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4
votes
1answer
65 views

Suggest a follow up book to Axler's Linear Algebra Done Right?

So I know that a similar question has probably been asked about alternatives or compliments to this book, but I think my situation is different enough to warrant slightly different advice. So I've ...
1
vote
1answer
76 views

Is it ill-advised to read books casually for entertainment? [closed]

I'm a student who has about a year (and a few months) to go before entering a university and I've been reading some math books recently. I'm on Chapter 6 on Rudin's PMA, Chapter 5 in Munkres' Analysis ...
2
votes
1answer
47 views

How to know what kinds of substitution can we do in math?

I have seen in many contexts that somebody out of the blue decides to put $x=y^2$, or $x=t/2$. So how do I know what kind of sustitution I'm allowed to do? Is there any necessary conditions or we ...
2
votes
0answers
63 views

Cognitive processes involved solving IMO level problems [closed]

I am currently 16 years old and, though I'm obviously not as good as most of the people on this site, I have always been considerably better than most of my classmates in mathematics. This, of course, ...
1
vote
1answer
36 views

Methods for factorising polynomials into irreducibles over finite fields

I was given a problem recently, part of whose solution was to factorise $x^{15}+1$ in $\mathbb F_2[x]$. It turns out that the factorisation is $$(x+1)(x^2+x+1)(x^4+x^3+x^2+x+1)(x^4+x+1)(x^4+x^3+1),$$ ...
11
votes
6answers
146 views

Where to find interesting integrals for a Calc III student?

I apologize in advance if this is a very soft question. I won't be surprised or offended if I can't get a good answer. One of my favorite things to do in my spare time, when I'm feeling analytical of ...
0
votes
0answers
45 views

Real or Complex Analysis next quarter?

I apologize if this is the wrong place for to ask a question about a math course I should take, but I didn't see a better stack-exchange forum for this. Long story short, I am a fourth-year chem PhD ...
3
votes
0answers
32 views

Criterion for Improvement

I was recently asked the question "How do you know when you've become a better mathematician/better at mathematics?," and I realized that at that moment I did not have a valid answer, since I have ...
0
votes
0answers
12 views

Good introductory texts on modular forms/L-functions

I am relatively new to these areas but would like to gain some understanding through an introductory text. I am an undergraduate math major so ideally these books should be accessible to someone with ...
5
votes
0answers
67 views

Is it worth proving every theorem you learn?

Ever since I determined mathematics - mainly set theory and number theory - was my main passion, and I began learning mathematics formally outside of the curriculum posed within secondary schools I ...
6
votes
0answers
89 views

Why are the fundamental theorems of calculus usually associated to the Riemann Integral?

I am writing a "textbook" on Analysis, and I've reached the time I must talk about integrals. I prefer to approach directly the Lebesgue Integral theory. This question is not about the status of this ...
0
votes
1answer
40 views

A problem in understanding principal root in the complex plane.

We know that every complex number has exactly $n$ $n$-th roots in the complex plane, and we usually take (if the context where we are working doesn't tell us more) the one with real and imaginary part ...
2
votes
2answers
57 views

Recommendation of multivariable calculus books

I am looking for some suggestions on a good calculus book I shall keep on hand all the time. I am a graduate student who will be commencing research in the area of theoretical PDE (nonlinear). ...
2
votes
2answers
81 views

Algebra Text Recommendations [closed]

I am looking for any recommendations or suggestions for a good book covering an introduction to the following; Relation , sets and functions, divisibility theory and modular arithmetic , groups, ...
0
votes
1answer
119 views

Latest episode of the big bang theory, vanity card.

I usually don't read these, but this time I did, and this was the card: Does the last mathematical symbols have any meaning? I get that the equal 150.6+V, is there any more meaning behind this?
1
vote
3answers
116 views

Definition of the mathematical proof

How do we define a mathematical proof? Is it a series of arguments? Is it a series of conclusions? Is it manipulation of formulas? Is it a mixture of laws of logic and axioms,theorems or ...
1
vote
0answers
19 views

Prove that $c_1 \phi_1 + c_2 \phi_2$ solves the IVP

Theorem: If $\{ \phi_1, \phi_2\}$ is a fundamental set of solutions of $$x''+p(t)x'+q(t)x=0,$$ then for any initial values $x(t_0)=x_0, x'(t_0)=y_0$, there are constants $c_1$ and $c_2$ so ...
6
votes
1answer
66 views

When vectors act on scalars.

Background. I've been struggling through an introduction to differential geometry this semester. Recently, a tiny part of what we've been learning "clicked" for me, and to solidify this, I'd like ...
1
vote
0answers
44 views

How to teach Mathematical Induction mathematically? [migrated]

I am exhausted of teaching Mathematical induction to my little brother. I have given him many examples, Domino effect, aligned shops of hot dogs etc and every time he says that he got it but when I ...
0
votes
0answers
64 views

Comments on Eilenberg and Steenrod's “Foundations of algebraic topology” and other similar books for recomendation

The biggest obstacle for me to learn geometry and topology is the haziness of textbooks. I took algebraic topology last semester and the textbook we used in class was Rotman's "An introduction to ...
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
33 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
89 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
120 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
30 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
55 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
115 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
73 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
9 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 ...
1
vote
1answer
37 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
15 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
70 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
38 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
50 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
335 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
76 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 ...
14
votes
5answers
239 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
63 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 ...
-1
votes
0answers
32 views

What does the derivative of acceleration represents?

The derivative of a distance function, represents instantaneous velocity. The derivative of the velocity function, represents instantaneous acceleration. What does the derivative of the acceleration ...
5
votes
0answers
39 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
19 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
127 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
1answer
56 views

Themes in Mathematics [closed]

My professors have alerted me to some themes throughout the subject. One that I've found useful is "abstraction and generalization": when studying rings, for instance, I initially saw nothing but a ...
1
vote
0answers
20 views

Can boosting be thought of as a genetic algorithm? [migrated]

Can boosting be classified as a genetic algorithm or as an instance of simulated annealing? Or, is it a completely different paradigm? Essentially, I'm trying to rectify discrete optimization ...
2
votes
2answers
25 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 ...
12
votes
1answer
184 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 ...
1
vote
0answers
29 views

Enough for Grad School? [migrated]

I just finished my 3rd year in a combined pure/applied math program at a Canadian university. I have been leaning towards pure math, but I'm not sure if I'll have done enough to go to grad school for ...
10
votes
4answers
184 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
49 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 ...