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2
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2answers
35 views

When is it insufficient to treat the Dirac delta as an evaluation map?

The Dirac delta "function" is often introduced as a limit of normal distributions $$\delta_a(x)=\frac{1}{a\sqrt{\pi}}e^{-\frac{x^2}{a^2}}\text{ as }a\to\infty.$$ Obviously, this sequence of functions ...
-2
votes
0answers
40 views

What's the origin of the word “mathematics”?

I'm curious to know what the word "mathematics" literally mean. I think this word originated from greek, and the word itself has a meaning. For example, the chinese&japanese vocabulary for ...
1
vote
0answers
18 views

Infinite strings and infinite theorems - Is there a theory on these stuffs?

I can have an alphabet $\mathcal{A}$, a set of axioms $\mathcal{X}$ which are finite strings of $\mathcal{A}$ and a set of rules $\mathcal{R}$. Every finite strings produced by applying a finite ...
4
votes
0answers
40 views

Are there examples of mathematical problems proven by abduction?

Proof by deduction is a simple principal. For example: All humans are mortal, and Bill is a human; Therefore, Bill is mortal. However, proof by abduction is a bit different. A famous example: ...
0
votes
1answer
55 views

Mathematicians average in student life but later became significant

What are the examples of mathematicians who were below the average in their student life (say, upto university level but it may be less) but later in life became significant mathematicians. Up until ...
3
votes
2answers
25 views

Why is convergence w.r.t $\mathcal L^p$-norm of a sequence $(f_n)$ of $\mathcal E-\mathcal B(\mathbb R)$-functions called “convergence in $p$-mean”?

In measure-theory, why is convergence with respect to the $\mathcal L^p$-norm of a sequence $(f_n)_{n \in \mathbb N}$ of $\mathcal E-\mathcal B(\mathbb R)$-measurable functions called "convergence in ...
10
votes
3answers
453 views

Bad at computations… but not math?

Very generally, the question I'm trying to ask is: Can someone be a good professional mathematician without being particularly good at (or even being - relatively - bad at) computations? More ...
2
votes
1answer
42 views

Order of a study

So i just recently had to drop two math courses, topology, math logic, because my math maturity wasn't up to the level needed to excel in them. I intend on taking them again, but not without first ...
2
votes
0answers
49 views

Second reading on set theory?Any recommendations?

I have in past six-ish months studied through the Herbert Endertons Elements of set theory book. Up to the point the book is great,I loved most parts of it and learned almost everything up to the ...
0
votes
0answers
19 views

Soft: Interesting examples of applications of the Itô–Nisio theorem

I'm writing up a presentation on the Itô–Nisio theorem and I'm looking for a simple (nontrivial) example showcasing it. I was thinking of a 2-dimensional simple random walk, but that seems ...
1
vote
3answers
52 views

What does inner product actually mean?

What does inner product actually mean? So far most of the cases that I encounter seems to suggest that dot product is the only useful inner product. I mean most of the things that we discuss about ...
0
votes
2answers
55 views

Why non-real means only the square root of negative?

Once in 1150 AD, an Indian mathematician Bhaskara wrote in his work Bijaganita (algebra) that, There is no square root of a negative quantity, for it is not a square However later on in 1545 an ...
0
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0answers
12 views

(soft question) Kreyszig's Functional Analysis or Rudin's Real and complex analysis for an introduction into more advanced analysis?

I realise they are quite different in their approach and material covered, but they share the central stuff like normed/Banach/Hilbert spaces, Hahn-Banach theorem etc. Not really understanding what ...
1
vote
1answer
27 views

Connections between SDE and PDE

I have encountered a number of situations where the solution of a PDE and a certain expectation associated to a Markov process are equal. Two examples include: The heat equation $u_t = \Delta u$ ...
6
votes
3answers
146 views

Mathematical results that were generally accepted but later proven wrong?

I am giving a presentation on mathematical results that were widely accepted for a period of time and then later proven wrong, or vice versa. This talk is geared towards undergraduates who are likely ...
1
vote
0answers
25 views

Why real variable methods can take place of complex variable methods in harmonic analysis' research?

We know that one complex variable methods have been largely used in the early research of harmonic analysis. But, as is known to me, there is much difficulty when mathematicians attempt to generalize ...
7
votes
3answers
64 views

How do I decide what problems and how many problems to do when I try to self study?

I am a math major at a relatively small college with barely any choice of classes to choose from so I have to supplement my studying with a lot of self studying. I usually have no problem getting ...
0
votes
0answers
13 views

Basic Topology: Armstrong

I am currently reading basic topology by Armstrong and he references "thickening" a tree. I am not sure what this means. Can anyone briefly explain?
3
votes
1answer
78 views

What is the incorrect proof by Euler that $\pi = 0$ (or something like that)?

I seem to remember a proof by Euler, involving infinite series, which was really complex (for a maths hobbyist). I believe it was sent in a letter to someone, and that it ended up with $\pi = 0$ or ...
3
votes
0answers
39 views
+50

Reference request regarding calculus exam

I'm currently a first year computer science student and I'm deeply interested in calculus . That being said, what we studied so far consists of: Cantor sets, sequences and a brief introduction to ...
0
votes
0answers
29 views

Maturity and Proficiency in calculus, linear algebra for successful research

Will the high level maturity and proficiency in basic calculus, linear algebra (both calculation and theorem aspects) be required or recommended as an important factor to be successful in mathematical ...
0
votes
1answer
30 views

Discovering the mathematical nature of Nature - Galileo's inclined plane experiment

In 1638 Galileo published Two New Sciences, in which he described his inclined plane experiment. He discovered that the acceleration of gravity was uniform, and could be modeled mathematically by the ...
1
vote
3answers
52 views

Problem about a bijective map from $\mathbb R^2 \rightarrow (0,1)$ [on hold]

Does there exist a bijective map from $\mathbb R^2 \rightarrow (0,1)$? What will be the mapping?
1
vote
3answers
50 views

Question about $e^x$

Let $ p(x)=1+x+x^2/2!+x^3/3!+....+x^n/n!$ where $n$ is a large positive integer.Can it be concluded that $\lim_{x\rightarrow \infty }e^x/p(x)=1$?
1
vote
2answers
73 views

What branch of analysis deals most with sequences and series?

I'm really interested in sequences and series (mainly series). What kind of math branch should I look more into? I understand that sequences and series mostly point toward analysis, but what ...
1
vote
2answers
48 views

Idea behind definitions in math

Maybe this is not such a great question for this site as it is a bit open ended, but I will ask it anyways. I was wondering if anyone has any insight into the motivation for quote unquote "major" ...
4
votes
2answers
122 views

Are all calculus textbooks “the same”?

I'm not satisfied with my calculus textbook,[1] and because of that I have searched for books by other authors. The problem is: all the books I have taken a look at are almost the same, even the ...
0
votes
1answer
61 views

calculus first impressions

I just did the first lecture on differentiation what the derivative is, and how is it calculated I didn't find it to be difficult at all it's just some forms of algebraic calculations when will ...
5
votes
0answers
75 views

Can a Power Series tell when to stop?

The naive description of the radius of convergence of a complex power series is as the largest radius so that the ball avoids poles and branch cuts. This makes sense in a world where analytic ...
2
votes
1answer
32 views

Other Useful Series Tests

So after taking calculus II, or maybe a first course in analysis, everyone learns a few series tests. They learn 1) Divergence Tests 2) Integral Test (from which we deduce things like $p$-series. ...
29
votes
1answer
367 views

What did mathematicians study as an undergraduate/graduate before modern mathematics such as modern algebra and analysis?

I am curious as to what mathematicians such as Leibnitz and Gauss and the Bernoulli's studied when they were students in university. I find it fascinating how we are taught calculus and abstract ...
2
votes
2answers
35 views

Proving arithmetical properties for non-natural numbers

Sorry if my question is dumb but here it is: I know how to prove all of the arithmetical properties such as $(a^{m})^{n}=a^{mn}$ and $a^{m}a^{n}=a^{m+n}$ and $a(b+c)=ab+ac$ etc. For numbers that ...
0
votes
0answers
41 views

Favourite proofs by induction?

I am searching for nice proofs by induction, that can be used to teach. I remember this example, that my analysis professor presented to us in first semester and I am searching for more such easily ...
1
vote
2answers
32 views

How to calculate row sums of a power of a matrix

Let $P $ be an $n\times n$ matrix whose row sums $=1$.Then how to calculate the row sums of $P^m$ where $m $ is a positive integer?
0
votes
0answers
50 views

Easiest Book to start Manifolds.

Ok, so here is the thing. I do not like analysis,Calculus or topology much, nor are they my strong point, but next week I have an exam on manifolds which I have a course First time in my life. I have ...
0
votes
0answers
30 views

A linear algebra textbook that is advanced enough as a prerequisite to read time series and econometric textbook?

A linear algebra textbook that is advanced and comprehensive enough as a prerequisite to read time series by Hamiliton and econometric by Hayashi? If possible, please also answer on which statistics ...
-4
votes
0answers
63 views

What's the most abstract field of math? [closed]

What's more abstract: category theory or model theory? Both seem to deal with abstracting math as much as you can.
1
vote
0answers
59 views

Have most popular(famous) mathematicians been determinists? [closed]

From what I have seen, most mathematicians are/were hard determinists. Has someone done research on this topic? Later edit: The mathematicians which I've seen as hard determinists are the ...
-1
votes
0answers
63 views

What actually happened to Cantor?

I saw this answer and its comments while browsing Math SE, and it made me tempted to ask: What actually happened to Cantor? Did he really, as it's usually claimed, be called blasphemous and ridiculed ...
2
votes
1answer
75 views

True but unprovable?

I would like to ask a question about Gödel's Incompleteness Theorems which I've had in the back of my head for some time. Since I'm a student working in a completely different area of maths (my usual ...
2
votes
2answers
49 views

How to make a sum vanish?

This is a very very basic question but I just cannot think of a way to tackle it for some reason. Say I have three numbers $a,b,c$ with the sum $a+b+c\neq1$. Now if I want to make this sum equal to 1 ...
4
votes
1answer
61 views

Is the book “Naive Set Theory” from P. R. Halmos still up-to-date?

My question is, if Halmos' book "Naive Set Theory" is still up-to-date concerning contemporary mathematics, that is, is it outdated or not? I really love the books so far, and while it's clear the ...
4
votes
2answers
107 views

What happens to a great mathematician's unpublished works when they die?

When a great mathematician dies, they often leave plenty of unpublished and incomplete works in their manuscripts. As we assumed that they were a really good mathematician, most of the ideas in these ...
3
votes
0answers
61 views
+50

Pre-requisites and references for $K3$ surfaces

I would like to know the "roadmap" to study $K3$ surfaces. Perhaps, my background might be helpful: I am an undergraduate student, who knows the basics of Differential Geometry, Topology, Complex ...
0
votes
5answers
65 views

Guessing on the SATs, is it ever better to leave it blank than to guess?

On most SAT questions, there are 5 answers of which exactly one is correct and exactly four are wrong. If one answers correctly you get $1$ point. If you answer incorrectly, you receive $-\frac14$ ...
7
votes
4answers
104 views

Is abstract algebra (mostly?) restricted to $2$-ary operators?

This may be due to my own pure ignorance but it's my experience that all abstract algebra I've been introduced to, both in actual courses and in self-studies only exclusively deals with algebraic ...
4
votes
2answers
38 views

Does $A^i \cap A^j = \emptyset, $ if $ i \neq j$?

I'm doing a bit of set theory and, of course, I'm confused. How true is it that if we have a series of cartesian products of a set, say $A^n, n< \omega$, then it necessarily holds that $A^i \cap ...
10
votes
3answers
151 views
+50

Value in retracing mathematicians' steps (specifically Galois)?

So I'd like to learn Galois Theory, which I am probably not "qualified" for in an ordinary sense (I've never done abstract algebra, and I'm just now learning linear algebra in my vector calculus ...
5
votes
5answers
139 views
+50

Mathematicians' manual of style

I know that there are many styles to write citations and footnotes and that they are all equally good (as long as the reference is complete), but I would like to know if mathematicians follow some ...
0
votes
0answers
55 views

Mathematics only with physics? What about biology and chemistry?

In The Mathematical Mechanic, the author "reveals how physics can simplify proofs and lead to quicker solutions and new theorems, and how physical solutions can illustrate why results are true in ways ...