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2answers
69 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 ...
1
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0answers
27 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 ...
2
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2answers
57 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 = \frac{1}{2} ...
6
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3answers
227 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 ...
2
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1answer
49 views

Why can real variable methods 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 (real variable) harmonic analysis. But, as is known to me, there is much difficulty when mathematicians ...
7
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3answers
90 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
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0answers
31 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
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1answer
109 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 ...
7
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2answers
184 views

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 ...
1
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0answers
70 views

Is High Maturity and Proficiency in Calculus and Linear algebra necessary for successful research?

Is it the case that most successful mathematicians have very high Maturity and Proficiency(do it without thinking) in Calculus and Linear algebra, both calculation part(double/line integrals, ...
0
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1answer
38 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 ...
2
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3answers
66 views

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

Does there exist a bijective map from $\mathbb R^2 \rightarrow (0,1)$? What will be the mapping?
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3answers
53 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
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2answers
102 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
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2answers
58 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
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2answers
163 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
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1answer
62 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
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0answers
87 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
33 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. ...
30
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1answer
553 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
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2answers
38 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
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0answers
47 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
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2answers
35 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
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0answers
59 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
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0answers
40 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 ...
2
votes
1answer
114 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
53 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
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1answer
93 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
122 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 ...
6
votes
1answer
128 views

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
73 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
113 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
40 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 ...
17
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4answers
248 views

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 ...
12
votes
8answers
252 views

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 ...
1
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0answers
70 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 ...
4
votes
2answers
123 views

Will computers one day start creating and proving math for us? [closed]

I suspect in the future we might be able to build computers that research math for us. And I also suspect they will probably be way more efficient at doing research than we are. I do think this ...
1
vote
0answers
48 views

Best way to learn material dealing with cosets, quotient groups and the isomorphism theorems

I'm self studying abstract algebra from Abstract Algebra by Dummit and Foote. I've been able to get through the first few chapters and do problems without any issue, until I hit the material on ...
0
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1answer
38 views

Why quotient space is needed?

I was wondering why quotient space is so important? Let say for vector space why quotient space is needed? Please explain!
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0answers
52 views

Normal distribution in nature

I applied for a job as a mathematician. In one of the test questions they asked the following: Why normal distribution is so common in nature? What do you think?
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0answers
11 views

Resource Request- Good classifcation of shapes by number of vertices?

There are many websites that classify various aspects of mathematics. For example: Integral table, a huge list of integrals oeis.org/ www.encyclopediaofmath.org/index.php/Main_Page ...
1
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2answers
78 views

How can we explain the discrepancy between $\rightarrow$ (IF-THEN) and $\setminus$ (A-BUT-NOT-B)?

Let $\mathbb{B} = \{0,1\}$ denote the Boolean domain, ordered in the usual way. Then $\mathbb{B}$ is a lattice. It has a join operation $\vee$ that coincides with "OR," a meet operation $\wedge$ that ...
1
vote
1answer
69 views

Is Linear Algebra the foundation of Applied Mathematics?

I've lately taken an interest in foundations of my field. While there are many important areas that contribute to Applied Mathematics (diffeqs, probability & statistics, numerical methods, ...
4
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0answers
59 views

Quantifying infinitely large sums such as $\sum_{x\in\mathbb{R}^+} x$

I thought of this as a student in calculus years ago, and it may be a silly kind of question. I wondered if there were notions of different sizes of infinity a series might sum to, which then lead me ...
1
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0answers
16 views

Is there a relationship between the integrand in Green's Theorem and the test for finding an integrating factor for a differential form?

Green's Theorem has the formula $$ \int_C Mdx+Ndy=\int\int_D\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)dxdy $$ There is also a well known test for finding an integrating ...
2
votes
1answer
54 views

Problem about problem solving

I am having some problems on how to solve a problem.When I read a chapter on say group theory or real analysis,I feel that I have grabbed the concepts quite well,but when I start solving exercises ...
0
votes
1answer
132 views

Platonist research on the cardinality of the reals

Apologies to any formalist! Here's the basic thought: $\mathbb{R}$ is a well-defined concept with unambiguous meaning in reality. Everyone can imagine an infinite series of digits (signifying the ...
1
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1answer
77 views

Differential-geometry textbook with solved problems

I'm looking for a textbook in differential geometry which inside has exercises with (at least) final answers. Since it's my first course in differential geometry it doesn't have to cover material (we ...
6
votes
2answers
211 views

Why is there apparently a consensus on the P = NP question?

So through my years of education I have heard a lot about the famous $\mathrm{P}=\mathrm{NP}$ problem. I have seen that a significant number of mathematicians believe that this result is false (and ...
1
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1answer
30 views

Difference between contradiction and paradox?

In multivalued logic one can distinguish at contradictions (of the type $P\wedge\neg P$) and paradoxes (of type $P\leftrightarrow \neg P$). How about in mathematics? Does the appearance of ...