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140

Theoretical computer science could certainly be considered a branch of mathematics. This branch of computer science deals with computers and computer programs as mathematical objects. Theoretical computer scientists could be described as computer scientists who know little about computers. However, when people say "computer science" they usually include ...


54

Let $C$ and $M$ be the set of all things which are considered "computer science" and "mathematics", respectively. If I understand you correctly, your question is: Is $C\subset M$? If this is the case, your question is not well posed because neither $C$ nor $M$ are well defined. How do you draw the line between what is math and what is not math without ...


49

We will use the Mellin transform technique. Recalling the Mellin transform and its inverse $$ F(s) =\int_0^{\infty} x^{s-1} f(x)dx, \quad\quad f(x)=\frac{1}{2 \pi i} \int_{c-i \infty}^{c+i \infty} x^{-s} F(s)\, ds. $$ Now, let's consider the function $$ f(x)= \frac{x}{e^{\pi x}+1}. $$ Taking the Mellin transform of $f(x)$, we get $$ ...


22

I would say that computer science is a branch of mathematics. Donald Knuth is a famous computer scientists and is also considered a great mathematician. He wrote a series of books called "The Art of Computer Programming" which is extremely rigorous and mathematical. Edit: To make my position more clear since it is apparently controversial. Almost all ...


20

According to http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Abstract_linear_spaces.html, the scalar product was introduced by Grassmann in 1844, in a work that was rejected by other mathematicians for many years. Peano was apparently the first to give a clear and precise account of Grassmann's ideas in his 1888 book Geometrical Calculus (see ...


19

I doubt that we will ever know the exact integral that vexed Feynman. Here is something similar to what he describes. Suppose $f(z)$ is an analytic function on the unit disk. Then, by Cauchy's integral formula, $$\oint_\gamma \frac{f(z)}{z}dz = 2\pi i f(0),$$ where $\gamma$ traces out the unit circle in a counterclockwise manner. Let $z=e^{i\phi}$. Then ...


18

I imagine one could write books on the subject, but probably the first moment one encounters homological algebra in algebraic geometry is the following. If $f: X\to Y$ is a proper map of varieties, then given a coherent sheaf $\mathcal{F}$ on $Y$ one can form the pull-back $f^{*} \mathcal{F}$. Now, the pull-back functor $f^{*}: \mathcal{C}oh(Y) \rightarrow ...


18

There is a nice survey of the subject area available in pdf: Survey: Computational Group Theory, which while somewhat dated, gives a nice introduction to the field and provides some historical insights. Here's a very nice Introduction to Computational Group Theory. It's a brief but fascinating survey by Ákos Seress, published in the AMS Notices (1997: 06). ...


17

This doesn't answer your question, but: why don't you just start reading papers in algebraic geometry? You will quickly be forced to come to grips with "reality" in this way. My basic point is that, if you have read (a lot of) Hartshorne and Liu, you don't need more textbooks; you just need to start reading some research mathematics. But regarding your ...


17

It is not uncommon to hear ideas along the lines that computer science is computer programming without practical constraints theoretical computer science is computer science without physical constraints mathematics is computer science without finiteness constraints Each subject in the chain is seen as a limiting case of the one before, where some ...


16

You might want to start at the Mathematical Atlas. It gives a visual map of domains of mathematics, and how they are inter-related: you can click on any "bubble" to learn more about each area. There is also a menu to the right-hand side: for example, see A Gentle Introduction to the Mathematics Subject Classification Scheme, also referred to as MSC. The ...


16

We want to compute $$S=\sum_{m=1}^{\infty}\frac{(-1)^{s(m)}}{m},$$ where $s(m)$ counts the number of appearances of primes of the form $4k+1$ in the prime decomposition of $m$. Note that $$S=\sum_{n=0}^{\infty}\frac{(-1)^{s(2n+1)}}{2n+1}+\frac{S}{2}\quad\Longrightarrow \quad \frac{S}{2}=\sum_{n=0}^{\infty}\frac{(-1)^{s(2n+1)}}{2n+1}.\tag{1}$$ But the ...


15

I suggest you take a look at the website The Art of Problem Solving. There are links to resources, to articles, to competition preparation books, an online AoPS competition problems ("For the Win"), and more: all geared to bright students who love math, are looking for challenging problems, and it is particularly aimed to those students who participate in ...


15

That depends on whether you consider software engineering to be computer science. I don't. The theory of computation is absolutely a branch of mathematics, and one of the most difficult. Forget P vs. NP, we can't even decide the Collatz conjecture, which can be understood by the average third-grader in a minute or two. On the other hand, software ...


15

Euler apparently had some trouble deriving the Jacobian used in change of variables for double integrals. He began by considering congruent transformations consisting of (affine) linear functions, and got something like $$\mathrm{d}x\,\mathrm{d}y=m\sqrt{1-m^2}\,\mathrm{d}t^2+(1-2m^2)\,\mathrm{d}t\,\mathrm{d}v-m\sqrt{1-m^2}\,\mathrm{d}v^2$$ which he ...


14

I recommend the Handbook of Computation Group Theory, especially chapter 2 (omitting 2.4.4 and 2.5) and chapter 8.1 and 8.7. Outline Constructing groups of order $n$ is inductive: we assume that one has constructed all groups whose order properly divides $n$ and have them sorted nicely. To construct the group $G$ we try a few things: If $G$ is solvable ...


14

The calculation of the Mellin transform of $f(x)$ is not present in the above answer, so I will show it here. $$\mathfrak{M}\left(\frac{1}{e^{\pi x}+1};s \right) = \int_0^\infty \frac{1}{e^{\pi x}+1} x^{s-1} dx = \int_0^\infty \frac{1}{e^{\pi x}} \frac{1}{1+e^{-\pi x}} x^{s-1} dx \\= \int_0^\infty \frac{1}{e^{\pi x}} \sum_{q\ge 0} (-1)^q e^{-\pi q x} ...


14

It is a fundamental problem in algebraic number theory to determine asymptotically how many number fields there are of a given degree $d$ of discriminant $\leq X$, and there is a lot of research on this question. This paper of Ellenberg and Venkatesh is devoted to this problem. In the introduction it discusses a conjecture of Linnik that (for $d$ fixed) ...


13

Barring any more information, I'd suggest you acquire Michael Spivak's Calculus, and begin working your way through the material. There are plenty of exercises and examples, but you'll also deepen your conceptual understanding at the same time, which will help ensure the material "sticks with you" over the long-run, which I presume is a goal of yours, as you ...


13

Let's start with $$ \sum_{n=0}^\infty x^n=\frac1{1-x}\tag{1} $$ Differentiating $(1)$ and multiplying by $x$, we get $$ \sum_{n=0}^\infty nx^n=\frac{x}{(1-x)^2}\tag{2} $$ Taking the odd part of $(2)$ yields $$ \sum_{n=0}^\infty(2n+1)x^{2n+1}=\frac{x(1+x^2)}{(1-x^2)^2}\tag{3} $$ Using $(3)$, we get $$ \begin{align} ...


13

Calculus, while useful, is not as important to Computer Science as other branches of maths of the more discrete kind. Consider: graph theory game theory boolean algebra numerical methods statistics linear algebra, matrices, etc. You're probably better of with a solid grounding in this stuff, rather than differential/integral calculus.


12

I'd recommend that you visit the Art of Problem Solving's (AoPS) website: I've linked you to their "resource" page with articles you can download (they are freely accessible.) The website is a "hub" for very motivated students of mathematics, many of whom engage in competition math. The site hosts mathematics resources, curricula, on-line forums, and a ...


12

Well there are different levels of "correct". Most of the bits of this answer have been given separately in other answers or comments but I'd like to submit a "complete" answer if I may. The most correct would be to write the whole name the Hungarian way: Erdős Pál In Hungarian, surnames come first, given names come last. This is also the case in ...


12

Yes, thank you, I'm alive and well, and currently writing a book on how to teach mathematics at high school and elementary level. The book was originally published in 1986, and then the second revised edition came out in 1998, which is a long time ago. Since then, the resources of the Internet. In this context, have exploded – as the reference to Plouffe's ...


12

There is some pedagogical value to understanding how theorems/definitions apply to and playout with particular groups (i.e., through examples). While it is admirable to yearn for a thorough theoretical understanding of group theory, try to think of studying and probing the examples as a means of testing your understanding of the theories and definitions, as ...


12

If $V$ is a Banach space we call $V'$ the dual space (see continuous dual space on wikipedia), i.e. the space of linear continuous functionals $\xi \colon V\to \mathbb R$. Then it is well known that there exists a natural injection $$ J \colon V \to V'' $$ defined by $$ J(v)(\xi) = \xi(v) $$ for all $\xi \in V'$. We know that $J$ is an isometry, in ...


11

Ramanujan's Collected Papers are available at http://www.imsc.res.in/~rao/ramanujan/collectedindex.html. The paper you seek is available at http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper18/page1.htm but it's from 1916 not 1918.


11

I have three suggestions: Mac Lane, S., and Moerdijk, I., "Sheaves in Geometry and Logic: A First Introduction to Topos Theory" Kashiwara, M., and Schapira, P., "Categories and Sheaves" The first is my favorite. The latter is more advanced, and doesn't really start talking about sheaves until late in the book. It's a quality text nonetheless. Finally, ...


11

The short answer is: it depends! To do differential geometry you don't really need category theory at all, and the same could (nearly) be said for some flavors of algebraic geometry. That said, some people (myself included) learn things best from a categorical standpoint. If you get excited whenever people mention universal properties, and are happiest ...



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