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2
votes
3answers
750 views

notation of distribution

I have a question: Does $$N(0, x)$$ mean a normal distribution with mean $0$ and variance $x$? Or standard deviation $x$? The notation seems ambiguous sometimes.
1
vote
2answers
978 views

Struggling to Bridge the Gap (to Rudin's Principles of Mathematical Analysis)

After taking an introductions to proofs course and abstract algebra, I have been trying to study from Rudin's Principles of Mathematical Analysis. Unfortunately, I still find it very very difficult to ...
10
votes
1answer
616 views

Undergrad Student Trying to Figure Out What to Study

this is my first time on stack exchange and I am seeking advice for my future studies. Some background first; I am a undergraduate student pursuing a degree in mathematics and I hope to pursue ...
4
votes
1answer
623 views

What Are R-Modules Used For?

Kind of a simple question, but what exactly are R-modules used for? Do they have any engineering applications? EDIT: If it helps, I'll give some more context to the question... I am a graduate ...
2
votes
2answers
725 views

Recommendation for a precalculus textbook

I'm a high school senior interested in pursuing a major in quantitative economics, which I understand is heavily math-intensive. However, as it stands, my academic strengths are more verbal (780 on ...
9
votes
4answers
2k views

What do I need to know to understand the Riemann hypothesis

Which kinds of fields of mathematics do I have to know about in order to understand the Riemann hypothesis millenium prize problem?
0
votes
1answer
240 views

Combination Conceptual Understanding

I'm currently studying some combinatorics and I'm trying to understand combinations ${{n}\choose{r}}$ conceptually. I don't have trouble understanding permutations (n!) and r-permutations P(n,r) ...
7
votes
4answers
50k views

How to figure out the log of a number without a calculator?

I have seen people look at log (several digit number) and rattle off the first couple of digits. I can get the value for small values (aka the popular or easy to know roots), but is there a formula. ...
5
votes
6answers
2k views

A book for self-study of matrix decompositions

I am a third year math student and I noticed that there are many uses for decomposing a matrix (I mean decompositions like SVD, LU etc'). Is there a good book for self-study of the subject ? Note ...
5
votes
3answers
205 views

Dealing with many entities that need a symbol

What does one do when one needs a lot of symbols and one has exhausted the useful symbols of the latin and greek alphabets? (I say useful symbols because letters like iota (ι) and upsilon (υ) seem too ...
1
vote
6answers
194 views

Taking the derivative of $y = \dfrac{x}{2} + \dfrac {1}{4} \sin(2x)$

Again a simple problem that I can't seem to get the derivative of I have $\frac{x}{2} + \frac{1}{4}\sin(2x)$ I am getting $\frac{x^2}{4} + \frac{4\sin(2x)}{16}$ This is all very wrong, and I do not ...
6
votes
3answers
358 views

Characterizations of primes

Let $\mathbb{P}$ be the primes set. We know from Wilson's Theorem that $$(p-1)!\equiv-1 \pmod p \iff p \in \mathbb{P}$$ What another formulas we have with an if and only if ($\iff$) statement to ...
0
votes
1answer
222 views

Name of probability distribution

Does this distribution have a name: $f(x) = yx^{y-1}$ for $0 < x<1$ and $y>0$? It looks like an exponential distribution. Or is it a nameless distribution?
18
votes
16answers
4k views

Explaining Horizontal Shifting and Scaling

I always find myself wanting for a clear explanation (to a college algebra student) for the fact that horizontal transformations of graphs work in the opposite way that one might expect. For example, ...
11
votes
2answers
286 views

Is there a way to tell whether a paper has been rigorously peer-reviewed and is completely valid?

When is it safe to assume that important theorems in a paper you're referencing are valid? I have a paper in a field in which I am not an expert, so could only check their proofs after months or ...
1
vote
1answer
233 views

Reference: Compendium of interesting graphs

I've been writing a little about some results on graph theory, and I want some nice examples of applying the results to some interesting finite connected graphs to show how the results might be ...
12
votes
2answers
348 views

When does “pairwise” strengthen and when does it weaken?

"Pairwise disjoint" is stronger than "disjoint"; it sometimes happens that $\displaystyle\bigcap\limits_{i\in I} A_i=\varnothing$ but for every $i,j$, or at least for some, one has $A_i \cap ...
38
votes
8answers
4k views

How important is programming for mathematicians?

I'm a freshman student in mathematics, and I'm considering whether or not to take a programming course. How important is programming for mathematicians? Do working mathematicians use programs to aid ...
18
votes
4answers
1k views

Why is the axiom of choice separated from the other axioms?

I don't know much about set theory or foundational mathematics, this question arose just out of curiosity. As far as I know, the widely accepted axioms of set theory is the Zermelo-Fraenkel axioms ...
7
votes
2answers
610 views

What are Free Objects?

I've read the wikipedia article, but couldn't grasp the concept. Is there an informal definition? Are there examples of uses of free objects in calculus? Are free objects somehow connected to ...
10
votes
3answers
2k views

Is Foundational Research a Dead Field?

I'm a second year mathematics major at a pretty good school. Ever since I became a math major I have been most interested in set theory and logic, which I guess can be lumped into the category of ...
8
votes
4answers
226 views

Encyclopedia of Groups

I was wondering whether or not there was an online encyclopedia of groups--finite or infinite. If there isn't, would you suppose that such a thing would be useful?
2
votes
2answers
721 views

What is an effective way to teach children the Cartesian coordinates?

My nephew is preparing for a $4$-th grade state test. They need to learn topics like reflection about $x$ or $y$-axis of a point( say $(3,5)$ reflected about the $y$-axis). I tried to explain but ...
7
votes
2answers
3k views

Some basic practical applications of Calculus

I am currently studying Calculus on my own for fun. I enjoy different components of math and how they can be used to solve so many problems. Many people, however, think I am crazy because I am ...
4
votes
1answer
119 views

An analogy between group actions and group represenations

I was trying to make a 'dictionary' between group action and group representation terms using the $\mathbb{C}[-]$ functor. I immediately found that if the set $Y \subset X$ is invariant under the ...
77
votes
19answers
12k views

How do you explain the concept of logarithm to a five year old?

Okay I understand that it cannot be explained to a 5 year old. But, how do you explain the logarithm to primary school students?
18
votes
4answers
706 views

A “clean” approach to integrals.

Many fields in mathematics start from the "dirty" approach. In calculus we do all sort of $\epsilon$-$\delta$ stuffs, until topology gives an elegant formulation using open sets. A first course in ...
4
votes
1answer
429 views

Learning approach of simultaneously enhance creation and imagination skills instead of 'follow' approach

I am a math-major bachelor student. And I want to get some advice about the approach I'm trying now for learning maths, not for efficiency, but for depth and fully-mastered. Firstly, I want to know ...
2
votes
1answer
126 views

How does one formulate a minimal set of axioms? [closed]

Suppose the study of some kind of mathematical object has evolved in an "organic" fashion, until it reaches the level of maturity that one wants to take an axiomatic approach to the study of those ...
14
votes
5answers
9k views

Which calculus text should I use for self-study?

I am 36 years old, and have forgotten a lot of math from high school, of which I only took up to Algebra 2. However I am teaching myself mathematics and am now, as an adult, completely fascinated ...
8
votes
2answers
3k views

How do I get good at Math?

How do I get good at math? I'm a freshman in college, and I've always done OK in math. I never had any good teachers in High school, and I always have done the bare minimum. Over the course of this ...
3
votes
2answers
201 views

Is $\infty$ enough or do I need to write $+\infty$

This is a question of notation. I have seen in many articles that people often denote $+\infty$ when talking about 'positive infinity' of the real numbers. Is that a convention, or it can be written ...
7
votes
2answers
241 views

terminology: what is meant if someone writes “calculus of ..”?

This question might be a little soft as it does not have a definite answer, so I hope I do not break the conventions of this forum by posting it here. I have now come across the term "calculus of ...
7
votes
0answers
155 views

Why are injective $\mathscr{O}$-modules flasque?

Let $X$ be a topological space, and let $\mathscr{O}$ be a sheaf of rings on $X$. It is easy to verify that the functor $\Gamma (U, -) : \textbf{Mod}(\mathscr{O}) \to \textbf{Ab}$ is representable, ...
7
votes
0answers
296 views

(Mathematical) Applications of Quantum Group

What are some mathematical applications of Quantum Groups? I have tried researching and found that it is used to find solutions to Yang-Baxter equation. Any other applications? Thanks a lot.
36
votes
4answers
3k views

A Young Math Enthusiast's Fear

Having been an avid lover of Mathematics, it is my dream to become a mathematician one day. I have been learning some "Advanced Mathematics" (Real Analysis and some Abstract Algebra mostly, and a ...
1
vote
1answer
197 views

Formal statement of theorem about perfect numbers?

I cannot seem to find the formal statement of the theorem if there are infinite perfect numbers in Wikipedia or online. I searched this site but the closest is the generalization of perfect numbers ...
8
votes
1answer
305 views

How can one visualize topological quotients or develop intuition for handling them?

This is a very open-ended question. I regret that -- I would like to be able to make it more precise, but I don't know how. I would appreciate comments on how to improve this question. I had my first ...
6
votes
1answer
500 views

How do I get into a masters course in pure mathematics?

My question is as stated in the title and to elaborate more: I would like to know if there are any standardized international exams to enter a masters course in pure mathematics (besides GRE Math) ...
6
votes
3answers
295 views

Connections/motivations of “Sums of Two Squares” to/from other fields of math.

I am to teach section 18 of "Elementary Number Theory" (Dudley) - Sums of Two Squares - to an undergraduate Number Theory class, and am having trouble cultivating anything other than a rote dissection ...
3
votes
1answer
235 views

Intuition on the definition of “rational maps”

I'm studying some representation theory on $S_n$ and $GL(V)$ and tensor spaces, and have come across a lot of material involving rational representations. I'm not really an algebraic geometer by ...
2
votes
3answers
559 views

How to fill up the gap between a typical advanced undergraduate algebraic curve course and High school basic geometry/precalculus course?

Based on this question i asked recently: A question about geometry of plane curve books, i think it is too advance for a HS student/ typical second or third year undergraduate math majors to read on ...
3
votes
1answer
156 views

A Formal and Precise treatment of Simplification?

I am looking to gain a deeper understanding of, and increase my own skill in "Mathematical Simplification". But I've been finding the concept overly vague and haven't been able to find any good ...
11
votes
2answers
344 views

Is there a geometric interpretation of $F_p,\ F_{p^n}$ and $\overline{F_p}?$

I have been doing some exercises about finite fields lately and I think I've obtained some understanding of what they are. What seems to be missing though is some kind of picture. Learning to work ...
4
votes
0answers
287 views

Divisibility notation history

I'm writing a paper project for school about divisibility, so I'd like to include a bit of history about that subject. I'm mostly interested in notation of $|$ sign used in past, but everything else ...
5
votes
4answers
6k views

What is the purpose of the standard deviation?

I don't have any knowledge of statistics beyond high school common sense. Why is the standard deviation usually seen in combinatorics textbooks, and why is the standard deviation defined ...
7
votes
2answers
556 views

Derivations in a ring. What applications do they have outside algebra?

INTRODUCTION Let $R$ be any ring, possibly without unity. We call a function $d:R\longrightarrow R$ a derivation on $R$ if it satisfies the following conditions. $(1)$ It is an endomorphism of the ...
13
votes
1answer
598 views

Does every closed curve contain the vertices of a square?

This is the question on Futility Closet Is there really no answer?
2
votes
2answers
193 views

How do you filter through published papers and find the ones you should read?

Here's a paper that answered a long standing problem: PRIMES is in P so it would go at the top of the list for someone working in that area. So, how do you find the most important results in some ...
6
votes
1answer
590 views

Why don't they teach Fundamental Theorem of Algebra in High School? [closed]

I am currently in AP Calculus BC and one more year to go, I have heard about Fundamental Theorem of Algebra several times, and with the resources that is out there today I tried to search and study ...