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10

In English, I've almost always heard mathematicians say "We now differentiate $f$ to get ...". Occasionally I've heard "derive," but in English (my native language!), that's generally used to mean "work out", as in "Ralph couldn't derive a proof of the intermediate value theorem from the information he had at hand." It's also used in generating one thing ...


8

The beautiful rhythmic patterns of trigonometric identities. The fact that things like Fourier series exist. Look at what it led to: http://en.wikipedia.org/wiki/List_of_Fourier_analysis_topics And look at this list of cycles The uses of trigonomety.


8

The answer is both yes, and no. $\Bbb C$ and $\Bbb R^2$ are both sets with the same cardinality, and they have a very natural bijection between them which preserves a lot of nice properties. So much that we can almost say that these two sets are the same for a lot of purposes. But these two carry very different structure as a natural structure. $\Bbb C$ is ...


7

You can define the set of complex numbers in different ways. One of those ways defined $\mathbb C$ to be $\mathbb R^2$ and then goes on to define the algebraic structure of the complex numbers. If that is the way you define the complex numbers, then it is certainly correct to write $\mathbb C = \mathbb R^2$ as sets.


7

If you're forgetting certain topics in math, then I suggest the best method is to do math problems daily or at least 5 days in a week. Usually when you learn math and stop doing it for over 2 weeks or more, you start to forget, lost your confidence, etc. (Personal experience). If you're looking for problems to do, I suggest: Buy a book from AOPS Visit ...


4

I can warmly recommend getting a Nintendo handheld device and a copy of one of the "Professor Layton" titles. A famous series of puzzle games and logic problems. The games are wrapped in an entertaining story too. Try a demo version first, if you don't believe me. :-)


4

It depends on what you mean by "equal". They are "equal" as sets, in the sense that both can be seen as a Cartesian product of $\mathbb R$ with itself. They are equal as vector spaces, where equality is interpreted as a linear isomorphism (both are real vector spaces of dimension 2) They are not equal when you consider $\mathbb C$ as a field, because when ...


4

It is quite amazing but such maps do exist! Call a map between two metric spaces (say, Riemannian manifolds) $f: M\to N$ a path-isometry if it preserves lengths of all paths. Recall that if $p: [a,b]\to M$ is a path (say, Lipschitz-continuous), then its length is defined as $$ \int_a^b |p'(t)|dt. $$ Example. $f: R\to R$, $f(x)=|x|$. This map is not a ...


3

If it's publishable material, you could could write an account of it and put it on the arxiv (google the word "arxiv"). You could also consider submitting it to be considered for publication in a refereed journal. As far as I know, nobody calls that "posting" (the word used in your question) even though some such journals exist only on the web and not in ...


3

Galois theory is the theory of the duality between profinite groups associated to fields and closed subgroups which arise as dual to field extensions of the original field. It's about the algebra of polynomials over a field and how that helps to understand other fields constructed algebraically from the original field, i.e. from roots of polynomials over a ...


3

The paragraph you refer to is about probably 50th and 60th, and I am not well aware of the book from that period. However, I would like to point out that starting from 1980 and till 1992 a series of math and physics books was published under the title "Библиотечка Кванта" (Kvant's library). Some of these books are translations of very insightful books, but ...


3

Generally what I think makes such series so good is that the format forces the authors to explain non-trivial and often non-elementary mathematics in accessible and inspiring way. The concentration of mathematically "clever" and "cool" both fascinates and challenges. They also often expose parts and perspectives of mathematics that are largely missing in ...


2

There was a time when the Mathematics and Physics departments weren't separate to begin with - one just studied "Natural Philosophy". During the 20th century, these disciplines separated and become quite specialized, obscuring the relations between them. The links between mathematics and physics are very broad as was championed by Atiyah, Witten, Verlinde, ...


2

No, it's not, not unless you play some semantics games to make it so. A complex number $c$ can be defined as $c = a + bi$, and indeed $\{a, b\} \in \mathbb{R}$. But what is $i$? That's the imaginary unit, $i = \sqrt{-1}$. And $i \not\in \mathbb{R}$. That's kinda the point of calling it "imaginary." So you do have a pair of real numbers, but one of those ...


2

This is a comment (since I cannot write comments yet): If $$\Bbb C=\Bbb R^2,$$ how can one write, for example, $i$? If you say $$i=(0, 1),$$ then how $$(0, 1)^2=-1.$$


2

That is a question for a native speaker, I fear. In German both are used to differentiate = differenzieren (determing the derivative) to derive = ableiten -> Ableitung (derivative) In English literature, I think I only saw differentiate for the operation. In German you can use "Herleitung" to stress more that it is about taking conclusions. In English ...


2

This isn't a complete answer by any means. A couple of weeks ago there was a conference based on the work of William Thurston. There were several references made to an idea he used (I believe) of having levels of understanding. When you first meet something, you can read the theorems, and get a first level of understanding. But as you come back to it, in ...


2

Want to build something with tools? You'll need trig to get the measurements right. Want to create a game? You'll need trig to understand the math. Want to work with electronics? You'll need trig to understand all sorts of things Want to do astronomy? You'll need physics which uses LOTS of trig. Want to pass high school and get into a college STEM course? ...


2

As indicated in Mochizuki's Report concerning activities devoted to the verification of IUTeich in December 2013, Go Yamashita is going to give a three-week lecture series starting in September 2014 in Kyushu University. All together it is planned that he spends 63hours, see here. Maybe his 200-300 pages detailed survey will be finished aswell at that time.


2

Claude Shannon's master's thesis, a seminal contribution to Boolean algebra and electrical engineering, used the notation of addition and multiplication for the two operations that we now think of as AND (multiplication) and OR (addition), applied to the elements 0 and 1. In this case, not only does multiplication distribute over addition, $x(y+z)=xy+xz$, ...


2

You statment needs a little additional specification : in second-order logic it is possible to define equality by Leibniz's law : $∀x∀y[x=y→∀P(Px↔Py)]$. In Second-order Logic we interpret the universally quantified predicate variables as meaning “for every property of objects (in the universe of discourse)”. The semantics for second-order ...


1

I agree with StrangerLoop's response. I learned what I know about Galois theory from Artin's Algebra, and it had a reasonable amount of depth while being understandable at the same time. Now that I've actually completed two semesters of undergrad abstract algebra, I've found Dummit and Foote to be a rather approachable text as well. I can see how certain ...


1

As mentioned in the comments, at $T\to\infty, f=\frac{FS}{N}$ for $N>S$ and for $N\leq S$ your code will start double pulsing periodically. However, that's the the limit. If you aren't looking at long term behavior, this is what you'll follow: Define $r\equiv N\pmod{S}$. (i.e. $r$ is the remainder after dividing $N$ by $S$.) The number generation will ...


1

Experience of others Many top notch mathematicians say that excellent teachers played a significant role in their success; the book Mathematicians: An outer view of the Inner world made that clear to me. Here is a small sample of some of the things that are recorded there: "Interaction with others has always been an important source of ideas for me." - ...


1

I have always been intrigued by the fact that $\sin$ and $\cos$ appear "automatically" if you put an imaginary number into the power series of the $e^x$.


1

To answer your question succinctly, a first course on linear algebra should cover the basic computational tools: row reduction, determinants, and eigenvalues. A more advanced course should force the students to come to terms with more abstract language (vector spaces over an arbitrary field), and it should contain a sophisticated treatment of the spectral ...


1

In my experience learning mathematics is a lot like learning a language. You need that basic vocabulary, but in order to really have a conversation you need a deep understanding of what all the words really mean and how they fit together and interact, and all the subtleties therein. Once we are proficient at a language we no longer worry about what each ...


1

I think what Poincare calls "certain order" can also be called the mathematical idea (behind the subject you are studying). To understand a mathematical idea, the following items are important: A mathematical idea is a dynamic creature and it continually evolves according to its applications. Take for example "continuity". It started as a notion for ...


1

I would like to add a name that nobody has mentioned: Alexandre Grothendieck. The name comes to my mind not only for what he achieved in his active career, which was itself pathbreaking and quite visionary, but rather for the Esquisse d’un Programme, in particular for how it is described by people that worked on it. For example, this is what Leila Schneps ...


1

With the discussion of what is actually meant by "ahead of their time" in mind, and even if he is not the "father of computing" as is often claimed, I still think that Alan Turing deserves a mention for the intuition that nonlinearities may give rise to complexities and unstable vacuum solutions, and for the intuition that such phenomena (through ...



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