# Tag Info

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In differential topology we discuss tangent spaces to smooth manifolds- it turns out that these tangent spaces are vector spaces essentially "sitting" on top of manifolds. Their dimension matches the dimension of the manifold. A lot of properties crucial to classifying manifolds boil down to the action of linear maps which transform the tangent spaces of ...

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More: numerical analysis, cryptography. Less: set theory, general topology. FORTRAN keeps being used for number crunching with C/C++ growing in popularity. Mathematica/Maple/Matlab are used almost everywhere and you will see Python, Perl, Ruby, Haskell, LISP...

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Write $z=x+\mathrm{i}y$. Then $$\text{Im } z < 2|z|\Rightarrow y<2\sqrt{x^2+y^2}\ ,$$ which (in the plane $(x,y)$) has the solution: $y<0$ $\forall x$ or $$y>0\ \wedge y^2<4(x^2+y^2)\Rightarrow 4x^2+3y^2>0\ ,$$ which is verified by any point $(x,y)$ in the upper half-plane. So, in summary every complex number $z\neq 0$ is such that ...

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Ed Nelson had demonstrated in his book Radically Elementary Probability Theory that a good part of probability can be made clear to freshmen without appealing to measure theory by using the Mises frequency approach. There are other gains for various courses. But above all nonstandard analysis allows students to understand better the triumphs and tragedies ...

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"Software engineering" is an enormously broad term. Let's assume you end up writing software in industry, somewhere (like me). The kinds of mathematics that are useful will depend very much on the applications/functionality of the software. Some examples: Graphics/games: People will tell you that linear algebra is used in these fields. It is, but it's ...

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If you decide to take up any classes in regards to numerical analysis for differential equations, you'll see that in both finite difference methods and finite element methods, things seem to always come down to solving a massive system of linear equations. Obviously diff eq's have their place all over engineering.

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A simple way to see this is by considering, for $z\ne a$, $$\varphi(z)=\frac{f(z)-f(a)}{z-a}-f'(a)$$ and noting that $$\lim_{z\to a}\varphi(z)=0$$ because $f$ is differentiable at $a$. Then write $$f(z)=f(a)+f'(a)(z-a)+\varphi(z)(z-a)$$ Now take the limit: $$\lim_{z\to a}f(z)= \lim_{z\to a}(f(a)+f'(a)(z-a)+\varphi(z)(z-a))=f(a)$$ Another way (but ...

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A limit when $z\to a$ should always be taken through values outside $a$. True that many times we forget to write $0<|z-a|<\delta$ instead of $|z-a|<\delta$.

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I am totally not into this topic, but once I came across the book "Topology for Computing" written by Afra Zomorodian. He uses morse theory, homotopy theory, group theory, topology and much more complex stuff in graphics and surface analysis. The good thing about his book is that mathematics in it is kept with perfect rigour. You should take a look at it.

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I'm highly interested about 3D modelling in software, and I know that it has some deep mathematics behind it too. You give no indication of what aspect of 3D modelling or what application domain you are interested in. The field is vast. Lots of software is offered, even more for engineering. On the other hand, what specific topics in these areas ...

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For the Axiom of comprehension the source is traditional logic. See Port Royal Logic: [for] Port-Royal [...] the significance of general ideas has two aspects: the comprehension [la comprehension] and the extension [l'étendue]. The comprehension consists in the set of attributes essential to the idea. For example, the comprehension of the idea ...

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One useful concept is the Leibnizian distinction between assignable and inassignable number. In Robinson's framework this is implemented in terms of a distinction between a standard and a nonstandard number. Thus, ordinary real numbers are standard, whereas infinitesimals and infinite numbers are nonstandard. The sum $\pi+\epsilon$ where $\epsilon$ is ...

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For ones who read German, I strongly recommend Harro Heuser's 'Lehrbuch der Analysis Teil I'. There is also 'Teil II'. I tried couple of other German text books, but gave up continuing due to many errors or lack of completeness, etc. Then a person recommended me this book. This book is self-contained and proofs are quite error-free as well as well-written ...

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Every one of the ones in that list which only take values $1,-1,0$ has the property that every range $[a,b]$ is isomorphic to some range in $\mathbb N^k$, the product of $k$ copies of the poset $\mathbb N$. We can show that the Möbius function for $P_1\times P_2$ is the product: $$\mu_{P_1\times P_2}([(a_1,a_2),(b_1,b_2)]) = ... 0 While I agree with the importance of Linear Algebra, calculus/analysis, and probability, in order to make really groundbreaking improvements Virtual Reality at this stage, depending on your interests, there is a wide variety of subjects you may want to look into. For example, augmented reality is becoming a very exciting field related to virtual reality. ... 1 I am currently in computer graphics course at my uni. The mathematics we need is mostly linear algebra on a rather computational level. You need linear algebra to describe e.g. points, vectors, transformations (matrices) in three dimensions. Further, some knowledge in calculus/analysis is helpful. For example, we learnt about graphic filters (e.g. gaussian ... 2 If you want to program then Logic. Anyway Linear Algebra will be useful (but just vectors, rotations of vectors and simple matrix transformations rules). The term Quaternion that appears in Unity3D hints just to that. 0 In algebraic geometry, there is a notion of a presheaf of abelian groups on a topological space X. To every open set U of X, there is associated an abelian group \mathcal F(U) such that a bunch of properties hold. But of course one is interested also in sheaves of rings, and sheaves of algebras over a ring. Since so many of the results can be ... 1 Not sure this is the answer you expect, but here is an important application of Higman's lemma in formal language theory. Let A be a finite alphabet and let A^* be the free monoid on A. The shuffle product of two languages U and V over A is the language \begin{multline*} U \operatorname{ш} V = \{ w \in A^* \mid w = ... 1 This story is told in the Encyclopedia Britannica about Sophus Lie, who invented, or studied, Lie groups. 3 It's an apocryphal story about the Russian mathematical physicist and Nobel Laureate Igor Tamm. In the story, he was said to stopped by highwaymen and was released upon answering a quiz question about Maclaurin Series. The story is recounted here, though the quoted source is down. At this site we see the same story, but this time he is suspected of being a ... 1 When in doubt, use parentheses. I would codify your observations thus: Parentheses may be omitted only under these circumstances: When the function symbol provides its own bracketing. For example, you can write \sqrt{1 + x}. Of course when you don't have TeX like we do here, you may have to write something like sqrt(1 + x), even though in this ... 2 In general, reducing complexity of notation is always good. That is, unless it introduces ambiguity into your statement. As such, I would advise omitting parentheses, unless complexity of the expression dictates a necessity. For instance, many authors use  Tx to denote  T(x), where T is usually a linear transformation. But sometimes you might have a ... 0 If we look at the particle level, heat trasnfer is kind of brownian motion. See how heat transfer. It is not a suprise that both equations look like the same. 0 My first recommendation (although not a MOOC) has got to be Khan Academy's Linear Algebra section. Note that according to the site, this is an advanced course normally taken by science or engineering majors after taking at least two semesters of calculus (although calculus really isn't a prereq). Alternatively, if I could recommend a book to you, Linear ... 1 Looking at your interests I think you might really like to do your thesis in Fourier Series. You can start by reading this incredibly fabulous treatment of the subject: http://www.amazon.com/gp/product/0486633179 It involves analysis at a beautiful but classical level and it will really round out your skills and give you some deep perspective. 2 Depends what are your aims. Indeed I think that the best thing you could study as freshman in physics is basic group theory. In Italy we do that in first year of Mathematics taking a two semester course called "Algebra", while often is not in a first year of Physics. Anyway modern Physics relies deeply on Algebra and so knowing a deeper notion of groups, ... 0 To answer your question, category theory is not ambiguous. It is a viable alternative to set theory and an alternative foundation for all of mathematics. It was first proposed over half a century ago and is a generally accepted framework (unlike other theories such as fuzzy sets for example, which still seem to face an uphill battle). It has been very ... 4 I'm sure there will be a better answer, but here are my thoughts. It happens that books on mathematics talk a lot and miss some rigorous treatment. I see two reasons for that. Firstly, usually careful rigorous explanation is useless for the purposes of the theory; neither does it provide insight, nor makes things or intentions clear. Secondly, the reader is ... 1 As one who started in physics major and ended up being in math, my experience is that math is much more difficult than physics at undergraduate level. That said, basic calculus and linear algebra education should be the same in either division. At this level, any textbook would pretty much serve well enough. As long as you get decent grades, you in a good ... 1 About building a quaternion extension of Q (or more generally of any number field) : the problem can be solved completely and explicitly starting from a biquadratic field, using techniques of embedding problem. Actually, given a field K containing a primitive p-th root of unity (where p is any prime), the more general problem of embedding a Galois extension ... 1 The (nonstandard) notation is probably what's confusing you. It would be better, and more customary, to write$$ \sigma\colon \mathcal{P}(V)\to S(V)\colon S\mapsto span(S), $$not S\to span(S). To write it another way, \sigma\colon \mathcal{P}(V)\to S(V) is defined by \sigma(S) = span(S). The statements a) and b), together with the fact that ... 1 L^2 function spaces arose out of Parseval's identity for the Fourier series, an identity that was known by the late 1700's:$$ \frac{1}{\pi}\int_{-\pi}^{\pi}|f(t)|^2dt = \frac{1}{2}a_0^2+\sum_{n=1}^{\infty}a_n^2+b_n^2, $$where the Fourier series for f is$$ f(x) \sim \frac{a_0}{2}+\sum_{n=1}^{\infty}a_n\cos(nx)+b_n\sin(nx). $$... 0 The characteristic polynomial of g over \mathbb F_2 is p(t) = t^3 + t + 1 which is irreducible. The splitting field of this polynomial has 2^3 = 8 elements, so any nonzero element of it satisfies r^7 = 1. The eigenvalues of g^7 are all 1, and g^7 must be diagonalizable over the splitting field since g is, so we must have g^7 = I. Now ... 0 I believe that you are referring to to the celebrated Picard-Lindelöf theorem which is an existence and uniqueness theorem for ordinary differential equations of the form y'(t) = f(t,y(t)). Here f should be continuous and uniformly Lipschitz continuous in the spatial variable(s) in a neighborhood of the initial condition. This second condition is ... 1 I have a silly one, which is mine and even if it's not really correct, it's quite cute! Be: \pi the famous constant we all know \phi the golden ratio \gamma = Euler-Mascheroni constant Thence$$e \approx \frac{\pi + \phi + \gamma}{\pi\phi\gamma -1}$$Anyway, as I said it's not really correct because$$\frac{\pi + \phi + \gamma}{\pi\phi\gamma -1} ...

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I would add to the list Dobiński's formula for the $n^{\rm th}$ Bell's number (the number of partitions of an $n$-element set) which is given by $$B_n = {1 \over e}\sum_{k=0}^\infty \frac{k^n}{k!}.$$ When I first saw this formula I was amazed by an appearance of $e$ in a formula for a very concrete natural number.

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What is the radius of convergence of the power series $$f(x)=\sum_{n=0}^\infty \frac{n! x^n}{n^n}?$$ Answer. Exactly $\mathrm{e}$.

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In section 1.3 of Mathematical Constants by S.R. Finch we find this connection to prime number theory \begin{align*} \lim_{n\rightarrow \infty}\left(\prod_{{p\leq n}\atop{p \text{ prime}}}p\right)^{\frac{1}{n}}=e \end{align*} and also some Wallis-like infinite products \begin{align*} e=&\frac{2}{1}\cdot\left(\frac{4}{3}\right)^{\frac{1}{2}}\cdot ...

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$e$ appears in the basic Stirling's approximation for the factorial. $$n!\approx\left(\frac ne\right)^n$$ hence $$e\approx\frac n{\sqrt[n]{n!}}.$$

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Throw $N$ balls into $N$ bins at random. The probability that any given bin is empty is $e^{-1}$, and thus with high probability the fraction of empty bins is $e^{-1}$.

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Using finite differences involves a considerable complication of the procedures of the calculus. Such an approach may be constructively more satisfactory but may not be appropriate for a first acquaintance with the subject. The technical complications of the epsilon-delta approach are involved enough, and it is how certain things are defined (and it is way ...

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For algorithms, the best book I have read so far is - Cormen_Leiserson_Rivest_Stein-Introduction_to_Algorithms Try it. It's easy to understand, and you get all the basics.

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Yes, it is to the students' advantage to learn about infinitesimals. But Keisler is not the king's road. Nelson's approach in his Radically Elementary Probability Theory is much better.

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In response to your question about calculus for physicists, I would suggest you study the book by Keisler entitled Elementary Calculus and freely available online here. The book uses an intuitive approach exploiting infinitesimals and is particularly well-suited to a future physicist who is less concerned with epsilon-delta techniques that many math majors ...

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Go back and review and upgrade your Calc 1 and 2. These course are foundational. You are building on a house of cards if you lack comfort here. Get a Schaum's Outline or some programmed workbook. You can't express supply and demand functions without them. Linear Algebra (lots of applications in econometrics) Operations Research (very business oriented ...

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I see nothing "magical". A topological space is a set with certain additional properties. The category of topological spaces respects all properties of the category of sets.

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The arithmetic mean of the first $N$ positive integers is about $N/2$. This is easy to justify without any computation. Less obviously, the geometric mean of the first $N$ positive integers is about $N/e$.

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This is likely a picture of the lorenz attractor:

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Ultimately, you should hopefully be able to prove the vast majority of the mathematics you know. You do not necessarily need to prove things as you learn them, nor is it necessary or of benefit to do so. Oftentimes you will find that the proof for what you are learning is of a greater level and complexity than you can handle. But as you learn new ...

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