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Open University pure maths study and research blog
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A highly composite number is a positive integer with more divisors than any positive integer smaller than itself.
page 2 of the paper |
Hardy (l) and Ramanujan (r). |
Many schools include a research project as part of the graduation requirements for their mathematics majors. But most students are at a loss to create their own research questions, leaving this task to their advisors. It would be better if the student came up with their own research question that involved significant mathematical investigation and the creation of original mathematics. This is a daunting task: most graduate students are unable to do this, and rely on their advisors to frame a suitable area for investigation. The task is further complicated by the fact that many questions relating to undergraduate mathematics have “already been solved,” while many of the unsolved questions require so much specialized background to understand or so much existing research to review that the preparation needed to tackle the problem is itself a major project. - Jeff Suzuki in "But How Do I Do Mathematical Research?” on MAA website.
... the intensity of his interest in mathematics led him to pay scant attention to the other subjects in which he was obliged to show some facility. ... On each occasion, his interest in his own mathematical researches was so all consuming that he neglected his more quotidian studies, with the result that he failed his examinations and lost his scholarships. p118 ...no wonder he doesn't understand how to do a proof! p125 - The Indian Clerk
f[n_] := Map[# // MatrixForm &, Map[SparseArray[{i_, i_} \Rightarrow 1, {n, n}][[#]] &, Permutations[Array[# &, n]]]]maps $n$ to $S_n$, the symmetric group of $n$ elements, and displays its elements as a list of permutation matrices.
How did Srinivasa Ramanujan perceive the mest world? |
Lecture 1: Introduction |
Let $a$ be a natural number and let $g: \mathbf{N} \times \mathbf{N} \rightarrow \mathbf{N}$ be a function. The function $h: \mathbf{N} \rightarrow \mathbf{N}$ is said to be defined by primitive recursion from the constant $a$ and the function $g$ if
- $h(0) = a$,
- $h(n+1) = g(n, h(n))$.
Example:
- $h(0) = 1$
- $h(n+1) = g(n, h(n)) = (n+1) \times h(n)$
is the well-know faculty function.
( Open University, M381 ML-1 )
In its classical nineteenth-century form, the tripos was a distinctive written examination of undergraduate students of the University of Cambridge. Prior to 1824, the Mathematical Tripos was formally known as the "Senate House Examination".From about 1780 to 1909, the "Old Tripos" was distinguished by a number of features, including the publication of an order of merit of successful candidates, and the difficulty of the mathematical problems set for solution. By way of example, in 1854, the Tripos consisted of 16 papers spread over 8 days, totaling 44.5 hours. The total number of questions was 211. The actual marks for the exams were never published, but there is reference to an exam in the 1860s where, out of a total possible mark of 17,000, the senior wrangler achieved 7634, the second wrangler 4123 and the lowest scoring candidate 237. ( W'pedia )
Click to enlarge |
Click to enlarge |
PS Tricks graphic example |
Students of literature read Shakespeare; students of music listen to Bach. In mathematics such a tradition is, if not entirely absent, at least fairly uncommon. This book is meant to address that situation. Although it is not intended as a history of the calculus, I have come to regard it as a gallery of the calculus. - (William Dunham)
$$\sum_{d/n}\frac{\mu(d)}{d} = 1 -\sum \frac{1}{p_i} + \sum \frac{1}{p_i p_j} - \sum \frac{1}{p_i p_j p_k} + \cdots$$
Apostol, Chapter 2: Arithmetical Functions
We start with a "sound wave" that is "noisy" at the prime numbers and silent at other numbers; this is the von Mangoldt function. Then we analyze its notes or frequencies by subjecting it to a process akin to Fourier transform; this is the Mellin transform. Then we prove, and this is the hard part, that certain "notes" cannot occur in this music. This exclusion of certain notes leads to the statement of the prime number theorem. - Tao
Schatz discovered in 1929 that the Platonic solids could be inverted, and one of the products of the inversion of the cube was the oloid.
Exercise, show that for $n>1$: $$\sum_{d/n} \mu(d)=0,$$ where $\mu$ is the Möbius function.
Exercises in a text generally have two functions: to reinforce the reader's grasp of the material and to provide puzzles whose solutions give a certain pleasure. Here, the exercises have a third function: to enable the reader to discover important facts, examples, and counterexamples. The serious reader should attempt all the exercises (many are not difficult), for subsequent proofs may depend on them; the casual reader should regard the exercises as part of the text proper.
(J. Rotman - An introduction to the theory of groups )
float InvSqrt(float x){ float xhalf = 0.5f * x; int i = *(int*)&x; // store floating-point bits in integer i = 0x5f3759d5 - (i >> 1); // initial guess for Newton's method x = *(float*)&i; // convert new bits into float x = x*(1.5f - xhalf*x*x); // One round of Newton's method return x; }
Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before
(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)