For questions whose answers can't be objectively evaluated as correct or incorrect, but which are still relevant to this site. Please be specific about what you are after.

learn more… | top users | synonyms

14
votes
10answers
2k views

Math problems that are impossible to solve [closed]

I recently read about the impossibility of trisecting an angle using compass and straight edge and its fascinating to see such a deceptively easy problem that is impossible to solve. I was wondering ...
2
votes
0answers
35 views

Is there a measure theoretic version of Stokes's theorem?

Is there a way to generalize Stokes's theorem on manifolds to general measure spaces? This idea came from trying to generalize the fundamental theorem of calculus to general function/infinite ...
3
votes
0answers
106 views

How to study multiple books per math subject?

S.E advisers, I am a college sophomore in US with double majors in mathematics and microbiology. I apologize for this sudden interruption but I wrote this email to seek your advice regarding to ...
6
votes
2answers
128 views

Mathematics that is not taught in college [closed]

Among the mathematics that is not usually taught in high school and college (for pure math majors), what are some: Important Interesting Elegant mathematical ideas, tools, ...
1
vote
2answers
36 views

Does Convexity play a role in topological equivalence?

Does convexity of two shapes play a role in them to be topologically equivalent? For example, a circle and a heart (as asked in another question I posted). They are homeomorphic, but the circle is ...
8
votes
1answer
60 views

Circle and heart homeomorphic?

Is a circle and heart homeomorphic to one another? Intuitively, I can picture that the one can be "morphed" into the other by bending and stretching and not breaking. But I am unsure if that is ...
7
votes
6answers
119 views

Intuitively understanding $\sum_{i=1}^ni={n+1\choose2}$

It's straightforward to show that $$\sum_{i=1}^ni=\frac{n(n+1)}{2}={n+1\choose2}$$ but intuitively, this is hard to grasp. Should I understand this to be coincidence? Why does the sum of the first ...
7
votes
1answer
102 views

Proving the Riemann Hypothesis and Impact on Cryptography

I was talking with a friend last night, and she raised the topic of the Clay Millennium Prize problems. I mentioned that my "favorite" problem is the Riemann Hypothesis; I explained what it posits ...
1
vote
0answers
12 views

Good resources for learning to recognize word problems in statistics?

I've got a number of books and resources for statistics theory, but I've always had problems with the approaches needed in answering questions, specifically for probability theory where counting ...
2
votes
1answer
40 views

The Monty Hall problem - convincing a skeptic

I have lost count of the number of times that I have been debating the solution of the Monty Hall problem with someone. Recently I had a long conversation with a colleague, who didn't seem to buy ...
0
votes
1answer
17 views

Algorithm for finding Complex Eigenvectors?

I'm wondering if there's a fairly easy algorithm by which one can, by hand, find eigenvectors corresponding to complex eigenvalues for small matrices. Of course, one can always row reduce, but it can ...
7
votes
3answers
75 views

Formal writing in math: equations

What is the formally correct way to solve a bunch of equations in math? Is it \begin{align} 42x = 4324 \\ x = 4324/42 \end{align} or \begin{align} 42x = 4324 \\ \Rightarrow x = 4324/42 \end{align} or ...
2
votes
0answers
44 views

Machine Learning and Probability/Stochastics

Main question: What connections are there between machine learning and stochastics (Probability theory, analysis, processes, SDEs)? Background: I've just been accepted into a master's programme for ...
1
vote
0answers
55 views

Do mathematicians visualise what 4D would look like when they are working with abstract 4D concepts?

I was wondering if mathematicians do truly have a sense of visualising what the fourth dimension would look like in general, such as the fourth dimension as found in hypercubes. I know that we, as 3D ...
11
votes
4answers
362 views

Elementary problems that would've been hard for past mathematicians, but are easy to solve today? [closed]

I'm looking for problems that due to modern developments in mathematics would nowadays be reduced to a rote computation or at least an exercise in a textbook, but that past mathematicians (even famous ...
-1
votes
2answers
77 views

What are the prerequisites required if I have to do induction to prove a certain theorem

I have always been fascinated by mathematical induction. The idea of induction is itself such a great analogy. But sometimes induction makes me feel that it is very messy. My professor keeps on saying ...
15
votes
5answers
265 views

Continuing math on my own? [closed]

I am in 6th grade and neither of my parents are mathematicians. I feel that at school though my teacher is great, the stuff I am studying (Pre Algebra) is just a little too elementary. I often find ...
3
votes
1answer
91 views

What are the primary disadvantages of Dummit and Foote's abstract algebra text (3rd ed.)?

I have done a fair amount of research concerning which abstract algebra book to "settle down into"; that is, I wanted to pick an algebra text and really commit to it as my "primary text," more or ...
8
votes
3answers
1k views

I thought the | symbol meant “divides by”, but in set theory, does it mean something different?

I thought the | symbol meant "divides by", but in set theory it seems that it means "such that." However, I thought we wrote "such that" as ...
7
votes
1answer
59 views

Informal interpretation of meager sets

I've been wondering if there is a nice informal interpretation of meager sets akin to the respective interpretations I give below to other notions of "small" sets. The general setup to tease out ...
1
vote
0answers
28 views

Heuristic: Daniell integral vs. Lebesgue integral

What are the advantages of the Daniel Integral over the Lebesgue integral and visa-versa? Heuristically speaking, I was wondering why this axiomatic operator is less popular besides the fact that it ...
2
votes
1answer
137 views

Famous Problems the Experts Could not Solve [closed]

After Yitang Zhang stunned the mathematics world by establishing the first finite bound on gaps between prime numbers, it got me thinking about the following question: $\underline{\text{Question}}:$ ...
2
votes
2answers
27 views

Algorithmic procedure to establish the possibility of finding a proof of a conjencture

I was always puzzled by these conjectures which can be stated quite simply, yet finding a proper proof is a monumental task even for the most brilliant mathematicians . Consider the following ...
13
votes
1answer
234 views

How to fill my mathematical gaps?

To do the story short, I became interested in mathematics in a serious way like two years ago, I'm currently in graduate school, but the problem is that my mathematical background is not as good as ...
10
votes
4answers
199 views

Is there fundamental goal of mathematics? [closed]

I did not ask this question before scaring of down-voting but could not stop the curiosity and cannot find the answer by searching the web. In physics we are looking for say smallest mass or particle, ...
2
votes
2answers
60 views

How do I get good grades in an exam?

I study hard throughout the year and I am able to solve most problems in the text assigned to us and I am frequently the only one who can solve the hardest problems in the assignments or the problem ...
1
vote
3answers
78 views

Definition of a compact operator

Operator compactness is characterized by maps the send the unit ball to relatively compact sets. Does anyone have a good justification for why we call this property compactness? The best ...
8
votes
0answers
313 views

Overview of nonlinear analysis, differential equations (ODE and PDE), dynamical systems, and mathematical physics, and their relationships [closed]

The fields of (i) nonlinear analysis, (ii) ODE and PDE, (iii) dynamical systems, and (iv) mathematical physics (e.g., electromagnetism, general relativity, gravitation, etc,...) are very huge, ...
5
votes
1answer
60 views

How to get the most out of attending a conference talk or a research presentation?

I'll say up front that I imagine there's a resource for this somewhere, but I was unable to find it. I've been attending conference talks and research presentations for about a year at my university, ...
2
votes
1answer
58 views

Background required to understand the mathematical definition of knots and their transformations

What are the concepts of math required as a prerequisite to understand Knot Theory? I'd like to be able to make a humble beginning by being able to mathematically define knots and the non-rigid ...
2
votes
1answer
29 views

Is there a way to extend operations as integration for summation?

I've read a few times that integration is a sort of extension of summation so my point is: is there any sensated way to extend other operations in the same way ? If this extension exists what branch ...
4
votes
1answer
172 views

Which mathematical topics should an applied math major know to be employable in industry? [closed]

Question I'm a junior majoring in applied math computation at UCLA, and I was wondering what exactly constitutes a viable mathematics education? That is, what kinds of mathematical topics should an ...
1
vote
0answers
31 views

Beyond the Basics:Complex Analysis Topics/textbooks Suggestions

I am currently taking a semester long Graduate course in Complex Analysis. We have covered Basics of Complex Analysis,Automorphisms of Disc and Upper Half Plane,Riemann Mapping Theorem,Weierstrass and ...
12
votes
6answers
282 views

What are some examples of generalizations to higher dimensions which do not hold? [duplicate]

My professor said (hesitantly) $\textit{If it works in 1-D, it is likely that it also works in N-D}$ Hesitantly because, she remarked, it is not true for everything (which is expected). After some ...
3
votes
1answer
54 views

Why is a cartesian morphism called cartesian?

I am reading about fibred categories. After reading the definition of "vertical" morphism, I can imagine why they are named like that. What about "cartesian" morphisms? What is cartesian about them? I ...
1
vote
1answer
65 views

Where can I learn to define mathematical terms?

For example, take the following: The radian measure of a central angle of a circle is defined as the ratio of the length of the arc the angle subtends, s, divided by the radius of the circle, ...
2
votes
1answer
94 views

How to understand cocategories

$\newcommand\CC{\mathsf{C}}$The notion of a category is well-known. There are multiple equivalent definitions; small categories can be seen as an internal category in $\mathsf{Set}$, that is a ...
14
votes
2answers
87 views

Knowing when to ask for help

I'm an undergrad with weekly problem sets due. Often if I begin the problem set early and think about it a bit each day, I can solve all of the problems. But other times it will quickly become ...
7
votes
1answer
84 views

Error function etymology: Why the name?

I've recently been introduced to the error function: $$\operatorname{erf}(z) =\frac{2}{\sqrt{\pi}}\int_0^z e^{-t^2}dt$$ Naturally, I wondered about the origin of its name: The error of... what? I'm ...
2
votes
1answer
87 views

When is $0^{0}=0$ useful?

Are there mathematical areas/situations in which defining $0^{0}$ as $0$ is useful/sensible and convention (in contrast to the "common" definition as $1$) ?
1
vote
0answers
69 views

Three theorems for the price of one? (like duality)

Why the notion of "duality" (when we get two theorems for the price of one) are ubiquitous in mathematics (order and lattice theory, category theory, group theory), but "triality" (three theorems for ...
0
votes
1answer
23 views

Why is it important to recognize CRT establishes an isomorphism between $Z/p\#Z$ and $Z/qZ$

As I understand it, it is well known that CRT can be used to show that $Z/p\#Z$ as a ring is isomorphic to the product of the finite fields $Z/qZ$ where $q$ ranges over the primes up to $p$. For ...
3
votes
2answers
62 views

What's the significance of the Church-Turing Thesis?

My understanding is that the thesis is essentially a definition of the term "computable" to mean something that is computable on a Turing Machine. Is this really all there is to it? If so, what makes ...
5
votes
3answers
89 views

What would be an effective way to learn group theory on my own?

I've read the basics of this branch and I found it extremely interesing, and I would really love to learn more about it. I want to study as much as I can on my own, as my course doesn't have group ...
0
votes
0answers
46 views

Some notes on $D_n, S_n$ and $A_n$

http://www.stat.uchicago.edu/~lekheng/courses/repth/sol2.pdf In these solutions it refers to See "Some notes on $D_n, S_n$ and $A_n$". Does anyone know where these notes can be found? They sound ...
6
votes
5answers
242 views

Proof writing: how to write a clear induction proof?

What is an effective way to write induction proofs? Essentially, are there any good examples or templates of induction proofs that may be helpful (for beginners, non-English-native students, etc.)? ...
2
votes
0answers
94 views

Good starting point for learning noncommutative geometry?

Currently, I am attempting to learn noncommutative geometry. My interests mostly lie on the boundaries of pure mathematics and theoretical physics, so I am not only interested in the mathematical ...
5
votes
3answers
142 views

Real Analysis book with pictures and ideas of proofs

I am taking real analysis course in my graduate class of Maths. My classes will start in 3 months. I have studied real analysis but not very rigorously. Whenever I see theorem I have no idea on how ...
0
votes
2answers
66 views

Looking for good intro book on differential equations

I am looking for a good book to study ordinary differential equations. My background is that I have successfully completed calculus 1 through 3. So this included derivatives and integrals, ...
0
votes
1answer
37 views

How to evaluate a squared sum?

I need (the name of) the formula to evaluate $(x_1 + x_2 + ... + x_n ) ^2$ . I know the question is not very interesting, but I am stuck and WolframAlpha also doesn't get my input. Thanks in advance