University Degree: Mathematics

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  1. The Mandelbrot set is a set of points in a complex plane, c, of the orbit, which is how the function operates under the iteration of zn+1 = zn2 + c around 0.

    This can repeated by letting c be x for the next iteration in the original equation, and this will yield c2 + c. You can continue repeating putting the previous answer in for x into the equation and the result will be (c2 + c) 2 + c. Doing this continuously will create a list of complex numbers and if these complex numbers are increasing in size and thus further away from the origin then 'c' is not in the Mandelbrot Set.

    • Length: 603 words
  2. Bridges of Knigsberg & Eulerian circuits

    The way Euler solved it was by thinking of the river and bridges in terms of graph theory. This was achieved by removing anything unnecessary to the problem and so therefore ended up drawing a picture primarily consisting of points that represented the islands and separate lines showing the bridges that connected the islands.2 Figure 2 shows the picture of what Eular might have drawn up. Fig. 2 As you can see the problem now looks much simpler and one could attempt to solve by just using a pencil and going around the lines looking for a solution but to find a systematic approach to is incredibly tedious laborious.

    • Length: 690 words
  3. Maths Economics

    • Length: 3572 words
  4. A function is a relation between a set of input values

    s(8) = 4 s(10) = 0 s(345) = 5 s(27,560) = 0 s(738,954,683,012) = 4 Using mental arithmetic I can calculate 82 and 102 quite easily. However calculating 738,954,683,0122 is very challenging to say the least. So how can I be confident that my answers are correct? Using the Arabic-Hindu numeral system of hundreds, tens, units etc. and the following simple theory of multiplication I am confident of my answers being correct. Using a three digit number and each number is represented by letters. Let the first three digits of the first number be 'a' (hundreds digit)

    • Length: 1054 words
  5. Essay on Professor V Lakshmikantham - A great mathematician

    He published a book "Theory of Fuzzy Differential Equations and Inclusions" as an author. This was when he was a Professor and the Head of Applied Mathematics at the Florida Institute of Technology in Melbourne. This book was about complicated differential equations were applied to real-world problems in subjects like engineering, computer science, and social science. His proofs and solutions made quick and recent developments. Also these equations were presented in a detail way and includes the basics for "fuzzy set theory".

    • Length: 922 words
  6. Banach-Tarski Paradox Talk

    A1 ? A2 ? ... ? An = S (assembling all the pieces yields the original sphere S) There exist T1, T2, ... Tn, where each Ti represents some finite sequence of rotations and translations, such that if we apply each Ti to each Ai (let's call the result Ai'), then: A1' ? A2' ? ... ? Am' = S (a subset of the original pieces forms S) Am+1' ? Am+2' ? ... An = S' (the remaining pieces forms a copy of S) where m is some number between 1 and n, and S' is S translated by some finite amount so that S and S' are disjoint.

    • Length: 1927 words
  7. Development of Calculus

    Newton and Leibniz clearly owed much to their immediate predecessors, but their mutual influence during the crucial periods when they were making their own original inventions was minimal.1 A controversy arose from the respective claims made by Newton and Leibniz and their supporters to priority in the 'invention' of the calculus. This controversy was extremely bitter and its repercussions were not without influence on the history of mathematics in the 18th century alienating British mathematics from the developments in the rest of Europe.

    • Length: 2333 words
  8. Greek Maths

    This was to stop people claiming his work to be their own and he is quoted as saying, "those who claim to discover everything but produce no proofs of the same, may be confuted as having pretended to discover the impossible".2 The Method is a treatise containing a collection of Archimedes methods of discovery which was unexpectedly found in Jerusalem in the late nineteenth century.3 The Method includes Archimedes' methods of discovery by mechanics of many important results on areas and volumes.

    • Length: 2502 words
  9. Egytptian Numbers

    It was based on powers of ten from one to one million. The number one was illustrated by a short length of rope. The number ten was shown as a horseshoe of a longer piece of rope. One hundred, a coil of rope. Rope was used as a hieroglyphic in different shapes and lengths to signify the key role of the "rope stretchers" of ancient Egypt.2 The hieroglyphic for one thousand was a lotus flower, a finger represented ten thousand, one hundred thousand was pictured as a tadpole/frog, and anything as large one million or more was illustrated by a man with raised arms.

    • Length: 2421 words
  10. OPTIMAL PATH PLANNING USING AN IMPROVED A* ALGORITHM FOR HOMELAND SECURITY APPLICATIONS

    The output of a search algorithm is either a failure or a solution. The efficiency of a search algorithm can be evaluated in four ways [1] * Completeness: Is the algorithm guaranteed to find a solution when it exists? * Optimality: Does the strategy find the optimal solution? * Time Complexity: The time taken to find the solution * Space Complexity: The memory needed to perform the search 1.1 Background Search algorithms have been used extensively to solve various problems like the Queens problem, the dining philosophers problem etc.

    • Length: 3816 words
  11. 'The debates between Keynes and the "classical" economists are only of interest to economic historians; they are of no relevance to modern policymakers.'

    i The above quote shows how Adam Smith described this by using his theory of the 'invisible hand' to back it up. The classical model of the economy suggests that all markets are always at equilibrium. The labor market failing to reach equilibrium level cannot exist in the classical model because of the competitive exchange equilibrium. This way quantities and prices can adjust accordingly. The classical model is for a closed economy and the variable are employment, real and nominal wages, prices levels, interest rates and real output.

    • Length: 1835 words
  12. Lines, Regions and Crossovers

    I also plan to find the relationships / sequence between any two of these characteristics of the line(s). I will accomplish this by using the following formula: A+B (n+1) +0.5(n-1) (n-2) C. A = 1st term of the sequence. B = the difference between the first two terms. C = the 2nd difference (the difference of the 1st difference). Thirdly I will put all my findings onto to a table consisting of total regions, closed regions, open regions, crossover points and all of the formulas. Lastly I will try to summarize all my findings by writing up a conclusion and evaluation.

    • Length: 1728 words
  13. In general terms, biological pest control is the use of a specially chosen living organism to control a particular pest. This chosen organism might be a naturally occurring parasites,

    Unfortunately biological pest control methods alone are rarely sufficient. Research suggest that an integrated approach, using pesticides, biological pest control and other techniques may be the most affective. In general terms, biological pest control is the use of a specially chosen living organism to control a particular pest. This chosen organism might be a naturally occurring parasites, predator, or disease that will attack a harmful insect.

    • Length: 1378 words
  14. Case Analysis: Forecasting Food and Beverage Sales

    145 152 110 130 152 206 2106 2 263 238 247 193 193 149 157 161 122 130 167 230 2250 3 282 255 265 205 210 160 166 174 126 148 173 235 2399 Total: 787 728 744 576 587 449 468 487 358 408 492 671 6755 Mean: 262.3333333 242.66667 248 192 195.6667 149.6667 156 162.3333 119.33333 136 164 223.6667 2251.666667 Variance: 400.3333333 116.33333 273 183 174.3333 100.3333 111 122.3333 69.333333 108 117 240.3333 21464.33333 StDev: 20.0083316 10.785793 16.52271 13.52775 13.20353 10.01665 10.53565 11.06044 8.326664 10.3923 10.81665 15.50269 146.5071102 Analyze the Data There are two main goals of time series analysis: (a)

    • Length: 3082 words
  15. Analyzing First Order Kinetics - cis-trans isomerization of mercury dithizonate

    ) / 2 From the data, y0 = 0.2969208 y? = 0 y t1/2 = (0.2969208 + 0) / 2 = 0.1484604 From Graph 1, At absorbance 0.1484604, t1/2 = 21.5s By kobs = ln2 / t1/2 , kobs = ln2/ 21.5 = 0.0322 s-1 Semi-log Method A graph of Z(t) = ln[yt-y?] against time is plotted. (The value of ln[yt-y?] refer to Appendix 3) Graph 2 From the equation, y= -0.0249x - 1.5439 Slope of graph = -kobs = -0.0249 ? kobs = 0.0249 s-1 Guggenhein's Method Take ?t = 1 A graph of Z(t) = ln[y(t+?t)-yt] against time is plotted. (The value of ln[y(t+?t)-yt] refer to Appendix 3)

    • Length: 730 words
  16. This paper intends to examine the words starting with <sn> given in the Oxford English Dictionary, and if possible categorise them, according to shared meaning. It will examine briefly the academic background to Sound symbolism

    Clustering is concerned with words which share sounds that have meaning at vowel and consonant level; Magnus (1997) refers to this as the Semantic Association. An example of Semantic Association given by Magnus is that if in any language the given word for house begins with an <h> or a <t> or even an <i> then by the process of clustering it would be expected that a disproportionately large number of words concerning housing would also being with that letter, <h> or <t> of <i>. So therefore in English we have house with the other house words including hut, home, hacienda, and hovel.

    • Length: 3578 words
  17. This paper presents a model of a four-cylinder spark ignition engine, which shows the capability of Simulink to model an S.I. engine from the crankshaft output down to the throttle.

    manifold * Mass flow rate * Compression stroke * Torque generation and acceleration Additional components can be added to the model to gain greater accuracy in the simulation. Throttle The simulation on the throttle body is based on the control input of the angle of the throttle plate. The rate in which the model induces air into the intake manifold can be expressed as the outcome of two functions. One is the function of the throttle plate angle and the other is an atmospheric function of the manifold pressure.

    • Length: 1122 words
  18. Wireless Local Area Network

    Th? wir?l?ss nod?s in th?s? n?tworks do not talk dir?ctly to ?ach oth?r. Inst?ad, ?ach nod? is conn?ct?d to a bas? station through which it communicat?s with oth?r nod?s. Infrastructur?-l?ss wir?l?ss n?tworks ar? usually call?d multi-hop wir?l?ss mobil? ad hoc n?tworks. In th? r?st of this pap?r w? will us? th? t?rm ad hoc n?twork. "Ad hoc" oft?n m?ans "improvis?d", or "for th? n??ds of th? mom?nt". (Siva Ram Murthy & Manoj). In comput?r n?tworking, w? think of an ad hoc n?twork as a wir?l?ss n?twork without any pr?-?xisting infrastructur?.

    • Length: 3559 words
  19. TILINTARKASTUS - STANDARDIEN VAI INFORMAATION TOTEUTTAJA?

    (83/349/ETY). Direktiivi 86/635/ETY: Neuvoston direktiivi, annettu 8 päivänä joulukuuta 1986, pankkien ja muiden rahoituslaitosten tilinpäätöksestä ja konsolidoidusta tilinpäätöksestä. (86/635/ETY). Hallituksen esitys n:o 295 (1993 vp.) Hallituksen esitys Eduskunnalle tilintarkastuslaiksi ja eräiksi siihen liittyviksi laeiksi. (HE 295/1993). Hex Oyj - Keskuskauppakamari - Teollisuus ja työnantajat: Suositus listayhtiöiden hallinnointi- ja ohjausjärjestelmistä (Corporate Governance). Helsinki 2003. (Hex Oyj - Keskuskauppakamari - Teollisuus ja työnantajat). Horsmanheimo, Pasi - Steiner, Maj-Lis: Tilintarkastus - asiakkaan opas. Helsinki 2002. (Horsmanheimo - Steiner). Kauppa- ja teollisuusministeriö: Corporate governance. Osoitteesta: http://www.ktm.fi/ index.phtml?menu_id=201&lang=1. Viittauspäivä 15.10.2004. (Kauppa- ja teollisuusministeriö: Corporate governance). Kauppa- ja teollisuusministeriö: Tilintarkastuslakityöryhmä - yhteenveto saaduista lausunnoista 8/2004: Kauppa- ja teollisuusministeriön julkaisuja 8/2004.

    • Length: 7186 words
  20. Edgar Allan Poe's The Raven is 108 lines long, and is written introchaic octameter, therefore having 16 syllables per line, less thelast line of each stanza, which is trochaic, but having only seven

    Poe uses alliteration in writing, "doubting, dreaming dreams no mortal ever dared to dream before." Assonance, consonance, and alliteration pervade The Raven, creating a melodic and rhythmical facet of the poem. The second, fourth, fifth, and final line of every stanza end with the "ore" sound, with such words as implore, nevermore, Lenore, lore, and yore. Every stanza contains 6 lines, there are no perturbations from this pattern. The poem is nearly acatalectic- a few of the deviate from the octameter, such as line 62, which contains a mere 15 syllables. The atmosphere created by Poe especially appeals to the readers senses- the narrator apprises the reader that it is during the month of December,

    • Length: 1982 words
  21. Hange of sign, Newton-Raphson and the rearrangement method and are going to use them to find roots of different equations, and hence

    = 96>0 the graph has 2 turning points. As the graph shows there are three roots. I calculated each of the f(x) values in order to see where the change in sign was, as shown in the table on follow page 3 : There is a root in the interval of [1, 2] which I have chosen to investigate. Now I use excel to calculate the change values of f(x) by taking increments in x of size 0.1 for the equation that we are using. As the graph and the table show, there is a change of sign between x=1.5 and x=1.6.

    • Length: 2068 words
  22. Lesson Plan for Physical Education Year GroupKS1 AreaSports hall/gym with apparatus ThemeTravelling in different directions

    Ask them to come up with different ways that they could do this. For example running, walking, skipping, hopping etc. Then ask them to move using both their hands and their feet - see how many different ways they can come up with. Encourage them to keep moving, move into space and to change direction. (8 minutes) Ask the children to come up with three or four ways of travelling using a combination of hands and feet remembering to think about direction. Using mats they should practice these three or four travelling movements. They should also think of a starting and finishing position. (15 minutes) Supervise the children putting out their designated apparatus.

    • Length: 635 words
  23. Concept of Limit in Real Analyses

    On the other hand, even if one could intuitively estimate, that the terms of the above sequence tend to zero (converge to zero), as one chooses a term far enough, it is still unclear, how to justify one's intuitive guess, because he/she might have intuition different from others. The space '...' in (1) reflects the distance between first attempts of ancient scholars to halt 'leak' of information about incompleteness of rational numbers they discovered, and the modern era development of old concepts such as differential and integral calculus, which solved and interpreted ancient and modern enigmas.

    • Length: 3597 words
  24. FindingZeros of Functions

    = cos x. So from this, we can also state that that is the same thing as g(x1) = cos x1. Hence, if we make the substitution, it would result to x2= cos x1, where we can further substitute the x1 to 1, making the whole equation x2= cos 1. This would then be the first iteration. Subsequently, this entire substitution process would be repeated another four times, each time using the previous answer as a substitute. The calculations below will provide a better understanding of the concept. 1st iteration, n = 1 2nd iteration, n = 2 3rd iteration, n = 3 x1+1 = g(x1)

    • Length: 2566 words
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"Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not."

- G. H. Hardy

Have you ever wondered how we can be certain that there are infinitely many prime numbers? If you have, you should consider studying mathematics. Mathematicians will tell you, sometimes arrogantly, that maths is the most challenging subject in the world. They may have a point: it's hard to think creatively and rigorously at the same time, and for a university degree in mathematics, that's exactly what you have to do. That and a lot of problem sets.

But if you think that mathematics means no writing, youre mistaken. The only way to communicate results is through language: proofs and maths papers will require writing, and you'll probably be assigned a final year project as well. Luckily, Marked by Teachers can help. Study our marked papers, and it will only be a matter of time before writing well-structured arguments is second nature to you.

A mathematics degree will change the way you think forever. When you graduate, you'll be set to pursue a whole slew of careers including research, banking,economics, computer science, information technology, and statistics.

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