install so many North American men, supplies, and facilities in *maps project* Honduras that one expert called it the USS Honduras , a [stationary aircraft carrier of college essays sorts. (Walter LaFeber, Inevitable Revolutions (New York, 1989), 309.)
By December 1981, American agents--some CIA, some U.S. Special Forces--were working through Argentine intermediaries to set up contra safe houses, training centres, and base camps along the Nicaraguan-Honduran border. (Peter Kornbluh, Nicaragua, in Michael Klare (ed), Low Intensity Warfare (New York, 1983), 139.)
Despite strong public opposition, the Reagan Administration continued to install so many North American men, supplies, and facilities in Honduras that one expert called it the USS Honduras, a stationary aircraft carrier or sorts. 3.
In December 1981, American agents--some CIA Special Forces--were working through Argentine intermediaries to *bless me ultima essay*, set up contra safe houses, training centres, and **application** base camps along the Nicaraguan-Honduran border. 4.
3. Walter Lafeber, Inevitable Revolutions (New York, 1989), p. 309.

4. **Online Writing For Dummies**. Michael Klare (ed), Low Intensity Warfare (New York, 1983).
Students anxious about committing plagiarism often ask: How much do I have to change a sentence to be sure I#146;m not plagiarizing? A simple answer to this is: If you have to *application*, ask, you#146;re probably plagiarizing.
This is important. Avoiding plagiarism is *essay life vs. country*, not an exercise in inventive paraphrasing. There is no magic number of words that you can add or change to *college essays*, make a passage your own.

Original work demands original thought and organization of thoughts. In the following example, although almost all the words have been changed, the phd thesis, student has still plagiarized. An acceptable use of this material is also provided below.
Shortly after the two rogues, who pass themselves off as a duke and a king, invade the raft of Huck and Jim, they decide to raise funds by performing scenes from Shakespeare#146;s Romeo and Juliet and **essays** Richard III . That the presentation of on city Shakespeare in small Mississippi towns could be conceived of as potentially lucrative tells us much about the position of Shakespeare in *application essays* the nineteenth century. (Lawrence Levine, Highbrow, Lowbrow: The Emergence of a Cultural Hierarchy in America (Cambridge, 1986), p. 10)
Soon after the two thieves, who pretend they are a king and a duke, capture Huck and Jim#146;s raft, they try to make money by **book report essay** putting on **college application** two Shakespeare plays ( Romeo and **report** Juliet and Richard III ). Because the brainstorming college application, production of Shakespeare in tiny Southern towns is seen as possibly profitable, we learn a lot about the status of Shakespeare before the twentieth century.
As Lawrence Levine argues, casual references to Shakespeare in popular nineteenth century literature suggests that the identification of highbrow theatre is a relatively recent phenomenon. **On Data**. 5.
Note that this version does not merely rephrase or repeat the material from the passage cited above, but expands upon it and places it in *brainstorming application* the context of the student#146;s work.
Varieties of Footnotes.

The use of sources can be clarified in a number of bless me ultima essay ways through careful footnoting. Consider the different forms of documentation and acknowledgement in the following:
With the election of Ronald Regan, covert operations in Latin America escalated rapidly. 6 The influx of American funds, notes Peter Kornbluh, determined the brainstorming essays, frequency and destructiveness of writing for dummies contra attachs. 7 In the early 1980s, the Regan Administration increasingly used Honduras as a base for the contra war. The Administration set up a number of brainstorming application military and training facilities--some American, some contra , and some housing Argengine mercenaries--along the border between Nicaragua and Honduras. [T]he USS Honduras , as one observer noted, was little more than a [stationary] aircraft carrier. 8 These strategies seemed to represent both a conscious acceleration of American involvement in *mining* the region, and the inertia of past involvements and failures. 9.
6. The following paragraph is drawn from Walter Lafeber, Inevitable Revolutions (New York, 1989), p. 307-310; and Peter Kornbluh, Nicaragua, in Michael Klare (ed), Low Intensity Warfare (New York, 1983), pp. 139-149.

Note: FOOTNOTE 6 provides general background sources.
7. Peter Kornbluh, Nicaragua, in Michael Klare (ed), Low Intensity Warfare (New York, 1983), p. 139.
Note: FOOTNOTE 7 documents a quoted passage, noting the exact page location.
8. **Brainstorming College Essays**. Observer quoted in *book essay* Walter Lafeber, Inevitable Revolutions (New York, 1989), p. 309.
Note: FOOTNOTE 8 documents a secondary quotation.
9. Peter Kornbluh, Nicaragua, in Michael Klare (ed), Low Intensity Warfare (New York, 1983), stresses the renewal of brainstorming application essays counterinsurgency under Reagan; Walter Lafeber, Inevitable Revolutions , stresses the ongoing interventionism of the U.S. (New York, 1989), p. **On Data Mining**. 307-310.
Note: FOOTNOTE 9 distinguishes your argument from **brainstorming college essays** that of your sources.

Dr. **Online Essay Writing**. Colin H. **Essays**. Gordon.
(Department of History, UBC)
Professor Peter Simmons.
(President#146;s Advisory Committee on Student Discipline, UBC)

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Department of Mathematical Sciences, Unit Catalogue 2003/04.
Aims: This course is designed to cater for first year students with widely different backgrounds in *brainstorming college application* school and college mathematics. It will treat elementary matters of maps, advanced arithmetic, such as summation formulae for progressions and will deal with matters at a certain level of abstraction. This will include the *brainstorming college* principle of mathematical induction and *segovia thesis*, 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 essays, 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 on globalization, positive integer powers and 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 college application, 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 phd thesis, functions of a real variable at a level accessible to **application** 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 *segovia thesis* notions of 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. *Brainstorming Essays*. Review of differentiation: product, quotient, function-of-a-function rules and Leibniz rule. Maxima, minima, points of essay writing, inflection, radius of curvature. Graphs as geometrical interpretation of functions. Monotone functions. Injectivity, surjectivity, bijectivity.

Curve Sketching. Inequalities. *Application*. Arithmetic manipulation and *google*, geometric representation of inequalities. Functions as formulae, natural domain, codomain, etc. Real valued functions and graphs. Orders of magnitude. *Brainstorming Application Essays*. Taylor's Series and *essay on city life vs. country life*, Taylor polynomials - the error term. Differentiation of Taylor series. Taylor Series for exp, log, sin etc.

Orders of growth. Orthogonal and tangential curves. MA10003: Integration differential equations. Aims: This module is designed to cover standard methods of differentiation and integration, and the methods of solving particular classes of 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 basic formulae from trigonometry and algebra: polynomials, trigonometric and *college application*, hyperbolic functions, exponentials and logs. Integration by substitution. Integration of rational functions by partial fractions. Integration of papers, 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. *Brainstorming College*. Volumes and surfaces of revolution. Definition of the order of a differential equation. Notion of linear independence of solutions. Statement of theorem on number of essay, linear independent solutions. General Solutions. CF+PI . First order linear differential equations by **brainstorming application essays**, 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 *segovia thesis* concepts of logic that underlie all mathematical reasoning and the notions of set theory that provide a rigorous foundation for brainstorming college, mathematics.

A real life example of all this machinery at work will be given in the form of an *bless me ultima book report essay* introduction to the analysis of sequences of real numbers.
Objectives: By the end of this course, the students will be able to: understand and *college application essays*, 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. Predicates and relations. The meaning of the logical operators #217 , #218 , #152 , #174 , #171 , #034 , #036 . 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. Countability and uncountability. Maxima and minima of finite sets, max (A) = - min (-A) etc. *Google Project*. 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 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 *application essays* students will have a sound grasp of elementary matrix theory and multivariate calculus and will be proficient in performing such tasks as addition and 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 planes in two and three dimension. *Google*. Linear dependence and independence. Simultaneous linear equations. Elementary row operations.

Gaussian elimination. Gauss-Jordan form. Rank. Matrix transformations. Addition and multiplication. *Essays*. Inverse of a matrix. Determinants. Cramer's Rule. Similarity of essay for dummies, matrices. Special matrices in geometry, orthogonal and symmetric matrices. Real and complex eigenvalues, eigenvectors.

Relation between algebraic and geometric operators. Geometric effect of essays, matrices and the geometric interpretation of determinants. Areas of bless report essay, triangles, volumes etc. Real valued functions on IR #179 . Partial derivatives and gradients; geometric interpretation. *Brainstorming Application*. Maxima and Minima of functions of two variables.

Saddle points. Discriminant. Change of coordinates. Chain rule. Vector valued functions and their derivatives. The Jacobian matrix and *google maps essay project*, determinant, geometrical significance. Chain rule.

Multivariate integrals. Change of order of integration. Change of college essays, variables formula.
Aims: To introduce the theory of three-dimensional vectors, their algebraic and geometrical properties and their use in mathematical modelling. To introduce Newtonian Mechanics by considering a selection of research papers on data, 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 3D algebraic and geometrical problems. The student should also be able to use vectors to describe and model physical problems involving kinematics. The student should be able to **brainstorming application essays** 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 segovia thesis, vectors with respect to a scalar variable. Curvature.

Cartesian, polar and spherical co-ordinates. Vector identities. Dot and cross product, vector and scalar triple product and determinants from geometric viewpoint. Basic concepts of mass, length and time, particles, force. Basic forces of application essays, nature: structure of matter, microscopic and macroscopic forces. Units and dimensions: dimensional analysis and scaling.

Kinematics: the description of particle motion in terms of vectors, velocity and acceleration in polar coordinates, angular velocity, relative velocity. Newton's Laws: Kepler's laws, momentum, Newton's laws of essay writing, motion, Newton's law of gravitation. *College*. Newtonian Mechanics of Particles: projectiles in a resisting medium, constrained particle motion; solution of the governing differential equations for google project, a variety of brainstorming, problems. Central Forces: motion under a central force.
MA10031: Introduction to 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 *essay*, basic laws of application essays, 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 conditional distributions of bivariate discrete random variables; calculate and make use of 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. *Essay*. Partition Theorem. Bayes' Theorem. Independence of events. *Essays*. 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 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 essay life, continuous random variables. To introduce statistical modelling and *brainstorming college application essays*, parameter estimation and to 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. *Vs. Country*. To be able to **brainstorming essays** use the *on globalization* R statistical package for simulation and data exploration.
Definition of brainstorming application essays, 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 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 marginals). The distribution of a sum of research papers mining, independent continuous RVs, including normal and exponential examples. *Essays*. Statement of the central limit theorem (CLT).

Transformations of RVs. Discussion of the 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. Point estimates and *on globalization*, estimators. Estimators as random variables. Bias and precision of estimators.

Introduction to model fitting; exploratory data analysis (EDA) and model formulation. *Brainstorming College Application Essays*. Parameter estimation via method of moments and (simple cases of) maximum likelihood. Graphical assessment of goodness of segovia thesis, fit. Implications of model misspecification.
Aims: To teach the basic ideas of probability, data variability, hypothesis testing and of relationships between variables and 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 likely to follow a binomial, Poisson or normal distribution and *essays*, be able to carry out simple related calculations. They should be able to carry out a simple decomposition of segovia thesis, a time series, apply correlation and regression analysis and understand the basic idea of statistical significance.
The laws of Probability, Bayes' Theorem, Decision Trees. 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 *brainstorming college application essays*, interpretation in terms of maps essay project, variability explained. Idea of the *brainstorming college application* sampling distribution of the sample mean; the Z test and the concept of bless me ultima report essay, significance level.

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.
Numbers: Integers, Rationals, Reals. Algebra: Straight lines, Quadratics, Functions, Binomial, Exponential Function. Trigonometry: Ratios for brainstorming application, general angles, Sine and Cosine Rules, Compound angles. *On Globalization*. Calculus: Differentiation: Tangents, Normals, Rates of Change, Max/Min.
Core 'A' level maths. *College Application*. The course follows closely the essential set book: L Bostock S Chandler, Core Maths for bless me ultima report, A-Level, Stanley Thornes ISBN 0 7487 1779 X.

Integration: Areas, Volumes. 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 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 elementary use of brainstorming essays, 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 *phd thesis*, creating web pages.

Introduction to UNIX and *brainstorming application essays*, EMACS. Brief introduction to HTML. Programming in *segovia thesis* MATLAB and applications to mathematical and statistical problems: Variables, operators and *brainstorming college*, control, loops, iteration, recursion. Scripts and *papers*, functions. Compilers and *brainstorming*, interpreters (by example). Data structures (by example).

Visualisation. Graphical-user interfaces. Numerical and symbolic computation. The MATLAB Symbolic Math toolbox. Introduction to complexity analysis. Efficiency of algorithms. Applications. Report writing. Presentations.

Web design. Group project.
* Calculus: Limits, differentiation, integration. Revision of logarithmic, exponential and *segovia thesis*, inverse trigonometrical functions. Revision of college essays, integration including polar and parametric co-ordinates, with applications.
* Further calculus - hyperbolic functions, inverse functions, McLaurin's and Taylor's theorem, numerical methods (including solution of research mining, nonlinear equations by 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 brainstorming essays, 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 analysis of the elementary theory of sequences and series of real numbers and to extend these ideas to sequences of research on data mining, functions.

Objectives: By the end of the *brainstorming college application essays* module, students should be able to read and understand statements expressing, with the use of quantifiers, convergence properties of segovia thesis, sequences and series. 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. *College Application Essays*. Bounded sequences and monotone sequences. Cauchy sequences. Algebra-of-limits theorems.

Subsequences. *Me Ultima Report*. Limit Superior and Limit Inferior. Bolzano-Weierstrass Theorem. Sequences of partial sums of series. Convergence of brainstorming college essays, series. Conditional and absolute convergence.

Tests for convergence of series; ratio, comparison, alternating and nth root tests. Power series and radius of convergence. Functions, Limits and *book report essay*, Continuity. Continuity in terms of convergence of brainstorming college essays, sequences. Algebra of limits. Brief discussion of convergence of phd thesis on globalization, 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 *brainstorming college* applications. Students should know how to execute the Gram-Schmidt process.
Real and complex vector spaces, subspaces, direct sums, linear independence, spanning sets, bases, dimension. The technical lemmas concerning linearly independent sequences. Dimension. Complementary subspaces. *Phd Thesis*. Projections. Linear transformations.

Rank and *brainstorming application essays*, nullity. The Dimension Theorem. *Bless Essay*. 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 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. Foundations will be laid for more advanced studies in nonlinear differential equations and control theory.

Phase plane techniques will be introduced.
Objectives: At the 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 *application* control theory (such as stability, stabilization by feedback, controllability) and will be able to **research papers mining** solve simple control problems. The student will be able to carry out *college application essays* 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 first and higher derivatives, damping, delay; inversion by partial fractions; solution of 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 **segovia thesis** controllability and observability: definitions, rank conditions (without full proof) and *brainstorming*, examples. *On Data*. 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 this course students will be familiar with the fundamental results of vector calculus (Gauss' theorem, Stokes' theorem) and *application*, will be able to carry out line, surface and *segovia thesis*, volume integrals in *college essays* general curvilinear coordinates. They should be able to solve Laplace's equation, the wave equation and the diffusion equation in simple domains, using separation of variables.
Vector calculus: Work and energy; curves and surfaces in parametric form; line, surface and volume integrals. Grad, div and *phd thesis on globalization*, curl; divergence and Stokes' theorems; curvilinear coordinates; scalar potential. Fourier series: Formal introduction to Fourier series, statement of Fourier convergence theorem; Fourier cosine and *application*, sine series. *Online*. Partial differential equations: classification of 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 exercises, in 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 analytical methodology to problems in other disciplines, as they arise. They should have a thorough understanding of the abstract notion of an integral, and *brainstorming college*, a facility in the manipulation of integrals.
Weierstrass's theorem on continuous functions attaining suprema and *on city life*, infima on compact intervals.

Intermediate Value Theorem. *College Application Essays*. Functions and Derivatives. Algebra of derivatives. Leibniz Rule and compositions. Derivatives of inverse functions. *Online Essay Writing For Dummies*. Rolle's Theorem and *brainstorming application essays*, Mean Value Theorem.

Cauchy's Mean Value Theorem. L'Hopital's Rule. Monotonic functions. Maxima/Minima. Uniform Convergence. Cauchy's Criterion for Uniform Convergence. Weierstrass M-test for papers on data mining, series. Power series. Differentiation of power series. Reimann integration up to **college** the Fundamental Theorem of Calculus for 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 axioms derived. Objectives: Students should be capable of phd thesis on globalization, finding eigenvalues and 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 *brainstorming application essays*, eigenvectors. Geometric and algebraic multiplicity. *Segovia Thesis*. Diagonalisability. Characteristic polynomials. *Brainstorming College Application Essays*. Cayley-Hamilton Theorem.

Minimum polynomial and primary decomposition theorem. Statement of and motivation for the Jordan Canonical Form. Examples. Orthogonal and unitary transformations. Symmetric and Hermitian linear transformations and their diagonalisability. Quadratic forms. *Project*. Norm of essays, a linear transformation.

Examples. Group Theory: Group axioms and examples. *Segovia Thesis*. Deductions from the *brainstorming college essays* axioms (e.g. uniqueness of identity, cancellation). Subgroups. *Segovia Thesis*. Cyclic groups and *college essays*, their properties. *Papers*. Homomorphisms, isomorphisms, automorphisms. Cosets and Lagrange's Theorem. Normal subgroups and Quotient groups. Fundamental Homomorphism Theorem.

MA20013: Mathematical modelling fluids.
Aims: To study, by **essays**, example, how mathematical models are hypothesised, modified and elaborated. To study a classic example of mathematical modelling, that of online writing for dummies, fluid mechanics.
Objectives: At the end of the course the *brainstorming application essays* 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 essay, proposed model in light of discrepancies between model predictions and observed data or failures of the model to exhibit correct qualitative behaviour. *Brainstorming*. The student will also be familiar with the equations of motion of an ideal inviscid fluid (Eulers equations, Bernoullis equation) and *google maps project*, how to **college** 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 model components; self-consistency and structural stability. The three stages of modelling:
(1) Model formulation, including the use of empirical information,
(2) model fitting, and.
(3) model validation.

Possible case studies and projects include: The dynamics of 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. Mass conservation, incompressible flow, simple examples of book report, potential flow. Aims: To revise and develop elementary MATLAB programming techniques. To teach those aspects of brainstorming application essays, Numerical Analysis which are most relevant to a general mathematical training, and to lay the foundations for the more advanced courses in later years. Objectives: Students should have some facility with MATLAB programming. They should know simple methods for the approximation of functions and integrals, solution of initial and boundary value problems for phd thesis on globalization, ordinary differential equations and the solution of linear systems. 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 Convergence and Accuracy: Order of convergence, extrapolation and error estimation. Approximation of 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 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. *College*. Ability to carry out goodness-of-fit tests, analyse contingency tables, and 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. Examples. Neyman-Pearson lemma.

Distributions related to the normal: t, chi-square and F distributions. Inference for normal data: Tests and confidence intervals for normal means and *maps*, variances, one-sample problems, paired and unpaired two-sample problems. *College Application Essays*. Contingency tables and goodness-of-fit tests. *Essay Writing*. Non-parametric methods: Sign test, signed rank test, Mann-Whitney U-test.
MA20034: Probability random processes.
Aims: To introduce some fundamental topics in 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 Poisson processes.
Objectives: Ability to perform computations on 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. Random walks expected first passage times. *Application*. 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. *Bless Me Ultima Book Report Essay*. Understand the principles of statistical modelling.
One-way analysis of variance (ANOVA): One-way classification model, F-test, comparison of application essays, group means. Regression: Estimation of model parameters, tests and confidence intervals, prediction intervals, polynomial and multiple regression. *Essay Vs. Country*. Two-way ANOVA: Two-way classification model. Main effects and interaction, parameter estimation, F- and t-tests. *Brainstorming College Essays*. Discussion of experimental design.

Principles of modelling: Role of the statistical model. Critical appraisal of model selection methods. Use of residuals to check model assumptions: probability plots, identification and *google maps project*, treatment of outliers. Multivariate distributions: Joint, marginal and conditional distributions; expectation and variance-covariance matrix of 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 *brainstorming college*, 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. *Segovia Thesis*. 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 obtain interval estimates for 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 *brainstorming college essays* minitab package and show appreciation of the *segovia thesis* uses and *college essays*, 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 Student's t. Simple non-parametric tests and chi-squared tests. The probability of a type 2 error in the Z test and the concept of essay on city vs. country, power. Quality control: Acceptance sampling, Shewhart charts and the relationship to hypothesis testing.

The use of the minitab package and *application essays*, practical points in data analysis.
Aims: To teach the methods of segovia thesis, analysis appropriate to simple and multiple regression models and to common types of survey and experimental design. The course will concentrate on 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 **college** apply the methods of analysis of 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 *segovia thesis*, interactions. Experimental Design: Randomisation, blocking, factorial designs.

Analysis using the minitab package.
Industrial placement year.
Study year abroad (BSc)
Aims: To understand the *brainstorming college* principles of statistics as applied to **segovia thesis** 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 brainstorming application, average and variability. Discrete probability models - binomial, poisson. Continuous probability model - normal distribution. Poisson and normal approximations to binomial sampling theory. *Phd Thesis*. Estimation, confidence intervals.

Chi-squared tests for college essays, goodness of fit and contingency tables. One sample and two sample tests. Paired comparisons. 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 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 bless report essay, real-world phenomena. In this course students will consider how models are formulated, fitted, judged and modified in light of brainstorming essays, 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 use of computer packages.

Case studies will be drawn from *essay life* a wide range of biological topics, which may include cell biology, genetics, ecology, evolution and epidemiology. After taking this unit, the 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 the scientific method. Objectives of mathematical and statistical modelling; the iterative nature of application essays, 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 *research*, 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 *college*, data dredging.
Aims: This course develops the basic theory of rings and fields and expounds the fundamental theory of Galois on solvability of polynomials.
Objectives: At the end of the course, students will be conversant with the algebraic structures associated to **segovia thesis** rings and fields. Moreover, they will be able to state and prove the main theorems of Galois Theory as well as compute the Galois group of simple polynomials.
Rings, integral domains and fields.

Field of quotients of an integral domain. Ideals and quotient rings. *Brainstorming College Application*. Rings of polynomials. Division algorithm and unique factorisation of polynomials over *for dummies*, a field. Extension fields. *Essays*. Algebraic closure. Splitting fields. Normal field extensions. Galois groups. *Phd Thesis On Globalization*. 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 basic tools of the subject and *college essays*, more recent developments.
Objectives: At the end of the *essay on city life* course, students should be able to state and prove the main theorems of college application, 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 mathematics.
Topics will be chosen from the following: Review of 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 *essay on city life*, 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 *brainstorming college application essays* accessible introduction is given to an area of mathematics which has been the subject of active research for segovia thesis, over 200 years.
Objectives: At the end of the course, the students will be able to apply the methods of calculus with confidence to geometrical problems. *Brainstorming Essays*. They will be able to compute the curvatures of curves and surfaces and understand the geometric significance of 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 congruences. Curvature and torsion determine a curve up to congruence. Global geometry of curves: isoperimetric inequality; four-vertex theorem. Local geometry of bless book report essay, surfaces: parametrisations of surfaces; normals, shape operator, mean and *application*, Gauss curvature.

Geodesics, integration and the local Gauss-Bonnet theorem.
Aims: This core course is intended to **segovia thesis** be an *essays* elementary and accessible introduction to the theory of metric spaces and the topology of IRn for on globalization, students with both pure and applied interests.
Objectives: While the foundations will be laid for further studies in Analysis and Topology, topics useful in *essays* applied areas such as the Contraction Mapping Principle will also be covered. Students will know the fundamental results listed in the syllabus and have an instinct for their utility in analysis and numerical analysis.
Definition and examples of maps, metric spaces. Convergence of sequences. Continuous maps and isometries. Sequential definition of continuity. Subspaces and product spaces. *Essays*. 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 interior of sets. Topological approach to continuity and compactness (with statement of Heine-Borel theorem). Connectedness and *phd thesis*, 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 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. uniqueness.
Sturm-Liouville theory: Reality of eigenvalues.

Orthogonality of eigenfunctions. Expansion in eigenfunctions. Approximation in *brainstorming application* 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 three independent variables: Characteristics. *On Data*. 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 corresponding Cauchy problem (with data on a spacelike curve).

Aims: The course is **brainstorming college application** intended to provide an *online for dummies* 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. Contact with classical (Laplace-transform based) control theory is made in the context of realization theory.
Objectives: To instill basic concepts and results from control theory in a rigorous manner making use of elementary linear algebra and linear ordinary differential equations. Conversance with controllability, observability, stabilizabilty and realization theory in a linear, finite-dimensional context.

Topics will be chosen from the *brainstorming application* following: Controlled and observed dynamical systems: definitions and classifications. Controllability and observability: Gramians, rank conditions, Hautus criteria, controllable and unobservable subspaces. Input-output maps. 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 on globalization, 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. *Brainstorming College Application*. 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 *phd thesis* biology using ODEs and give a qualitative account of the type of solution expected. *Brainstorming College Application*. 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. Chaos. Application to 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 *google project*, instability. Lyapunov functions. *Brainstorming College Application Essays*. 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 *research mining* 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 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 elastostatics: Expansion of college, 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 **life life** composite materials; torsion of cylinders, Prandtl's stress function. *Brainstorming*. Linear elastodynamics: Basic equations and general solutions; plane waves in 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 essay vs. country life, modern iterative methods for solving linear systems and for solving the algebraic eigenvalue problem. They should be able to 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 brainstorming application essays, a real symmetric matrix to tridiagonal form. *Segovia Thesis*. Orthogonality properties of college application, Lanczos iterates. Iterative Methods for Linear Systems: Convergence of me ultima report, 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. Krylov subspaces. Convergence.

Connection with the Lanczos method. Iterative Methods for Nonlinear Systems: Newton's Method. Convergence in *college* 1D. Statement of algorithm for life, 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 example how one area of mathematics can enhance and *application essays*, enrich the *research papers on data* study of another.
Objectives: At the end of the course, the students will be able to state and prove the main theorems of Maschke and Schur and be conversant with their many applications in representation theory and *brainstorming college*, character theory.

Moreover, they will be able to apply these results to problems in 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 **research mining** the ideas of point-set topology culminating with a sketch of the classification of compact surfaces. As such it provides a self-contained account of one of the *college essays* 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 important notion of a topology and to familiarise them with the basic theorems of analysis in their most general setting. Students will be able to **phd thesis on globalization** distinguish between metric and topological space theory and to understand refinements, such as Hausdorff or compact spaces, and *brainstorming college application essays*, their applications.
Topics will be chosen from the following: Topologies and topological spaces.

Subspaces. Bases and sub-bases: product spaces; compact-open topology. Continuous maps and homeomorphisms. *Essay Writing*. Separation axioms. *College Application*. Connectedness. 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. 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 theory of segovia thesis, 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. The complex plane. Regions, paths, simple and closed paths. *Application Essays*. Path-connectedness. Analyticity and the Cauchy-Riemann equations. *Research Papers On Data*. 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. Cauchy's Residue Theorem and contour integration. Applications to real definite integrals.

Aims: To introduce students to the applications of advanced analysis to **brainstorming essays** the solution of segovia thesis, PDEs.
Objectives: Students should be able to **brainstorming college** 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 *online writing* two independent variables: Harmonic functions. Mean value property. Maximum principle (several proofs). *College Essays*. Dirichlet and Neumann problems. Representation of solutions in terms of Green's functions.

Continuous dependence of data for Dirichlet problem. Uniqueness. Parabolic equations in two independent variables: Representation theorems. Green's functions. *Writing*. 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. *College Essays*. Lagrange multipliers. Extrema for integral functions. Euler's equation and its special first integrals. Integral and non-integral constraints.
Aims: The course is intended to **segovia thesis** be an elementary and accessible introduction to **essays** 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 *essay* 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 discrete-time systems.
Objectives: An appreciation of the behaviour, and its potential complexity, of general dynamical systems through a study of discrete-time systems (which require relatively modest analytical prerequisites) and *college application*, 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. Local bifurcations and stability. The logistic map and chaos. Global properties. *Life*. Continuous-time systems. Periodic orbits and Poincareacute maps.

Numerical approximation of differential equations. Newton iteration as a dynamical system. Aims: The aim of the course is to introduce students to applications of 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 *brainstorming college application essays* terms of the original biological problem.
Topics will be chosen from the following: Partial Differential Equation Models: Simple random walk derivation of the diffusion equation. Solutions of the diffusion equation. *Maps Essay Project*. 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 *brainstorming application essays*, geometry effects.

Mode selection and dispersion relation. *On Globalization*. 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 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 college application, a viscous fluid. Topics will be chosen from the 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. 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 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.
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 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 **segovia thesis** their accuracy.

Normal linear model: Vector and *college essays*, matrix representation, constraints on parameters, least squares estimation, distributions of parameter and variance estimates, t-tests and *on globalization*, confidence intervals, the Analysis of application essays, Variance, F-tests for unbalanced designs. Model building: Subset selection and stepwise regression methods with applications in polynomial regression and multiple regression. Effects of collinearity in 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. *On Data Mining*. 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 brainstorming essays, 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 *life* properties of ARIMA and state-space models.
* choose an appropriate ARIMA model for brainstorming application, a given set of data and fit the model using an appropriate package.
* compute forecasts for a variety of linear methods and models.
Introduction: Examples, simple descriptive techniques, trend, seasonality, the *research* correlogram. Probability models for time series: Stationarity; moving average (MA), autoregressive (AR), ARMA and ARIMA models. Estimating the *college essays* autocorrelation function and fitting ARIMA models. *On Globalization*. Forecasting: Exponential smoothing, Forecasting from *brainstorming essays* ARIMA models.

Stationary processes in *on globalization* the frequency domain: The spectral density function, the periodogram, spectral analysis. State-space models: Dynamic linear models and *college application*, 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 ethical considerations and *research papers*, practical problems that govern medical experimentation,
(c) summarize medical data and *brainstorming college application essays*, spot possible sources of bias,
(d) analyse data collected from some types of clinical trial, as well as simple survival data and longitudinal data.

Ethical considerations in *maps essay project* clinical trials and other types of epidemiological study design. Phases I to **brainstorming college** IV of drug development and *segovia thesis*, 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 brainstorming, clinical trials: patient withdrawals, intent to 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 **online writing for dummies**, the ability to explain and *essays*, interpret numerical data to non-statistical clients.
MA30087: Optimisation methods of on globalization, 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 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 *brainstorming college application essays*, 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. *Online Writing For Dummies*. Interpretation of the optimal tableau. *Brainstorming College Essays*. Applications of LP. Duality. *Phd Thesis On Globalization*. Topics selected from: Sensitivity analysis and the dual simplex algorithm. Brief discussion of brainstorming application essays, Karmarkar's method.

The transportation problem and its applications, solution by **phd thesis on globalization**, Dantzig's method. Network flow problems, the Ford-Fulkerson theorem. Non-linear Programming: Revision of classical Lagrangian methods. Kuhn-Tucker conditions, necessity and sufficiency. *Brainstorming College Application*. Illustration by application to quadratic programming.
MA30089: Applied probability finance.

Aims: To develop and apply the theory of probability and stochastic processes to **on globalization** 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 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. Infinite horizon problems; discounted programming, the Howard Improvement Lemma, negative and *brainstorming college essays*, positive programming, simple examples and *phd thesis*, 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 Black-Scholes formula.
Aims: To develop skills in the analysis of application, multivariate data and *phd thesis on globalization*, study the related theory.
Objectives: Be able to carry out *brainstorming college* a preliminary analysis of multivariate data and select and apply an appropriate technique to look for structure in such data or achieve dimensionality reduction. Be able to **essay writing for dummies** 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 interpretation; approximate reduction of dimensionality; scaling problems. *Brainstorming College*. Multidimensional distributions: The multivariate normal distribution - properties and parameter estimation. One and two-sample tests on means, Hotelling's T-squared.

Canonical correlations and *essay vs. country life*, canonical variables; discriminant analysis. Topics selected from: Factor analysis. The multivariate linear model. Metrics and similarity coefficients; multidimensional scaling. *Application Essays*. 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. *On City Vs. Country Life*. Choosing an appropriate method of analysis, verification of assumptions. *Brainstorming College Essays*. Presentation of results: Report writing, communication with non-statisticians. Using resources: The computer, the library.

Project topics may include: Exploratory data analysis. *Vs. Country Life*. Practical aspects of college, sample surveys. Fitting general and generalised linear models. The analysis of standard and non-standard data arising from *google maps essay project* theoretical work in other blocks.
MA30092: Classical statistical inference.
Aims: To develop a formal basis for methods of statistical inference including criteria for the comparison of procedures. To give an in depth description of the asymptotic theory of application, maximum likelihood methods and hypothesis testing.
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 standard distributions.

Revision of standard distributions: Bernoulli, binomial, Poisson, exponential, gamma and normal, and *essay*, their interrelationships.
Sufficiency and *application essays*, 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. *Google Essay Project*. 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. *College Application*. 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 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 *essay* 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 decision theory. Quality Control: Acceptance sampling, single and double schemes, SPRT applied to sequential scheme. Process control, Shewhart charts for mean and range, operating characteristics, ideas of cusum charts.

Practical forecasting. Time plot. Trend-and-seasonal models. *Brainstorming Essays*. Exponential smoothing. Holt's linear trend model and Holt-Winters seasonal forecasting. Autoregressive models.

Box-Jenkins ARIMA forecasting. Introduction to 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. To apply results in *bless report* areas such genetics, biological processes, networks of queues, telecommunication networks, electrical networks, resource management, random walks and elsewhere.
Objectives: On completing the *brainstorming college essays* 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 essay, 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 college application essays, Erlang and Engset. *Research*. Models of interference in communication networks, the ALOHA model. Series of M/M/s queues. Open and *brainstorming college application essays*, closed migration processes. Explosions.

Birth-death processes. Branching processes. Resource management. Electrical networks. *Phd Thesis*. Random walks, reflecting random walks as queuing models in *brainstorming college essays* one or more dimensions. The strong Markov property. *Bless Me Ultima*. The Poisson process in time and *college essays*, space. 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 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 **research mining** derive and implement the finite element method for a range of standard elliptic and parabolic partial differential equations in one and several space dimensions. They should also be able to derive and use elementary error estimates for these methods.

* Variational and weak form of elliptic PDEs. Natural, essential and mixed boundary conditions. Linear and *college essays*, quadratic finite element approximation in one and *book report essay*, several space dimensions. An introduction to **college** convergence theory.
* System assembly and solution, isoparametric mapping, quadrature, adaptivity.

* Applications to PDEs arising in applications.
* Brief introduction to **life vs. country** time dependent problems.
Aims: The aim is to explore pure mathematics from a problem-solving point of 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 foundations of measure, proof techniques in algebra, analysis, geometry and topology.
Aims: This is an advanced pure mathematics course providing an introduction to **brainstorming application** classical algebraic geometry via plane curves. *Papers Mining*. It will show some of the links with other branches of mathematics.
Objectives: At the end of the course students should be able to **college** use homogeneous coordinates in projective space and to **on data** 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 brainstorming college application essays, plane cubic curves.
To be chosen from: Affine and projective space. *Essay On City*. Polynomial rings and homogeneous polynomials. Ideals in the context of polynomial rings,the Nullstellensatz. Plane curves; degree; Bezout's theorem. Singular points of plane curves. Rational maps and morphisms; isomorphism and birationality. Curves of 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 current research. In particular, this course provides the necessary prerequisites for post-graduate study of college essays, Algebraic Topology.

Objectives: At the end of the course, the *bless book essay* 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. *Brainstorming Essays*. Computation of the fundamental group and applications: Fundamental Theorem of Algebra; Brouwer Fixed Point Theorem. Covering spaces. *On Globalization*. 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 application, this course is to lay the basic technical foundations and establish the main principles which underpin the 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 *segovia thesis* a wide range of brainstorming college essays, 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 **writing for dummies**, handout. Lebesgue measure on **application essays**, IRn: existence; inner and *maps project*, outer regularity. Measurable functions.

Sums, products, composition, lim sups, etc; The Monotone-Class Theorem. Probability. Sample space, events, random variables. 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 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 **application essays** 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 *bless me ultima*, 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 *brainstorming college application* Lebesgue integral.
Objectives: By the end of the unit, students should be able to state and prove the principal theorems relating to 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 maps project, continuous functions on [0,1]. Abstract Stone-Weierstrass Theorem. Uniform approximation of continuous functions. Polynomial and trigonometric polynomial approximation, separability of C[0,1]. Total Boundedness. Diagonalisation. Ascoli-Arzelagrave Theorem.

Complete metric spaces. *College Application*. Baire Category Theorem. *Segovia Thesis*. Nowhere differentiable function. Picard's theorem for x = f(x,t). 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. *Brainstorming College*. Projections and subspaces. Orthogonal complements. Riesz Representation Theorem. Complete orthonormal sets in separable Hilbert spaces. Completeness of trigonometric polynomials in L #178 [0,1].

Fourier Series.
Aims: A treatment of the *google maps project* 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 the study of qualitative behaviour of dynamical systems. *Brainstorming*. An ability to investigate stability of equilibria and periodic orbits. A basic understanding and appreciation of bifurcation and chaotic behaviour.

Topics will be chosen from the *online for dummies* following: Stability of equilibria. Lyapunov functions. Invariance principle. Periodic orbits. *Brainstorming College Essays*. Poincareacute maps. Hyperbolic equilibria and orbits. *Online Essay Writing*. Stable and unstable manifolds. *Brainstorming Essays*. 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 prove qualitative properties of essay on city life life, 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 motivating much of modern physics. Objectives: Students will be able to state and prove general theorems for Lagrangian and Hamiltonian systems.

Based on **college essays**, these theoretical results and key motivating examples they will identify general qualitative properties of solutions of these systems.
Lagrangian and Hamiltonian systems, phase space, phase flow, variational principles and Euler-Lagrange equations, Hamilton's Principle of vs. country, least action, Legendre transform, Liouville's Theorem, Poincare recurrence theorem, Noether's Theorem.
MA40050: Nonlinear equations bifurcations.
Aims: To extend the real analysis of implicitly defined functions into **college application** the numerical analysis of iterative methods for computing such functions and to teach an awareness of practical issues involved in *me ultima* 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 brainstorming application essays, nonlinear equations: Newtons method for systems. Quasi-Newton Methods.

Eigenvalue problems. *Maps Essay*. Theoretical Tools: Local Convergence of Newton's Method. Implicit Function Theorem. Bifcurcation from the trivial solution. Applications: Exothermic reaction and buckling problems. Continuous and discrete models. Analysis of 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. Computation of solution paths for systems of nonlinear algebraic equations. Pseudo-arclength continuation. Homotopy methods. Computation of turning points. Bordered systems and their solution.

Exploitation of symmetry. Hopf bifurcation. Numerical Methods for Optimization: Newton's method for unconstrained minimisation, Quasi-Newton methods.
Aims: To introduce the *college application essays* 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 **maps essay** 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. *Brainstorming College Application Essays*. Banach spaces. Young, Minkowski and Holder inequalities. Examples - IRn, C[0,1], l p, Hilbert spaces. Riesz Lemma and finite-dimensional subspaces. The space B(X,Y) of bounded linear operators is a Banach space when Y is complete. Dual spaces and second duals.

Uniform Boundedness Theorem. *Essay Vs. Country Life*. Open Mapping Theorem. Closed Graph Theorem. Projections onto **college essays** closed subspaces. Invertible operators form an open set. Power series expansion for (I-T)- #185 . Compact operators on 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. *Essay On City Life Vs. Country*. 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 *brainstorming application* power of this elegant method in analysis and *research papers*, probability. Applications of the theory are at the heart of this course.
Objectives: By the *college application essays* end of the course, students should be familiar with the *segovia thesis* main results and techniques of discrete time martingale theory. *College Application Essays*. They will have seen applications of 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. Angle-brackets process, Leacutevy's Borel-Cantelli Lemma. Uniform integrability. UI martingales, the Downward Theorem, the *online* Strong Law, the Submartingale Inequality. *Brainstorming*. Likelihood ratio, Kakutani's theorem.
MA40061: Nonlinear optimal control theory.
Aims: Four concepts underpin control theory: controllability, observability, stabilizability and optimality. Of these, the *essay vs. country* first two essentially form the focus of the 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 *brainstorming application* theoretical and practical developments in an active and expanding discipline.

Objectives: To present concepts and results pertaining to robustness, stabilization and optimization of (nonlinear) finite-dimensional control systems in *for dummies* a rigorous manner. Emphasis is placed on optimization, leading to conversance with both the 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. Stability and robustness. Stabilization by feedback. Lyapunov-based design methods. *Brainstorming*. Stability radii. Small-gain theorem. *Life Vs. Country Life*. 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 *brainstorming application* general result with applications). Proof of the maximum principle for mining, 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. *College*. Foundations will be laid for advanced studies in dynamical systems and control. The material is also useful in *online writing for dummies* mathematical biology and numerical analysis.
Objectives: Conversance with existence theory for 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. 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 *brainstorming college* 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 numerical methods and assess the practical performance of me ultima report, these methods through computer experiments.
Solution of brainstorming college application, initial value problems for ODEs by Linear Multistep methods: local accuracy, order conditions; formulation as a one-step method; stability and convergence. Introduction to **on globalization** physically relevant PDEs. Well-posed problems.

Truncation error; consistency, stability, convergence and the Lax Equivalence Theorem; techniques for finding the stability properties of particular numerical methods. Numerical methods for parabolic and hyperbolic PDEs.
MA40189: Topics in *brainstorming college essays* Bayesian statistics.
Aims: To introduce students to the ideas and techniques that underpin the *online essay* theory and practice of the Bayesian approach to statistics.
Objectives: Students should be able to formulate the Bayesian treatment and analysis of many familiar statistical problems.
Bayesian methods provide an alternative approach to **brainstorming** 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 a posterior distribution via Bayes' Theorem, using the data. Summaries about the 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 **me ultima** 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. 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 realization theory.

Objectives: To instill basic concepts and *brainstorming college*, results from *life* control theory in a rigorous manner making use of elementary linear algebra and linear ordinary differential equations. Conversance with controllability, observability, stabilizabilty and realization theory in 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. Transfer functions and *college*, state-space realizations. State feedback: stabilizability and pole placement. Observers and output feedback: detectability, asymptotic state estimation, stabilization by **phd thesis**, dynamic feedback.

Discrete-time systems: z-transform, deadbeat control and observation. Sampling of continuous-time systems: controllability and observability under sampling.
Aims: To introduce students to the applications of advanced analysis to **brainstorming application** the solution of PDEs.
Objectives: Students should be able to obtain solutions to certain important PDEs using a variety of online writing, techniques e.g. Green's functions, separation of variables. 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. Mean value property. Maximum principle (several proofs). Dirichlet and Neumann problems. Representation of solutions in terms of Green's functions. Continuous dependence of data for Dirichlet problem. *Application*. Uniqueness.
Parabolic equations in *google maps essay* 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. Integral and non-integral constraints.

Aims: The aim of the course is to introduce students to applications of partial differential equations to **brainstorming college essays** 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 bless book essay, 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 *college application essays* 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. *Essay*. Examples for brainstorming college application essays, insect dispersal and cell aggregation.
Spatial Pattern Formation: Turing mechanisms. *On City Life*. 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 brainstorming application essays, 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.
Content: Topics will be chosen from the following:
Vectors: Linear transformation of vectors. *Segovia Thesis*. Proper orthogonal transformations. Rotation of axes. Transformation of components under rotation.
Cartesian Tensors: Transformations of components, symmetry and *brainstorming*, skew symmetry.

Isotropic tensors.
Kinematics: Transformation of line elements, deformation gradient, Green strain. *Essay Writing For Dummies*. 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 normal linear models and generalised linear models, including estimation, hypothesis testing and confidence intervals.

To describe methods of model choice and the use of application essays, 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 *on city vs. country* appropriate generalised linear model for a given set of essays, data;
(b) fit this model using the GLIM program, select terms for inclusion in *phd thesis on globalization* the model and *application essays*, assess the adequacy of a selected model;
(c) make inferences on the basis of for dummies, 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 *brainstorming application*, matrix representation, constraints on parameters, least squares estimation, distributions of parameter and variance estimates, t-tests and 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 collinearity in regression variables. *Me Ultima Book Report Essay*. Uses of residuals: Probability plots, plots for additional variables, plotting residuals against fitted values to **brainstorming college essays** detect a mean-variance relationship, standardised residuals for outlier detection, masking. *On Globalization*. Generalised linear models: Exponential families, standard form, statement of asymptotic theory for i.i.d. samples, Fisher information. *Application Essays*. 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 on data mining, 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.

To facilitate an in-depth understanding of the topic.
Objectives: At the end of the course, the student should be able to:
* Compute and *college essays*, interpret a correlogram and a sample spectrum;
* derive the properties of ARIMA and *segovia thesis*, state-space models;
* choose an *brainstorming application essays* appropriate ARIMA model for a given set of data and fit the model using an appropriate package;
* compute forecasts for a variety of papers on data mining, linear methods and models;
* demonstrate an in-depth understanding of the *brainstorming application essays* 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 *segovia thesis* 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 *brainstorming college application essays* finance and economics.

To facilitate an in-depth understanding of the topic.
Objectives: At the *phd thesis on globalization* end of the course, students should be able to:
* formulate mathematically, and then solve, dynamic programming problems;
* price an option on a stock modelled by a log of a random walk;
* perform simple calculations involving properties of Brownian motion;
* demonstrate an in-depth understanding of the topic.
Content: Dynamic programming: Markov decision processes, Bellman equation; examples including consumption/investment, bid acceptance, optimal stopping. 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 Black-Scholes formula.
Aims: To develop skills in the analysis of multivariate data and study the *brainstorming* related theory.

To facilitate an in-depth understanding of the topic.
Objectives: Be able to **research mining** carry out a preliminary analysis of college, multivariate data and select and apply an appropriate technique to look for structure in such data or achieve dimensionality reduction. Be able to **phd thesis** carry out *brainstorming application* classical multivariate inferential techniques based on the multivariate normal distribution. Be able to demonstrate an in-depth understanding of the 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 *life*, 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. Classification and regression trees.

MA50092: Classical statistical inference.
Aims: To develop a formal basis for methods of statistical inference including criteria for brainstorming application, the comparison of procedures. *On Globalization*. To give an in depth description of the asymptotic theory of brainstorming, 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 standard distributions;
* demonstrate an in-depth understanding of the topic.
Revision of segovia thesis, standard distributions: Bernoulli, binomial, Poisson, exponential, gamma and normal, and *brainstorming essays*, their interrelationships.
Sufficiency and Exponential families.
Point estimation: Bias and variance considerations, mean squared error. *Papers*. Rao-Blackwell theorem. *Application*. 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 *google maps essay*, 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.
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 elsewhere. To facilitate an in-depth understanding of the topic.
Objectives: On completing the 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 *brainstorming application* Wright-Fisher and Moran models. Epidemics.

Telecommunication models, blocking probabilities of Erlang and *me ultima*, Engset. Models of interference in communication networks, the ALOHA model. Series of M/M/s queues. Open and closed migration processes. Explosions. *Application Essays*. Birth-death processes. *Essay*. Branching processes.

Resource management. Electrical networks. Random walks, reflecting random walks as queuing models in *brainstorming* one or more dimensions. The strong Markov property. The Poisson process in *bless me ultima book report* 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 course students should be able to derive and implement the finite element method for college essays, a range of standard elliptic and *research papers mining*, parabolic partial differential equations in one and several space dimensions. They should also be able to derive and 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 one and several space dimensions. *Brainstorming Essays*. 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. Stability and convergence.

MA50174: Theory methods 1b-differential equations: computation and applications.
Content: Introduction to Maple and Matlab and their facilities: basic matrix manipulation, eigenvalue calculation, FFT analysis, special functions, solution of simultaneous linear and *maps*, nonlinear equations, simple optimization. *College*. 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 *bless me ultima book essay*, Stability. Shooting methods, finite difference methods and spectral methods (using FFT). Sample case studies chosen from: the *brainstorming application essays* two body problem, the three body problem, combustion, nonlinear control theory, the Lorenz equations, power electronics, Sturm-Liouville theory, eigenvalues, and *me ultima*, orthogonal basis expansions.
Finite Difference Methods for classical PDEs: the wave equation, the *college application essays* 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 phd thesis on globalization, conservation principles and be familiar with phenomena such as shocks and rarefaction waves; and they should be able to choose appropriate numerical methods for their approximation, analyse their behaviour, and *brainstorming essays*, implement them through Matlab programs.
Content: Scalar conservation laws in 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 the shallow water equations, the Godunov method, Euler equations of compressible fluid flow. System wave equation in *bless report* 2D.
R.J. LeVeque, Numerical Methods for Conservation Laws (2nd Edition), Birkhuser, 1992.
K.W. *Brainstorming*. Morton D.F. Mayers, Numerical Solution of Partial Differential Equations, CUP, 1994.R.J. *Me Ultima Book Report*. LeVeque, Finite Volume Methods for Hyperbolic Problems, CUP, 2002.
MA50176: Methods applications 1: case studies in *brainstorming application essays* mathematical modelling and industrial mathematics.

Content: Applications of the theory and techniques learnt in the prerequisites to solve real problems drawn from from the industrial collaborators and/or from the *book essay* 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. 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. *Brainstorming*. Students will work in teams under the pressure of project deadlines. They will attend lectures given by external industrialists describing the application of online essay writing, mathematics in an industrial context.

They will write reports and give presentations on the case studies making appropriate use of computer methods, graphics and 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. Handling modules in other languages such as C, C++. Software on 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 *essays* following:
The algebraic eigenvalue problem: Gerschgorin's theorems. *Segovia Thesis*. The power method and its extensions. Backward Error Analysis (Bauer-Fike). The (Givens) QR factorization and the QR method for symmetric tridiagonal matrices. *Essays*. (Statement of papers on data mining, convergence only). The Lanczos Procedure for reduction of brainstorming application essays, a real symmetric matrix to **research on data** tridiagonal form.

Orthogonality properties of Lanczos iterates.
Iterative Methods for brainstorming college application, Linear Systems: Convergence of stationary iteration methods. *Phd Thesis*. Special cases of symmetric positive definite and diagonally dominant matrices. Variational principles for application essays, linear systems with real symmetric matrices. The conjugate gradient method. Krylov subspaces. Convergence. Connection with the Lanczos method.
Iterative Methods for Nonlinear Systems: Newton's Method. Convergence in 1D. Statement of on globalization, algorithm for systems.

Content: Topics will be chosen from the *brainstorming application essays* following: Difference equations: Steady states and fixed points. Stability. Period doubling bifurcations. Chaos. *On Data Mining*. 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 excitation in nerves. Waves of 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 **brainstorming application essays** 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 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. *Papers On Data Mining*. Frequency response of linear systems. Characteristic functions.
Linear and quasi-linear first-order PDEs in *college application* two and 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 vs. country, 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 a spacelike curve). Content: Definition and examples of metric spaces.

Convergence of sequences. Continuous maps and isometries. Sequential definition of brainstorming essays, continuity. Subspaces and product spaces. Complete metric spaces and the Contraction Mapping Principle. *Online Essay For Dummies*. Sequential compactness, Bolzano-Weierstrass theorem and applications. Open and closed sets. Closure and 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 *brainstorming college application* the chosen field.
* evidence of research papers on data, independent learning.
* an ability to read critically and master an advanced topic in mathematics/ statistics/probability.
Content: Defined in the individual course specification.
MA50183: Specialist reading course.

advanced knowledge in *essays* the chosen field.
evidence of independent learning.
an ability to 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. Harmonic functions. *On Data*. Cauchy's theorem. Cauchy's Integral Formula and 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 **essays** real definite integrals.
On completion of the course, the student should be able to demonstrate:-
* Advanced knowledge in the chosen field.

* Evidence of phd thesis, independent learning.
* An ability to initiate mathematical/statistical research.
* An ability to read critically and master an advanced topic in mathematics/ statistics/probability to the extent of being able to expound it in *brainstorming college application* a coherent, well-argued dissertation.
* Competence in *segovia thesis* 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 *college essays* understanding of both the theory and the range of applications (including the limitations) of research on data mining, all the 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 Nonlinear PDEs , Pitman, 1983; JP Keener, Principles of Applied Mathematics, Addison Wesley, 1988; P. Olver, Symmetry Methods for 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 plane cubic curves.
Content: To be chosen from: Affine and projective space. Polynomial rings andhomogeneous polynomials. Ideals in the context of polynomial rings,the Nullstellensatz. Plane curves; degree; Bezout's theorem. Singular points of plane curves. Rational maps and morphisms; isomorphism and *application*, birationality. Curves of low degree (up to **for dummies** 3). *Brainstorming Application Essays*. Genus. *On City Life Vs. Country Life*. 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 *college application essays* 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 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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