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

0
votes
0answers
52 views

Collatz algorithm generalization try-out (Collatz k-algorithm)

Recently I have been reading about the Collatz conjecture here in Mathematics Stack Exchange, and also found the fantastic paper of professor Lagarias about it. Everything was so interesting (and I ...
1
vote
1answer
28 views

Question about optimization

I have a question about maximization/minimization problems. I have noticed that for almost all the practice problems that I have had that ask to find the sum of numbers and minimize product or ...
4
votes
2answers
107 views

How much does Proof writing improve over the years?

This is a very soft question. Just a bit of background: I'm a junior in high school taking Analysis I and II out of Baby Rudin at a very well-recognized university. I find quite a few of his ...
0
votes
2answers
40 views

How should trigonometric expressions be simplified?

I have been learning trigonometric identities and I am having trouble understanding how they should be applied to simplify expressions. For example, the expression $2\sin{x} \cos{x}$ is equal to ...
0
votes
0answers
12 views

Formula to convert currencies

I know this currency exchange rates: 1.000000 USD = 0.943837 EUR 1.000000 USD = 0.683463 GBP What would be the formula to find: 245 EUR = ??? GBP
34
votes
10answers
2k views

How do mathematicians find formulas? [closed]

How do mathematicians find formulas? For instance, the area of a triangle is $$\mathrm{area}=\frac{\mathrm{base}\times \mathrm{height}}{2}.\tag{1}$$ When I study maths, the book I am using tells ...
1
vote
2answers
44 views

Representative Pedagogical Examples of Groups, Real Functions, Modules, etc.

In the preface of Munkres's Topology, he writes, Fortunately, one does not need too many counterexamples for a first course; there is a fairly short list that will suffice for most purposes. Let ...
3
votes
2answers
49 views

Particular case of an Implication

Let's take the following propositions : 1 - "If Bill Gates is poor then Bill Gates is rich". 2 - "If Bill Gates is poor then the moon is made of cheese". Both propositions are inevitably true ...
1
vote
1answer
77 views

Learning mathematical concepts

Our teacher loves to test us on pure concept based questions and test if we really know what we are doing when learning a particular lesson. For example, when we first started learning about ...
2
votes
0answers
55 views

How does Godel Escher Bach support Artificial Intelligence? [closed]

Typically, Godel's Incompleteness theorems have been used to argue against the possibility that the human mind is essentially equivalent to a formal system. However, in Daniel Dennett's book "Darwin's ...
2
votes
0answers
48 views

How to remember a proof for a long time

A very basic question of mine: Whenever I read a proof I am able to remember it only for a couple of months. But I really want to remember it for at least one year or so if not more. Is it the ...
3
votes
1answer
33 views

Is the closure axiom necessary for algebraic structures defined via a binary operation?

Numerous algebraic structures are often defined as a set $X$ equipped with a binary operation $f:X\times{X}\rightarrow {X}$ that satisfies some set of axioms. Since the image of $f$ is always in $X$ ...
1
vote
0answers
28 views

What is the value of an Infinitesimal?

In the regular type of math "0.999..." is the same thing as the value 1. In some other different kind of math they say that "0.999..." is not the same thing as the value 1. Where 1 is > "0.999..." ...
46
votes
4answers
3k views

Is “A New Kind of Science” a new kind of science?

A couple of years ago I was reading "New Kind of Science" (NKS) by S. Wolfram, and it presented lot of interesting ideas for a young Physics undergraduate. Now that I am studying Mathematics however, ...
4
votes
1answer
108 views

Why is it called the Fundamental Theorem of Arithmetic?

The Fundamental Theorem of Arithmetic is easy enough to understand, saying that every integer greater than 1 is either prime or is the product of a unique combination of prime numbers. What I don't ...
3
votes
1answer
119 views

Mathematics books that tell you what is really happening? [closed]

Many book I've read teach you symblobic manipulations instead of pointing out what's really happening. So my question would be: what mathematics textbooks don't do that? Books that rather than listing ...
8
votes
1answer
110 views

Importance of Exercises in Mathematics for Self-Studying

I am a high school student wanting to major in Mathematics in the future. I started to like Mathematics recently, starting a year ago and I watched some interesting math videos on YouTube for fun (ex: ...
2
votes
0answers
59 views

How much math was “Broken” by Russell's Paradox?

As you know, the phrase "the set of all sets that don't contain themselves" caused a paradox that "broke" (made trivial) Naive set theory. How much mathematics had to be redone because of this? Most ...
1
vote
0answers
40 views

Is there a companion to the book 'A Synopsis of Elementary Results in Pure and Applied Mathematics' by George S. Carr?

A Synopsis of Elementary Results in Pure and Applied Mathematics by George S. Carr is as most of you probably know a book that was famously used by the great mathematician Ramanujan. It is said he ...
1
vote
0answers
40 views

Least upper bound and greatest lower bound of the void set.

Let $(L,\ge)$ a partially ordered set. Suppose that for avery $S \subset L$, there exists an element $LUB(S)= a$ such that $x \ge a \iff x \ge u \quad \forall u \in S$ and, with the obvious meaning of ...
2
votes
0answers
52 views

What are the current frontiers of mathematics? [closed]

In language suitable for an undergraduate student familiar with the basic objets of interest and classical techniques used in the major subfields of mathematics, what are the ''frontiers'' of modern ...
4
votes
1answer
53 views

Generalisation of posets

The notion of (set) monoid can be generalised to that of monoid object in a monoidal category. Can the notion of poset be generalised in a similar fashion? How?
2
votes
1answer
64 views

Where can I find Wielandt's original proof of Sylow's Theorem?

I have seen several proofs of Sylow's Theorem based on Wielandt's method. Everyone gives credit to Wielandt's proof of Sylow's theorem, but ironically everyone puts their own spin on it. Where can I ...
1
vote
3answers
69 views

Are there infinite sequences not reproducible by finite algorithms?

Let me know if this is a repeat question. I was thinking that sequence of integers we deal with (e.g., the digits of $\pi$, the prime numbers, the Fibonacci numbers, pseudorandom numbers) seem to be ...
35
votes
3answers
573 views

Is Stokes' Theorem natural in the sense of category theory?

Stokes' Theorem asserts that for a compactly-supported differential form $\omega$ of degree $n-1$ on a smooth oriented $n$-dimensional manifold $M$ we have the marvellous equation $$\int_M d\omega = ...
0
votes
2answers
74 views

High school looking to prepare for university

I am a high student and doing the general math course at my high school, it will cover: Geometry Graphs and Relations Matrices Statistics Next year I want to enroll in a science degree and major ...
1
vote
0answers
26 views

proof of Wiener’s criterion

I'm in my first course of PDE and I need to investigate the proof of Wiener's Criterion for Laplace Equation which says, if $\Omega \subset \mathbb{R}^n$$(n>2)$ is a bounded domain and $\partial ...
21
votes
0answers
596 views

This one weird thing that bugs me about summation and the like

Most of us know $$\sum_{n=a}^b c_n=c_a+c_{a+1}...+c_{b-1}+c_b$$ Some of us know $$\prod_{n=a}^b c_n=c_a \cdot c_{a+1}...c_{b-1} \cdot c_{b}$$ A few of us know ...
0
votes
1answer
80 views

What are eigenvalues and eigenvectors really?

I know how to determine the eigenvalues and eigenvectors of a given matrix $A$, but we were not really explained to what exactly ARE eigenvalues and eigenvectors, what is their purpose and what ...
5
votes
2answers
71 views

Is there any other constant which satisfy Euler formula?

Every body knows Euler Formula $e^{ix}=\cos x +i\sin x$ Is there any other constant beside $i$ which satisfies the above equation?
12
votes
4answers
2k views

Is it necessary for one to understand analysis?

Is it necessary for one to understand analysis in order to pursue a career in mathematics? Basically, I am very weak at analysis. But the problem is that most of the topics listed in the syllabus ...
1
vote
0answers
41 views

Going to graduate school in applied mathematics without having taken a topology course?

Due to a critical course conflict, I won't be able to take any topology or geometry courses before I graduate. However, I plan to go to graduate school in [applied] mathematics. Will this hinder me ...
3
votes
2answers
63 views

How can Bayesian and Frequentist approach be different?

Let's say I am trying to add numbers, like say one to ten. I can either add them in order, or I can notice that I can group them into five groups of eleven, so I suppose which method to use depends on ...
1
vote
3answers
39 views

Concerning $Frac((Frac \space D)[x])$ and $Frac(D[x])$ for an integral domain $D$

Is the fraction field of $\mathbb Z[x]$ a proper subfield (or isomorphic to a proper subfield) of the fraction field of $\mathbb Q [x]$ ? In general , what can we say about $Frac((Frac \space D)[x])$ ...
1
vote
1answer
67 views

What are Bernoulli numbers?

In my calculus class, my teacher said that if one was to try to calculate the maclaurin or taylor series of $\tan x$ by strictly using the definition , then you would run into many problems and your ...
1
vote
1answer
26 views

Can we characterize those Euclidean domains $D$ for which $D/I$ is finite for any ideal $I \ne \{0\}$ of $D$?

Let $I$ be any ideal of $\mathbb Z[i]$ , then as $\mathbb Z[i]$ is euclidean domain , so $I=(z)$ for some gaussian integer $z$ ; so we can write every element of $\mathbb Z[i] / I$ as ...
3
votes
0answers
104 views

Examples of categories which appear naturally without objects

Regarding the morphisms-only-definition of a category (which is equivalent to the usual one dealing with objects and morphisms), I would like to ask: Which examples of categories in practice appear ...
2
votes
1answer
34 views

Nice application of fixed point theorems

I need some nice application of fixed point theorem for some kind of divulgative seminar. About Banach fixed point theorem a nice application is the definition of self-similar fractals (and I could ...
1
vote
1answer
62 views

Problem sets for Combinatorics?

I'm taking a class in Discrete Math at the moment and our prescribed text doesn't have much in the way of problem sets and solutions so I'm finding it difficult to practice. Does anyone have any ...
0
votes
0answers
40 views

Looking for examples on Jordan Form

I am not asking a very specific question, but rather I am looking for any good examples that illustrate the following; $\mathbf{Theorem}: $ Let $T: V \to V$ be a linear operator with characteristic ...
93
votes
15answers
9k views

Has lack of mathematical rigour killed anybody before? [closed]

One of my friends was asking me about tertiary level mathematics as opposed to high school mathematics, and naturally the topic of rigour came up. To provide him with a brief glimpse as to the ...
1
vote
0answers
35 views

What is the mathematical intuition of what an $ \mu-$incoherent matrix is?

Recall the definition of what a $\mu$-incoherent matrix is. The columns of $A \in \mathbb{R^{n \times m}}$ are $\mu$-incoherent for all $i \neq j$ if: $$ | \langle A_i , A_j \rangle | \leq ...
6
votes
1answer
130 views

Why are Lie algebras “rigid” objects?

I read the following motivation for quantum groups on wikipedia: The discovery of quantum groups was quite unexpected, since it was known for a long time that compact groups and semisimple Lie ...
0
votes
1answer
272 views

Does one necessarily need an MS in Math before taking a PhD in Math? [closed]

I finished bachelor's in mathematical finance and am nearly finished with master's in mathematical finance (I am already done with thesis), and I plan to pursue a PhD not in mathematical finance but ...
11
votes
1answer
144 views

Has the age at which we teach Mathematics changed over the last two centuries?

My experience of learning Advanced Trigonometry and Calculus is that it was done to 17 and 18 year olds (School Curriculum in Australia). I assumed that it was similar in the UK, US and Europe. In ...
7
votes
4answers
483 views

Does it make sense to learn any other language except English, being a mathematician? [closed]

Would it be an advantage for a committed modern day research mathematicians to learn any foreign language? Are there works in Russian, French, or any other foreign language being produced or that have ...
1
vote
2answers
44 views

What is the rigorous justification for using inner products as a function of similarity between two vectors?

In machine learning, it is a common thing to define similarity measures, specially using the so call Kernel function. Kernel functions are defined though through inner products of feature vectors: ...
2
votes
3answers
41 views

How might I read “$\cos\left(\theta\right):\sin\left(\theta\right):1::x:y:r$”?

In the book I'm reading, A Course in Pure Mathematics, the author writes the following when introducing polar coordinates in section 22: ...
5
votes
1answer
105 views

Being a good listener in math [on hold]

I'm reasonably good at learning from textbooks, but I often find that when people explain mathematical ideas to me in person I don't understand what they are saying. I sense that there is actually a ...
1
vote
0answers
69 views

Examples of real world situations where mathemematical rigour is needed

I had a discussion with a friend about the need of mathematical rigour in the real world. He argues that little rigour is needed for the "application of mathematical results.Mathemticiians ...