Hot answers tagged

38

Here are some general pointers for gaining intuition in topology: Learn lots of examples early, and use them to guide your understanding. Take the definition of a topology, for instance. The original motivation for this definition comes from familiar topological spaces, such as the real numbers or, more generally, $\mathbb R^n$ or, more generally still, ...


32

The bad news is that mathematical notation can be regarded as an incredibly dense kind of compression: you take a very subtle and carefully evolved idea and choose some symbol to represent it. The "quick meaning" of a symbol will typically lose all the subtlety. There's a worse problem: typesetting is a pain, so it's pretty common for mathematicians to use ...


32

There's a few things I can think of which might fit the bill: We could work in a non-$\omega$ model of ZFC. In such a model, there are sets the model thinks are finite, but which are actually infinite; so there's a distinction between "internally infinite" and "externally infinite." (A similar thing goes on in non-standard analysis.) Although their ...


31

John Stillwell's Mathematics and Its History is a terrific book for precisely this purpose. See my Amazon review. I have also written some free history of mathematics course materials myself in a similar spirit.


29

The whole field of partial differential equations is an application (and origin of many problems) of functional analysis. Book: Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis. Lecture notes: Applied Functional Analysis by H. T. Banks. And a (big) bit of history: On the origin and early history of functional analysis ...


29

The concept you have defined is usually called "provable", not "true". This is indeed a quite central and important concept, but the point is that it is different from truth. For example, let $\psi$ be the formula $(\forall x)(\forall y)(x=y)$. Then $\psi$ is not provable in your system, and $\neg\psi$ is not provable in your system either (these two facts ...


27

These are all symbols that are usually learned in high school or introductory math in college. There is a wiki page that gives names and purpose for most symbols: https://en.wikipedia.org/wiki/List_of_mathematical_symbols Wikipedia may not be a reliable source for some topics, but with math it is usually very good for a run down in an unknown or new topic.


27

There are various ways to interpret the question. One interesting class of examples consists of "speed up" theorems. These generally involve two formal systems, $T_1$ and $T_2$, and family of statements which are provable in both $T_1$ and $T_2$, but for which the shortest formal proofs in $T_1$ are much longer than the shortest formal proofs in $T_2$. One ...


25

Carl Linderholm's Mathematics Made Difficult is quite interesting. It was described by Halmos (Linderholm's PhD adviser) as a sort of "mathematical in-joke." But you'll find reviews that, while unanimously positive, are all over the map. I think that officially makes it a work of art, since the meaning of the content is truly in the eye of the beholder ...


25

Well, there are a few notions of "infinite" sets that aren't equivalent in $\mathsf{ZF}.$ One sort is called Dedekind-infinite ("D-infinite", for short) which is a set with a countably infinite subset, or equivalently, a set which has a proper subset of the same cardinality. So, a set is D-finite if and only if the Pigeonhole Principle holds on that set. The ...


25

Don't read anything too advanced, i.e. you should be able to understand everything, so you don't waste your time. How to Prove It: A Structured Approach - Velleman Numbers and Geometry - Stillwell Calculus - Spivak Linear Algebra Done Right - Axler


24

I don't know where you live, but the libraries near me are part of the interlibrary loan system. That means that if they don't have a book I want, they can borrow it from a library that does. Check to see if your local library is a part of this system or a similar program. You can also find a lot of books freely & legally distributed online in a large ...


24

Starting from von Neumann and his contribution to economic theory (1937, existence of an optimal equilibrium in the model of economic growth ) The von Neumann model and the early models of general equilibrium There are lots of applications of functional analysis in Economic theory: Functional Analysis and Economic Theory In Financial Mathematics, in the ...


24

I think we can reasonably state that an ellipse is both $1)$ existent and $2)$ not a straight line. The reality is that, if we are $100\%$ rigorous in everything we say, any textbook would be thousands of pages long. Each statement would have to be proven from axioms, and nobody would want that. There are certain reasonable things we can and must assume. I ...


23

Here is a possible explanation. Let $\alpha \in [0, \pi/2]$ and define $\epsilon_1, \epsilon_2, \cdots$ by $ \epsilon_i = \operatorname{sgn}( \cos ( 2^i \alpha )) \in \{-1, 1\}$. Here, we take the convention that $\operatorname{sgn}(0) =1 $. Then applying the identity $2\cos\theta = \operatorname{sgn}(\cos\theta) \sqrt{2 + 2\cos(2\theta)}$ repeatedly, we ...


22

I don't know if there are any particular topics you're interested in, or at what level of detail you want them presented, but Knuth's Surreal Numbers is an example of the sort of book you mean: B. You mean we're actually using the New Math to decipher this old stone tablet? A. I hate to admit it, but that's what it looks like. Here the first ...


20

It has no practical applications in mathematics or any other field as far as I know. It is expected that there are an infinite number of Mersenne primes, so discovering a new one is completely in line with that. On the other hand, it would be much more interesting and perhaps somehow consequential if we failed to find a new one where one ought to be ...


19

The expression "vector space" is usually reserved to the case when the coefficients form a field. When the coefficients only form a ring (such as a skew field), the corresponding structure is called a module.


19

Another real world (theoretical physics) application is the Lagrange formalism of classical and modern mechanics which relies on the Euler-Lagrange Equation - which as you properly know is a fundamental result of functional analysis. A book on the topic: Lagrangian and Hamiltonian Mechanics Particularly I find that one of the real exciting parts of this ...


18

Perhaps A Book of Abstract Algebra by Charles Pinter? It's very cheap, and if I remember correctly, provides motivation for the theorems/corollaries/etc. It's a small book, and there's a lot of stuff it doesn't cover, but I think it would prepare the reader well for more advanced treatments of algebra.


18

Try these books by Morris Kline: Mathematical Thought from Ancient to Modern Times Mathematics: A Cultural Approach


17

There are infinitely many differentiable functions $F\colon [0,\infty)\to\mathbb{R}$ that satisfy $$ F'(x)=F(2x)\qquad\text{and}\qquad F(0)=1. $$ We will follow the argument outlined by @HenningMakholm in the comments. First, consider the substitution $G(x) = F\left(\dfrac{2^x}{\log 2}\right)$. Plugging this into the above equation gives $$ G\,'(x) = 2^x ...


17

Here are some rather unconventional books with focus on visual perception and guarantee for many interesting and amusing hours. Proofs without words - Exercises in visual thinking: The title is program. Volume I and II by Roger B. Nelson follow the motto: A picture is worth a thousands words and present graphical solutions without words to ...


16

We have the identity $$\sum_{n\geq0}\frac{r_{2}\left(n\right)}{\sqrt{n+a}}e^{-2\pi\sqrt{\left(n+a\right)b}}=\frac{1}{\sqrt{\pi}}\sum_{n\geq0}r_{2}\left(n\right)\int_{0}^{\infty}e^{-\left(n+a\right)x-\pi^{2}b/x}\frac{dx}{\sqrt{x}}. \tag{1}$$ In fact ...


16

The Number Devil: A Mathematical Adventure Is very good; It explains complex concepts in ways simple enough that young children can understand them, yet complex enough to maintain an adults attention with the revelations and connections it provides.


16

I'd recommend (elementary) number theory - save linear algebra for college. Dover offers many inexpensive titles; you could buy several and read about the same topics from different points of view. I particularly like Friedberg's offbeat Adventurers Guide to Number Theory. If you visit that book's page http://store.doverpublications.com/0486281337.html ...


15

This was first meant as a comment but got rather long, so I am posting it as an answer. Trying to learn abstract algebra without linear algebra is sort of like trying to dig a hole without a shovel. It's possible, sure, but you'll have plenty of moments when you'll go "man, I wish I had a shovel". Even worse, when you ask other people to show you how to ...


15

"A first course in Abstract Algebra" by Fraleigh is a great introduction to Algebra and it is packed with examples. My experience with this book is I learned Algebra on my own out of the first half of this book and took a course out of the second half. It is also available for free online if cost is a factor.


14

Sub $x=\tanh{u}$, $dx = \operatorname{sech^2}{u} \, du$. Then the integral is $$\int_{-\infty}^{\infty} du \, \frac{\operatorname{sech^2}{u}}{\pi^2+4 u^2} $$ Now, use Parseval. The Fourier transforms of the pieces of the integrand are $$\int_{-\infty}^{\infty} du \, \frac{e^{i u k}}{\pi^2+4 u^2} = \frac14 \frac{\pi}{\pi/2} e^{-\pi |k|/2} $$ ...


14

Approach 1: For the first integral \begin{align} 2\int^1_{0}\frac{{\rm d}x}{\pi^2+\ln^2\left(\frac{1-x}{1+x}\right)} &=-\frac{4}{\pi}\mathrm{Im}\int^1_0\frac{{\rm d}x}{(\ln{x}+\pi i)(1+x)^2}\tag1\\ &=-\frac{4}{\pi}\mathrm{Im}\int^1_{-1}\frac{{\rm d}x}{\ln{x}(1-x)^2}\tag2\\ ...



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