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1
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1answer
43 views

An introduction for integral tricks.

I wonder if there's a good book or internet page introducing integral tricks? For example integration by parts, and Feynman's trick. I'm not looking for an exercise book such as "Problems in ...
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4answers
331 views

Examples of “Non-Logical Theorems” Proven by Logic

I am still an undergraduate student, and so perhaps I just haven't seen enough of the mathematical world. Edit: In my personal opinion as well as a few mathematicians/logician to which I've talk to, ...
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3answers
100 views

What is subtraction?

Let $a, b \in \mathbf{R}$. It is an elementary fact that addition is a commutative binary operation on the reals, that is, $a + b \in \mathbf{R}$ and $a + b = b + a$. With the exception of ...
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1answer
33 views

Fibonacci Numbers in Nature

Supposedly the Fibonacci sequence appears naturally in nature, and my question is how, where and I guess why? I read that one way this is so is that it models the population of honey bees under ideal ...
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2answers
39 views

“Unclosure” on a set with binary operation

I was wondering if there is any usefulness to having a set that has no closure under a particular operation. For example, the set of prime numbers, $\mathbb{P}$ along with multiplication of integers ...
4
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0answers
42 views

Where to post a Calculus review guide?

I created a PDF document (using LaTeX) in which I wrote relevant review materials and Calculus problems for Calculus 1, 2, and 3. Is there an appropriate forum where I could try to post this to ...
2
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0answers
27 views

Asymmetry in definition of regular measure

In a Borel measure space $(X, \mathcal{B}, \mu)$, $\mu$ is outer regular at $E$ if \begin{equation} \mu(E) = \inf_{U \textrm{ open}} \{\mu(U): U \supseteq E\} \end{equation} and ...
0
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1answer
52 views

Have any one studied this binomial like coefficients before?

Note that the simillarities of following identities. $\dbinom{n}{r}=\dbinom{n}{n-r}$ $\dfrac{n}{n-r}\dbinom{n-r}{r}=\dfrac{n}{r}\dbinom{n-r-1}{r-1}$ $\dbinom{n}{r-1}+\dbinom{n}{r}=\dbinom{n+1}{r}$ ...
3
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0answers
56 views

Why not defining a measure as a function on functions?

A measure $\mu$ is a function to $\left[0,\infty\right]$ on the sets belonging to a $\sigma$-algebra. Then for integrable functions $f$ the integral $\int fd\mu$ comes in, having nice properties ...
4
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2answers
212 views

Is there any similar math limerick?

I found this one $$\frac{(12+144+20)+\left(3 \cdot \sqrt{4}\right)}{7}+(5 \cdot 11)=9^2+0.$$ Which is : ...
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1answer
40 views

Alternative function definitions

If you go to the wikipedia page on the sine function or the log function you'll find a number of different definitions of these functions. I know that what defines a function are it's values, for ...
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2answers
53 views

Books that integrate physical reasoning with mathematical reasoning? mathematicians?

As the title says, can anyone help me to find any book that shows how physical reasoning using concepts from classical/quantum mechanics and physics in general can enlighten us about mathematical ...
2
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1answer
131 views

Why do some universities offer mathematical logic in different departments?

I'm thinking of pursuing mathematical logic after my undergraduate work and I have noticed that some universities offer mathematical logic in their philosophy department while others offer it in their ...
1
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1answer
23 views

Is it possible that a randomized recursion has a nonzero probability of either converging or diverging?

I have very little "hands-on" experience with probability, but here is my context: I was looking at the random Fibonacci sequence: $$f_0=f_1=1, f_n=f_{n-1}+Xf_{n-2}$$ where $X$ is chosen randomly ...
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0answers
51 views

Connection between differential form of a manifold and Sheaf of relative differential of a map of schemes.

I was wondering whether there is a connection between the differential form of a manifold and Sheaf of Relative differential of a Scheme map. Definitions: Differential form on a manifold M is a map ...
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0answers
31 views

Formal Trigonometric Refrence

I'm Using a textbook for mathematic which is produced to learn for normal students. Here I'm giving the link of chapter of trigonometric functions of my textbook : ...
6
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1answer
104 views

Why is adding Cohen reals so “uninteresting”?

I read the following in this paper (Otmar Spinas, Proper products. Proceedings of the AMS, 137 (8), (2009), 2767–2772): $ \mathbb{L}^2 $ adds a Cohen real. Thus it was considered uninteresting ...
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1answer
82 views

Alternative set therories?

Is there a version of set theory that allows the existence of a set that does not admit the empty set as a member? I.e., reject the axiom $A\cup \emptyset = A$
1
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1answer
39 views

Software to easily draw 3d plots from functions

my problem is that I need a way to quickly check results of my, that is to say, homework. I think that the best way to do this is to draw a plot of a function to quickly see whether my solution is ...
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0answers
72 views

Why are Lie Groups so “rigid”?

This is probably a naive question, but here goes. To motivate my question, I'll consider a unit circle in $\mathbb C$ or $\mathbb R^2$. This is a compact Lie group equipped with the usual exponential ...
6
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1answer
475 views

Is calculus not rigorous?

While studying single and multivariable calculus during my first year some people complained that calculus wasn't rigorous enough, when I asked about this no one seemed to be able to really specify ...
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0answers
28 views

Uniform distribution on convex hull

Let $X=\{ x_1,\dots,x_n \} \subset \mathbb{R}^m$. Let $H(X)$ be the convex hull of $X$. Assume that $X$ is a convexly independent set, i.e. none of the $x_i$ are a convex combination of the others. ...
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2answers
45 views

Which letters to use as index in sequences?

Usually the latin letters $i,j,k,l,m,n$ are used as indexes in sequences or sets with $k$ elements ($A = \{ a_1,...,a_k \} $). But when we already used all these letters is there any recommendation? ...
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0answers
28 views

general local to global principle

Consider the Diophantine equation $f(x)=0$, where x is a vector of integers and $f: \mathbb Z^n \rightarrow \mathbb Z$ is a polynomial function. Is the following statement true? The structure of the ...
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32answers
6k views

What are some 'conceptualizations' that work in mathematics but are not strictly true?

I am having an argument with someone who thinks that it's never justified to teach something that is not strictly correct. I disagree: often, the pedagogically most efficient way to make progress is ...
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0answers
30 views

International scholarship for maths [closed]

Do you mathematicians know if there is a scholarship (or something similar) for maths awarded at international level regardless of the country and/or university from which you come?
18
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1answer
1k views

Mathematical Intuition Behind Schizophrenic Numbers?

Schizophrenic numbers (A014824) are numbers whose square roots "look" like rational numbers. They were first discussed in 2004 by Darling in the Universal Book of Mathematics (page 282), and I ...
2
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1answer
72 views

How to see 4th dimension? [closed]

Anyone has tricks or methods? Great help, thank you. P.S.: Abstractly of course, not actually seeing 4D, which I don't think is possible.
7
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1answer
87 views

Applications of Geometry to Computer Graphics

How is differential geometry (or any type of theoretical math) related to computer graphics and/or computer programming? A friend of a friend of mine has only a bachelors degree in pure math and got ...
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0answers
49 views

A city wants to encourage downtown

could you please help me with this ( part d ) A city wants to encourage downtown employees to use public transportation. Below is the time in minutes to get to work on one morning according to ...
6
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1answer
115 views

How do people on MSE find closed-form expressions for integrals, infinite products, etc?

I always wanted to ask this question since when I joined MSE, but because I was afraid of asking too many soft questions I never asked it. I've seen some pretty complicated integrals and infinite ...
8
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1answer
162 views

How much math is there? [on hold]

Among other things I teach high school-level math, and one question that often comes up is: "How long would I have to study math in order to know all of it?" I usually tell them that it's like ...
5
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0answers
45 views

How to take limit *along a path*

So in multivariable calculus for a limit of a function to exist, the limits of the function along all possible paths must exist and equal the same value. But how does one calculate the limit along a ...
0
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0answers
49 views

Is it necessary to know all the details in proofs of theorems you study in PDE's?

I've been studying PDE from the book by L.Evans for some time now. I came across some statements in the proofs which I couldn't justify. But to complete the exercises I didn't need to know all these ...
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2answers
38 views

Infinitesimal Unit of Measurement

This is just a question that popped into my head which I lack the knowledge to answer (or even to know whether there is an answer, honestly). Does the idea of an infinitesimal unit of measurement even ...
1
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1answer
72 views

How do you self-study Functional Analysis?

It would be very handy to know about function spaces, distributions and Fourier stuff. It looks like Rudin's Functional Analysis covers these things, but I do not yet have the foundation for it. (see ...
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5answers
6k views

How hard should a mathematician work?

Of course I would never compare myself to, and don't expect to be, one of the great mathematicians. However, I am curious as to how my work habits, dedication and passion differs from theirs. How ...
0
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0answers
25 views

What are the patterns in the number of divisors $d(n)$ of the highly composite numbers?

I am trying to understand the patterns in the number of divisors $d(n)$ of the highly composite numbers. The numbers marked with an asterisk are the superior highly composite numbers. The first ...
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0answers
63 views

How long should you study Mathematics each day if you want to get into a graduate school? [closed]

I'm a university student (Junior now!) and I was wondering how many hours fellow undergraduates and graduate students study a day. I hear posts about how time doesn't really matter and that it's about ...
0
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0answers
29 views

Text on Witt vectors that are accessible to undergraduate students

I am looking for a thorough text on Witt vectors that is accessible to an undergraduate student that have completed the following courses: Calc 1, 2, Linear Algebra and Abstract Algebra. (In Norway, ...
0
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0answers
58 views

Self-Contained Books / Series / Lectures for Comprehensive Introduction to College-Level Math for Someone with VERY Poor Math Foundation?

I've long been interested in various math related subjects (technology, philosophy, sciences, computer science, languages, etc.) without really invested time to actually any learn any of them. I ...
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2answers
115 views

Who invented or used very first the double lined symbols $\mathbb{R},\mathbb{Q},\mathbb{N}$ etc

Who invented or used very first the double lined symbols $\mathbb{R},\mathbb{Q},\mathbb{N}$ etc. to represent the real number system, rational number system, natural number system respectively?
1
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0answers
32 views

best introductory intuitive books for learning ODE

I want to know best introductory intuitive books for learning ODE (mainly interested in Picard' theorem, Gronowall's inequality and most importantly stability). I started with Philip Hartman. Not ...
6
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2answers
118 views

The best balance in studying Mathematics?

I'm a student studying Mathematics at a university level. I've completed Single Variable Calculus, Differential Equations, Multivariable Caculus, Real/Complex Analysis, and Linear Algebra and I've ...
5
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0answers
80 views

Helping a reluctant 12 year old. [closed]

My 12 year old daughter needs to strengthen her math skills. My strategy up until a year or so ago has been pretty relaxed. I subscribe to the idea that the best motivation for learning is the ...
0
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0answers
37 views

What is the meaning behind this mathematical rebus? [closed]

I believe I saw this 'inequality' in someone's profile description here on Math Stack Exchange. I think it expresses a message or has a meaning. What is it?
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0answers
65 views

Problem solving strategy?

This question has always bothered me, and I've never had a maths coach to guide me along the way and show me what to do next. Nevertheless, I did make top 50 nationally in my national mathematics ...
6
votes
1answer
106 views

Coming up with short “magical” proofs

I was reading the solution to this problem: Prove that $f(n) = 2n$ is the only non-constant solution to $2f (m^2 + n^2 ) = (f (m))^2 + (f (n))^2 .$ The solution used these identities, pulled out of ...
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2answers
239 views

Is it normal that a pure math student doesn't know vector analysis?

Today I was watching a series of online video lectures about electromagnetism. At some point of the lecture, the professor used this vector calculus identity: $$ ...
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0answers
115 views

“Deep” maths books in certain subjects [closed]

I would like a suggestion on the 'deepest' books in Calculus and analysis (something along the lines of Rudin's) Linear algebra Abstract algebra Geometry (and topology); (even something along the ...