For questions that don't admit a definitive answer. Please do not ask too many of these.
4
votes
0answers
43 views
Simplest examples of real world situations that can be elegantly represented with complex numbers
Mathematics could be defined as the study of formally defined abstractions. These abstractions may or may not be useful for describing real world phenomenon. Indeed, Physics could be defined as the ...
3
votes
1answer
107 views
Advice: Modern vs. Classics
First of all, my apologies if (well, I know I am but I don't know where to put it) I am posting this in the wrong place. So please feel free to move it to someplace else or to tag it differently if ...
4
votes
2answers
106 views
Are algebraic numbers analogous to group elements with finite order?
Would you say that the "elements with finite order" in group theory are analogous to "algebraic numbers" in field theory?
I thought this is the case since requiring an algebraic number $\alpha$ to be ...
0
votes
1answer
17 views
Characterisation of the spectrum of certain unitary representations on $L^2(G)$
I have been informed of the existence of theorems in harmonic analysis that will allow me to calculate the spectrum of given unitary operators $L^2(G)$ where $G$ is a locally compact group. So far I ...
6
votes
1answer
75 views
What are some good open problems about countable ordinals?
After reading some books about ordinals I had an impressions that area below $\omega_1$ is thoroughly studied and there is not much new research can be done in it. I hope my impression was wrong. ...
3
votes
4answers
63 views
Learning trigonometry on my own.
I have been self teaching myself math beginning with a grade 10 level for a while now and need learn trigonometry from near scratch.
I am seeking both books and perhaps lectures on trigonometry and ...
3
votes
2answers
60 views
Is there a symbol for the idea of the smallest value greater than zero?
I know that it isn't actually a number but I do think it's a concept in mathematics. So the question is, is there a symbol representing this concept? I thought maybe it was Phi but I couldn't find it ...
1
vote
2answers
93 views
Big Topics in Mathematics [closed]
My question is as follows: It is now the year 2013 as we know it, and I'm wondering what the "big topics" in mathematics are. What fields are of utmost interest and foundation in the modern era? How ...
1
vote
2answers
73 views
The maths required for an economics degree
I have a degree in computer science and I wanted to do another degree in economics.
However, my maths have been weak since high school always scoring slightly above passing rate. During my course of ...
7
votes
3answers
243 views
Why is 1 raised to infinity Not defined and not “1” [duplicate]
$1$ square is $1$, so is raised $1$ to $123434234$.
My maths teacher claims that $1$ raised to infinity is not $1$, but not defined. Is there any reason for this?
I know that any number raised to ...
1
vote
2answers
32 views
How do I find the residue of a function with a huge exponent?
How would I find the remainder of a function that has a huge exponent that would take ages to work out?
Say I have something like this:
$\frac{5x^{110} + x^4 - 7x^2 - 6}{x-1}$
I honestly don't know ...
0
votes
1answer
37 views
I think I solved it, but can someone check my solutions please? - Instantaneous Rate of Change
It's been awhile since I've done anything with rates of change and I'm struggling with deriving a formula in terms of 'x.'
From what I can recall.. the average rate of change and instantaneous rate ...
-5
votes
0answers
35 views
Mathematics and Language: An Outlet For Opinion [closed]
Why does mathematics exist as such a liquid language of thought? Is it really the universal language? Is this a naive question?
\begin{eqnarray}
0
\end{eqnarray}
4
votes
1answer
47 views
Why are germs of functions important?
Why is it necessary to define germs of functions (in my case, for foliations, but my question is in general)? does any inconsistency arises if instead of using a germ in some context, I use ...
6
votes
1answer
114 views
Which languages are preferable to study for a mathematician?
I am already fluent in portuguese and english, and I can also read spanish well. I have to read a text which is in french, and I'm having some difficulties. Knowing that there are a lot of people out ...
2
votes
2answers
63 views
Pascal's triangle and combinatorial proofs
This recent question got me thinking, if a textbook (or an exam) tells a student to give a combinatorial proof of something involving (sums of) binomial coefficients, would it be enough to show that ...
10
votes
4answers
218 views
What does it mean for a set to exist?
Is there a precise meaning of the word 'exist', what does it mean for a set to exist?
And what does it mean for a set to 'not exist' ?
And what is a set, what is the precise definition of a set?
1
vote
1answer
31 views
Extreme Value Theorem proof help
Extreme Value Theorem: If $f$ is a continuous function on an interval [a,b],
then $f$ attains its maximum and minimum values on [a,b].
Proof from my book: Since $f$ is continuous, then $f$ has the ...
5
votes
3answers
61 views
Analogy between Integration and Summation
There are many analogies between definite integral and Summation: $$\int_a^b \leftrightarrow \sum_a^b$$, This makes me wonder if there is analogous concept of indefinite integral, derivative and ...
0
votes
0answers
73 views
Math is a young man's game? [closed]
Do you agree with this quote from Hardy? Supposedly someone is in their prime between ages 18-25.I don't think I agree with this, since most of the people doing research and advancing math are ...
0
votes
1answer
33 views
How to get to these steps?
I found this question here (I reccomend that you read the question and the highest-voted answer there) How to solve for $x$ in $x(x^3+\sin x \cos x)-\sin^2 x =0$? and the math below is an answer. I ...
10
votes
2answers
188 views
Working habits in mathematics
Short version of my question: What are good and motivating working habits for a mathematician?
Note that, there are similar questions to mine: see this (reading books) or this or this. But none of ...
4
votes
2answers
140 views
Isn't seven bridges problem trivial? [closed]
What was the actual actual problem that led Euler to graph theory?
By looking even at non-simplified map like this
It is obvious that, if a landmass is connected by odd number of bridges, it ...
3
votes
1answer
27 views
Cohomology theories that arise in different fields of mathematics
During my studies in university I have encountered several cohomology theories. Part of them I've met in topology\differential geometry\analysis on manifolds courses (simplicial, singular, cell, ...
4
votes
0answers
41 views
Hopf Algebras in Combinatorics
I know that many examples of Hopf algebras that come from combinatorics. But I'm interested in knowing how Hopf algebras are applied in solving combinatorial problem.
Are there examples of open ...
1
vote
0answers
21 views
Notation for Restriction of Permutation
Suppose $\sigma$ and $\tau$ are permutations such that $\sigma(x)\not=x\implies \sigma(x)=\tau(x)$. Intuitively, I would like to think of $\sigma$ as a restriction (or projection) of $\tau$ onto a ...
4
votes
2answers
101 views
In what order should mathematical fields be learned?
This could be considered a broader version of this question, with all fields.
I know that when high-level maths are reached, the fields being to split quickly (i.e. specializing in this type of ...
1
vote
1answer
38 views
How important are the following undergrad courses when trying to pursue studies in chaos theory/dynamical systems?
I'm currently a physics major with a year left, and deciding whether to switch into mathematical physics, mathematics or applied mathematics. I'm definitely switching into one of them, as I can meet ...
2
votes
1answer
59 views
Materials for studying logic
I am looking for study and beginner material to study mathematical logic. I understand that it is a very broad topic but I would like to know what the best path there is to learning mathematical ...
1
vote
0answers
22 views
seek visual pictures or video on decomposition of manifolds
In my study of knot theory, I notice that I lack examples to show some classical decomposition theorems in 3-dimensional manifolds, such as JSJ decomposition theorem, Milnor's prime decomposition ...
16
votes
3answers
251 views
Why should “graph theory be part of the education of every student of mathematics”?
Until recently, I thought that graph theory is a topic which is well-suited for math olympiads, but which is a very small field of current mathematical research with not so many connections to ...
1
vote
1answer
35 views
Holonomy of the sphere
I saw an example in which the holonomy of $\mathbb{S}^n$ with the standard metric is calculated. I'm just starting to get familiar with holonomy groups and I wanted to know what could one do by ...
30
votes
14answers
529 views
How to entertain a crowd with mathematics? [closed]
I am a high school student who follows a university level curriculum, and recently my teacher asked me to hold a short lecture to a crowd of about 100 people (mostly parents of my classmates and such, ...
0
votes
3answers
73 views
When is it important to distinguish between an object in a category and that object's identity morphism?
When is it important to distinguish between an object in a category and that object's identity morphism?
I am wondering if the only reason that we consider objects at all is to avoid infinitely ...
5
votes
2answers
115 views
Analysis without algebra
I once heard someone say that analysis is $99 \%$ algebra. He was, of course, referring to the amount of algebraic manipulations in the exercises from any calculus course.
I know that in topology, ...
2
votes
5answers
71 views
How many types of functions are there [closed]
We have the following types of functions :
a) Logarithmic function
b) Rational Function
c) Irrational Function
d) Piecwise or modulus function
e) Smallest integer function or cieling function
f) ...
3
votes
2answers
49 views
Matrix Multiplication Definition
I'm sure everyone already thought about this at least one time.
Why matrix multiplication is not defined the way showed below?
$$\left( \begin{array}{ccc}
a_{11} & a_{12} & \ldots \\
a_{21} ...
2
votes
1answer
26 views
Which expressions in English should I use for a morphism having a certain source and target?
Say that $f: A \rightarrow B$ is an arrow in a category $\mathcal C$. Which verbs or expressions do we use to express in an alternative way that $A$ is the source of $f$ and $B$ its target? E.g., ...
-5
votes
1answer
94 views
Standing while doing math? [closed]
Just wondering, I get tired of sitting for hours while doing math, anyone use a standing desk for math?
Might build myself one but I would like to hear opinions from people that do math first. I ...
-1
votes
0answers
35 views
Most overlooked conventions of mathematics? Resources on Mathematic Conventisons [closed]
I am wondering about the conventions of mathematics. I am quite new to mathematics and I am wondering about what are the conventions associated with mathematics, and more importantly the ones that are ...
-1
votes
0answers
39 views
Does it ever happen to you that you cannot figure out a problem which you did earlier? [closed]
Does it ever happen to you that you cannot figure out a problem which you did some time ago? For example, I know I found a nice proof for $lim$ a$s$ $x$ $approaches$ $0$ $of$ $x^x$, but now just ...
5
votes
2answers
79 views
Why generalize the Euclidean metric?
It is well known that the Euclidean metric can be generalized to $\Bbb R^n$ by $\sqrt{(x_1-x'_1)^2+\cdots + (x_n-x'_n)^2}$, and that under this generalization it is still a metric and satisfies ...
0
votes
1answer
33 views
Similar textbook to Konigsberger's Analysis 2?
I am currently taking an introductory course to real analysis and my professor has decided to leave Rudin's "Principles of Mathematical Analysis" when teaching us the concepts of Lebesgue integration. ...
7
votes
1answer
63 views
Can we think of an adjunction as a homotopy equivalence of categories?
There is a way in which we can think about a natural transformation $\eta: F \rightarrow G$ as a homotopy between functors $F,G:\mathcal{C}\rightarrow \mathcal{D}$. Now, an adjunction $F \dashv G$ ...
0
votes
2answers
57 views
Polynomial Multiplication
I have been working with Lagrange polynomials and I have some very complicated calculations such as $$(x-1)(x-2)(x-3)(x-4)(x-5)(x-22)$$ Can you suggest an (possibly give url) online application which ...
2
votes
0answers
28 views
Soft Question: Scientific applications of ordinal arithmetic?
Are there any known scientific applications of ordinal arithmetic -- either direct applications or application of results in other areas that depend even indirectly on results from the study of ...
4
votes
0answers
27 views
Interpretation for the Functional Determinant
Let $S:V \rightarrow V$ be a linear operator on the function space $V$. It is possible to define a functional determinant for $S$ via the zeta function regularization process.
In specific we define ...
4
votes
2answers
102 views
Does $\mathsf{ZFC} + \neg\mathrm{Con}(\mathsf{ZFC})$ suffice as a foundations of mathematics?
I've heard people make the argument that:
$\mathsf{ZFC}$ suffices as a foundations of mathematics because almost all theorems in the mathematics literature can be proven using $\mathsf{ZFC}$, so ...
8
votes
0answers
81 views
Undergraduate thesis advice
I am presently in my senior year and I am considering fluid mechanics for my thesis. What area of research of fluid mechanics or models of transport mechanism in Biological systems which is purely ...
10
votes
2answers
96 views
What's the idea of an action of a group?
I know the formal definition of an action over a set. I'm not asking this.
What I'm asking is: what's the intuition of it? It is a way to define an algebra over a set? Since an action can exist in ...
