Oxford Mathematics Class 8 Solutions

[Sean Vaidhyanathan]

Oftenthe solution to a problem will require you to think outside its original framing. This is true here, and while you will see the second problem solved in your course, the first is far too deep and was famously solved by Andrew Wiles.

In applied mathematics we use mathematics to explain phenomena that occur in the real world. You can learn how a leopard gets its spots, explore quantum theory and relativity, or study the mathematics of stock markets.

Above all, mathematics is a logical subject, and you will need to think mathematically, arguing clearly and concisely as you solve problems. For some of you, this way of thinking or solving problems will be your goal. Others will want to see what else can be discovered. Either way, it is a subject to be enjoyed.

'Studying Mathematics at Oxford has been a massive change from A-Level but I have been really enjoying it, I especially love having tutorials as they are a fantastic opportunity to work through problems that specifically you and your tutorial partner are having difficulties with and ask questions that allow you to improve yourself as a mathematician.'

Tutorials are usually 2-4 students and a tutor. Class sizes may vary depending on the options you choose. There would usually be around 8-12 students though classes for some of the more popular papers may be larger.

Most tutorials, classes, and lectures are delivered by staff who are tutors in their subject. Many are world-leading experts with years of experience in teaching and research. Some teaching may also be delivered by postgraduate students who are usually studying at doctoral level.

Admission to Mathematics is joint with Mathematics & Statistics, and applicants do not choose between the two degrees until the end of their fourth term at Oxford. At that point, all students declare whether they wish to study Mathematics or study Mathematics & Statistics. Further changes later on may be possible subject to the availability of space on the course and the consent of the college.

The first year consists of core courses in pure and applied mathematics (including statistics). Options start in the second year, with the third and fourth years offering a large variety of courses, including options from outside mathematics.

The majority of those who read Mathematics will have taken both Mathematics and Further Mathematics at A-level (or the equivalent). However, Further Mathematics at A-level is not essential. It is far more important that you have the drive and desire to understand the subject.

Our courses have limited formal prerequisites, so it is the experience rather than outright knowledge which needs to be made up. If you gain a place under these circumstances, your college will normally recommend suitable extra preparatory reading for the summer before you start your course.

We don't want anyone who has the academic ability to get a place to study here to be held back by their financial circumstances. To meet that aim, Oxford offers one of the most generous financial support packages available for UK students and this may be supplemented by support from your college.

Living costs for the academic year starting in 2024 are estimated to be between 1,345 and 1,955 for each month you are in Oxford. Our academic year is made up of three eight-week terms, so you would not usually need to be in Oxford for much more than six months of the year but may wish to budget over a nine-month period to ensure you also have sufficient funds during the holidays to meet essential costs. For further details please visit our living costs webpage.

In 2024 Oxford is offering one of the most generous bursary packages of any UK university to Home students with a family income of around 50,000 or less, with additional opportunities available to UK students from households with incomes of 32,500 or less. The UK government also provides living costs support to Home students from the UK and those with settled status who meet the residence requirements.

Unistats course data from Discover Uni provides applicants with statistics about a particular undergraduate course at Oxford. For a more holistic insight into what studying your chosen course here is likely to be like, we would encourage you to view the information below as well as to explore our website more widely.

College tutorials are central to teaching at Oxford. Typically, they take place in your college and are led by your academic tutor(s) who teach as well as do their own research. Students will also receive teaching in a variety of other ways, depending on the course. This will include lectures and classes, and may include laboratory work and fieldwork. However, tutorials offer a level of personalised attention from academic experts unavailable at most universities.

You can also watch recent lectures, and see a real first-year tutorial on the Mathematics YouTube channel to get a feel for what studying here is like and find out about the department's research at the Oxford Mathematics Alphabet.

The P versus NP problem is a major unsolved problem in theoretical computer science. Informally, it asks whether every problem whose solution can be quickly verified can also be quickly solved.

Here, quickly means an algorithm that solves the task and runs in polynomial time exists, meaning the task completion time varies as a polynomial function on the size of the input to the algorithm (as opposed to, say, exponential time). The general class of questions that some algorithm can answer in polynomial time is "P" or "class P". For some questions, there is no known way to find an answer quickly, but if provided with an answer, it can be verified quickly. The class of questions where an answer can be verified in polynomial time is NP, standing for "nondeterministic polynomial time".[Note 1]

The problem has been called the most important open problem in computer science.[1] Aside from being an important problem in computational theory, a proof either way would have profound implications for mathematics, cryptography, algorithm research, artificial intelligence, game theory, multimedia processing, philosophy, economics and many other fields.[2]

In the game Sudoku, the player begins with a partially filled-in grid of numbers and attempts to complete the grid following the game's rules. Given an incomplete Sudoku grid, of any size, is there at least one legal solution? Proposed solutions are easily verified, and the time to check a solution grows slowly (polynomially) as the grid gets bigger. However, all known algorithms for finding solutions take, for difficult examples, time that grows exponentially as the grid gets bigger. So, Sudoku is in NP (quickly checkable) but does not seem to be in P (quickly solvable). Thousands of other problems seem similarly fast to check but slow to solve. Researchers have shown that many of the problems in NP have the extra property that a fast solution to any one of them could be used to build a quick solution to any other problem in NP, a property called NP-completeness. Decades of searching have not produced a fast solution to any of these problems, so most scientists suspect that these problems cannot be solved quickly; however, this is unproven.

The precise statement of the P versus NP problem was introduced in 1971 by Stephen Cook in his seminal paper "The complexity of theorem proving procedures"[3] (and independently by Leonid Levin in 1973[4]).

The relation between the complexity classes P and NP is studied in computational complexity theory, the part of the theory of computation dealing with the resources required during computation to solve a given problem. The most common resources are time (how many steps it takes to solve a problem) and space (how much memory it takes to solve a problem).

In such analysis, a model of the computer for which time must be analyzed is required. Typically such models assume that the computer is deterministic (given the computer's present state and any inputs, there is only one possible action that the computer might take) and sequential (it performs actions one after the other).

To attack the P = NP question, the concept of NP-completeness is very useful. NP-complete problems are problems that any other NP problem is reducible to in polynomial time and whose solution is still verifiable in polynomial time. That is, any NP problem can be transformed into any NP-complete problem. Informally, an NP-complete problem is an NP problem that is at least as "tough" as any other problem in NP.

NP-hard problems are those at least as hard as NP problems; i.e., all NP problems can be reduced (in polynomial time) to them. NP-hard problems need not be in NP; i.e., they need not have solutions verifiable in polynomial time.

From the definition alone it is unintuitive that NP-complete problems exist; however, a trivial NP-complete problem can be formulated as follows: given a Turing machine M guaranteed to halt in polynomial time, does a polynomial-size input that M will accept exist?[11] It is in NP because (given an input) it is simple to check whether M accepts the input by simulating M; it is NP-complete because the verifier for any particular instance of a problem in NP can be encoded as a polynomial-time machine M that takes the solution to be verified as input. Then the question of whether the instance is a yes or no instance is determined by whether a valid input exists.

The problem of deciding the truth of a statement in Presburger arithmetic requires even more time. Fischer and Rabin proved in 1974[17] that every algorithm that decides the truth of Presburger statements of length n has a runtime of at least 2 2 c n \displaystyle 2^2^cn for some constant c. Hence, the problem is known to need more than exponential run time. Even more difficult are the undecidable problems, such as the halting problem. They cannot be completely solved by any algorithm, in the sense that for any particular algorithm there is at least one input for which that algorithm will not produce the right answer; it will either produce the wrong answer, finish without giving a conclusive answer, or otherwise run forever without producing any answer at all.

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