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9
votes
5answers
495 views

Formulae of the Year $2016$ [closed]

Soon it's the year $2016$. Time to ponder how we can arrange the digits in 2016 to form a valid equation. Use any symbols you like (please explain the less obvious ones). Keep digits in the same order ...
3
votes
2answers
58 views

Does a random walk with infinite mean ever converge to anything?

Suppose we have a random walk on the real line whose step sizes have finite variance. We know that, when viewed as a function and suitably rescaled, this random walk will converge to a Brownian ...
4
votes
1answer
36 views

What is the minimum number of positive integer values at which uniquely determines a polynomial with integer co-efficients?

Let $p(x)$ be a polynomial with positive integer co-efficients; then what is the minimum number $k$ (depending on $p(x)$) of distinct positive integer(s) $r_1,..,r_k$ such that knowing ...
3
votes
3answers
110 views

What to teach first: Riemann sums or anti-derivative? [closed]

In some text books, I see that they teach Riemann sums first. In other texts, I see they teach anti-derivatives first. Is there any pedagogical preference? It seems to me that we should teach ...
0
votes
2answers
57 views

Are Standalone Introductory Linear Algebra Classes Bad? [closed]

The author of the textbook I will be using for my combination linear algebra/differential equations course next semester, Introduction to Differential Equations by Michael E. Taylor ...
1
vote
1answer
39 views

What is the relationship between the unit simplex and the nonnegative orthant?

I came upon this figure while reading something online. Pictured above is the intersection of the unit sphere in the nonnegative orthant and the unit simplex. Question: What is the relationship ...
5
votes
1answer
593 views

What are some major open problems in Galois theory?

Few days back one of my friend and I were discussing about Galois life and his ideas. Though we are not trained in Galois theory, but I am recently started to learn it by myself and hope to take up ...
8
votes
2answers
502 views

I like math, but can't keep up with the pace. [closed]

I'm a math major. I like math. I'm comfortable with it. I'm considering to do a PhD. The thing is I always fall behind. I can't keep up with the professor, can't turn in satisfactory homework in ...
23
votes
4answers
1k views

What's so special about characteristic 2?

I've often read about things which do not work in a field with a characteristic $2$, mainly things which have to do with factoring, or similar things. I'm not exactly sure why, but the only example of ...
10
votes
1answer
156 views

Is there a reason for different nomenclature on Calculus of Variations?

While sightseeing aspects of Calculus of Variations, the following fact elludes me: there is a plethora of new definitions which seem redundant to me. This phenomenom happens, of course, with other ...
0
votes
1answer
41 views

Catalan Numbers vs Bell Numbers

I did a lot of research on Catalan Numbers and I came across one interesting fact that the nth Catalan numbers never exceeds the nth Bell number. I know that the nth bell numbers counts the number of ...
0
votes
1answer
34 views

Techniques of division by numbers in base n

Our current number system is in base 10, so we have devised techniques when a number is divided by a power of 10. For example: $\dfrac{350}{100} = 3.5$, by moving the decimal by two places because 100 ...
4
votes
1answer
75 views

Are there any theorem about the linearization of PDE?

I am a beginner of PDE, and surprise that some nonlinear equation will become a linear equation after variable substitution,for example. So, I am curious that whether there are general theory making ...
39
votes
11answers
4k views

Does “Doing a thing to both sides of an equation” have a name?

A two part question. 1 True or False: when working with an equation or inequality, everything that you do is either: a substitution, or an operation performed on each side Note that algebraic or ...
5
votes
3answers
125 views

For the Sake of Argument

So someone sent me this xkcd as a reply to something I said online: https://xkcd.com/1432/ I frankly felt assaulted by the pop culture monster. Perhaps as a mathematician this is not uncommon. My ...
0
votes
0answers
24 views

Understanding tensor products and $R$-algebras

Sorry if this question isn't quite precise. Anyways, I'm reviewing for my algebra final right now and two things I don't think I quite grasp as well as I'd like are tensor products and $R$-algebras. I ...
1
vote
3answers
57 views

Applications of Multivariable Calculus

As part of the final for my Multivariable Calculus class, I am to create a project wherein I find an application of some multivariable calculus subject (up to and including Green's Theorem), and ...
37
votes
4answers
3k views

Must eigenvalues be numbers?

This is more a conceptual question than any other kind. As far as I know, one can define matrices over arbitrary fields, and so do linear algebra in different settings than in the typical ...
1
vote
2answers
101 views

Is there a well known nontrivial counterexample to this claim?

Suppose we have $A\subseteq\mathbb{N}$ with the property that if $B\subseteq\mathbb{N}$ and $B$ is finite, then $\exists a\in A\setminus\{ 1\}$, $\forall b\in B$, $\gcd(a,b)=1$. Are there any well ...
-1
votes
1answer
53 views

Creating the formula and understanding them [closed]

We have, for example, the formula for velocity is distance over time ($\upsilon = s/t$). How do we get this formula? Why not ($t/s$ or $s*t$)? For this example we need physics to explain velocity, and ...
7
votes
0answers
71 views

Undergraduate group theory Aha facts [closed]

I am learning group theory. I found out that there are a lot of (more or less) simple facts that are completely trivial for working mathematicians (e.g. my professor), but that are nowhere explicitly ...
5
votes
2answers
105 views

How do you stay sharp after graduating? [closed]

So I received my math bachelors back in 2013 and am now in the "real world". But my job doesn't require the upper level math I learned, i'm not in research. My notion is that there are a lot of people ...
0
votes
0answers
23 views

Why doesn't linear programs have analytical solutions?

In a lecture that briefly overviewed linear program, it was said that linear programs do not have analytical solutions Is that a succinct reason for why this is the case?
1
vote
0answers
93 views

Functional analysis and $\textsf{CH}$

Does anyone know any result in the field of Functional Analysis which holds only if one assume one of the following : $\textsf{CH}$ $\neg \textsf{CH}$ $\textsf{GCH}$
1
vote
1answer
31 views

Algebraic and geometric multiplicities

I'm aware of the formal definition of algebraic and geometric multiplicities, but I can't make much sense out of the names. What I mean is, if it were me, I would name these two quantities ...
2
votes
3answers
67 views

Looking for fractals which are computationally demanding and preferrably parallelizable.

Oh hello guys. I am in the middle of challenging myself to putting my computer and math skills together, trying to build a small hobby computational cluster. Being interested in fractals for a long ...
0
votes
4answers
88 views

If a quadratic polynomial $ax^2+bx+c$ has the form $(rx+s)^2$. Does it implies that $b^2-4ac=0$? [closed]

If a quadratic polynomial with integral coefficients $ax^2+bx+c$ is of the form $(rx+s)^2$ Can we say that the discriminant $$b^2-4ac=0?$$ If so how do you prove it?
2
votes
0answers
48 views

Difference between algebraic topology and geometric topology [closed]

What are the main differences between these two areas? Does geometric topology in general, use more analytic techniques? Which one would most consider harder? Is one more general than the other?
0
votes
1answer
35 views

Books on Zeta Regularization Product

Does anybody know some book on zeta regularization, and the zeta regularization product? I'm quite interested on the topic but I would need a book with some review...
2
votes
0answers
56 views

Help solving the quadratic equation $ax^2-4bx+4bc-\frac{d^2}{a}=0$

I have been struggling to solve this quadratic equation in the variable $x$ with integral coefficients: $$ax^2-4bx+4bc-\frac{d^2}{a}=0$$ $a\neq 0$ of course.How do I ensure that $x$ is an integer? ...
1
vote
3answers
70 views

Resources on Mathematics for the Educated non-Mathematician

I was wondering what the best books are for giving an educated non-mathematician the gist of real mathematics. Not to say that I am by any means a real mathematician, but it is one of my goals to ...
2
votes
7answers
108 views

In need of book recommendations (Soft Question) [closed]

I tried to label this with as many soft question labels as possible to avoid immediate downvotes. To give you a gauge of where I am math-wise, I'm a senior in high school who has gone through all the ...
6
votes
0answers
57 views

Can we express the roots of all polynomials in terms of roots of some special polynomials?

We can describe the roots of quadratic equations in terms of addition, subtraction, multiplication, division, and the square-root function $\sqrt a$ which computes a root of the special polynomial ...
1
vote
2answers
114 views

How do mathematicians come up with beautiful equations [closed]

In Linear regression for example, we can find weights as following: $\hat{\beta}=(X^{T}X)^{-1}X^{T}y$ how someone invented this? I mean how do they transform a problem to such an equation. And ...
0
votes
2answers
25 views

which phenomena follow normal distribution?

In nature so many phenomena following normal distribution such as human length, grass length etc. My question is what what natural phenomena must follow normal distribution? Is there any criterion ...
3
votes
3answers
103 views

Effective Methods of Studying in different areas of Math

I apologize if this question isn't appropriate for this site, but I am looking for advice that I think other math students might be better able to give me. I am an undergraduate math major entering ...
1
vote
1answer
32 views

Introductory book on Riemann Surfaces

What is a good textbook (or set of online lecture notes) to start learning about Riemann surfaces? They were briefly mentioned in a complex analysis course I am taking, and seem like a very ...
-1
votes
2answers
52 views

$a,x,t$ are non-zero integers. $x^2+2ax+at$ is a square for what value(s) of $t$?

For what values of $t$ does the diophantine equation: $$x^2+2xa+at$$ is a square? For me, the obvious solution is: $$t=a$$ However, by assuming that there exists an integer $r$ such that: ...
2
votes
1answer
46 views

What's the point with the Gelfand–Naimark theorem?

Does anybody can explain me in plain english what's the real point with the Gelfand–Naimark Theorem. I know it's crucial, but I think I'm missing how much it's crucial.
2
votes
1answer
44 views

Is there any operation that really exists beyond just a Rotation and a Reflection?

I'm looking for fundamental operations between numbers. Kind of really really essential operations which can be done on $\mathbb N$ and $\mathbb Z$, following Kroenecker motto "Naturals where invented ...
1
vote
0answers
16 views

Intuitive idea of Green's functions

We use Green's functions in differential equations. I also see them in electromagnetism in physics. I would like to know the intuitive idea behind their usage/existence and why they are useful. It ...
5
votes
3answers
33 views

Can we, in a certain way, quantify the measure of non-differentiability of functions that are continuous everywhere but differentiable nowhere?

I am not sure how to ask this question because it seems to me that my thoughts on this topic are not clear enough, but I will give it a try. What, really, do I want to know? Well, I would like to ...
0
votes
2answers
65 views

(Soft Question) Is it bad to use Sage built in functions instead of creating my own?

I've been doing Project-Euler just as a way to increase my competency in computer science. I'm currently a Pure and Applied Math major who recently adopted computer science as a minor in order to ...
4
votes
2answers
99 views

What does the case of $\operatorname{Spec}C^{\infty}(M)$ tell us about the relavance of scheme theory to general rings?

I guess a lot of people with previous exposure to differential geometry have had this naive question pop out in their mind when studying schemes for the first time. The category of compact smooth ...
0
votes
0answers
49 views

Learning multivariable/vector calculus through guided discovery

I am asking this question as a question similar to what has been asked previously for other topics as well as math in general. But I'd like to ask for text references specifically in the domain of ...
28
votes
10answers
10k views

How do I explain the Königsberg Bridge problem to a child?

I am going to demonstrate the Königsberg seven bridge problem in a science exhibition. I am also going to use a model for a more visual representation of the problem. Now, how do I explain this (the ...
5
votes
1answer
80 views

Reference Request for Topics in Group Theory

I'm taking an honors-level algebra class and we're getting into group actions, Sylow subgroups and semidirect products. These topics are fairly intimidating, but the lengthier discussions in the book ...
6
votes
1answer
44 views

What is the Benefit If a Stochastic Process is a Submartingale?

Suppose if I have a stochastic process which is a submartingale. What is the "practical" benefit from this property? I have a roughly idea that this submartingale property suggests a favorable ...
0
votes
0answers
27 views

Best source on learning how to do limits [duplicate]

I am just curious if there is some website or book comprehensively covering theme of counting limits of functions. I am interested in tricky limits, not an introductory ones.
3
votes
2answers
49 views

Find an example of a non trivial Think of a number game

Everyone knows the game where someone ask you to think of a positive integer, then ask you to do some elementary mental operations and finally predict the result. Example: think of a number $n$, ...