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1answer
26 views

Investigation Problem to challenge mathematical reasoning

I am a sophomore in college and currently enrolled in a Upper Division Problem Solving class. I was assigned a final project in which I need to come up with some sort of mathematical investigation ...
2
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2answers
24 views

Soft question about Lie Groups and 3D rotation

Let $R(\phi, \boldsymbol{n})$ be a member of Lie Group SO(3). According to Wikipedia If $R(\phi, \boldsymbol{n})$ denotes a counter-clockwise 3D rotation through an angle $\phi$ about the axis ...
4
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4answers
103 views

Should we or should we not take $1$ as a prime number? [duplicate]

I think I know that there were times in the past when it was convenient to look at a number $1$ as a prime number, and, as far as I can remember, even then it was dependent on who we ask is it prime ...
0
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1answer
29 views

Computing the “Mean Value” of a Point Sample From an Arbitrary Manifold

A friend of mine noticed that taking the "mean" of two points on the circle isn't as easy as just computing the arithmetic mean of their arguments: If one point has argument $-3.13$ radians and one ...
2
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0answers
27 views

Tricky divergent binomial expansions?

The binomial expansion of $(a+b)^n$, where $n\notin\mathbb{N}$, is given as $$(a+b)^n=a^n+\frac{n}{1!}a^{n-1}b+\frac{n(n-1)}{2!}a^{n-2}b^2+\cdots$$ In some situations, we can find the result of a ...
2
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0answers
52 views

What is the purpose of homomorphisms?

I know that a mapping $\phi:A\to B$ is a homomorphism provided that $$\phi(A*B)=\phi(A)\times\phi(B)$$ where $*$ and $\times$ are two operators on the algebraic structures $A$ and $B$ respectively. In ...
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0answers
18 views

Video Lectures on Multi variable Calculus in n dimensions

I am teaching myself multi variable calculus from various resources on the internet. I have completed the famous 18.02 offered at MIT. This is an introductory course on Multi variable Calculus and is ...
2
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0answers
61 views

What should I have learnt as an undergraduate? [closed]

I am getting my bachelor's degree this summer, and I feel like I don't know as much as I should in several fields in math (geometry, functional analysis, measure theory, abstract and linear algebra, ...
8
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1answer
419 views

What is the current state of formalized mathematics?

Russell and Whitehead famously tried to actually create and use a formal system to explicitly develop formal mathematics in their work, "Principia Mathematica." Much more recently, with the aid of ...
2
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2answers
85 views

About the Words “recursion” and “recursive”

According to Wikipedia, Recursion is the process of repeating items in a self-similar way. On the other hand, the word "recursive" is an adjective and is often used as a synonym of "computable" when ...
3
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3answers
50 views

Does the PNT establish a connection between primes and the logarithm?

The prime number theorem states that $$\pi(x) \sim \frac x {\ln(x)}$$ Morally, this seems to suggest that there is a fundamental connection between primes and the natural logarithm. But since we're ...
3
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1answer
118 views

Vladimir Blinovsky's Union-Closed Sets Conjecture Proof

Recently, Vladimir Blinovsky published an article (http://arxiv.org/pdf/1507.01270v6.pdf) claiming that he proved the union-closed sets conjecture (https://en.wikipedia.org/wiki/Union-...
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0answers
60 views

Example of a collection that is not a set?

Just out of curiosity (the subject is way out of my league at the moment) I have been reading a little about set theory, and I came across Russel's paradox. From what I understood, Russel's paradox ...
2
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0answers
39 views

Abelianization and analysis of Fundamental Groups

I am working through Hatcher on my own, and currently doing problem $9$ on $p53$. This problem brings up the strategy of abelianization of groups to solve problems of fundamental groups and ...
1
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1answer
147 views

Which numbers are necessary?

The Greeks were initially convinced that all numbers were rational until upon pain of contradiction were forced to accept that $\sqrt{2}$ was irrational and needed to be included in our number system ...
2
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0answers
44 views

What is the motivation for $l^p$ space?

I'm studying $L^p$,$l^p$ spaces recently, but I don't see the motivation for this. The only application I know is that $l^2$ can be used to characterize Hilbert spaces up to dimension. Is $L^p$ spaces ...
8
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0answers
156 views

Open mathematical questions for which Numerical Search promising

I am fascinated by mathematical conjectures, especially those that are believed to be false. Now I wonder, are there some mathematical conjectures for which the numerical search would be promising? ...
3
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3answers
58 views

What is the motivation for normed division algebras?

The famous Hurwitz theorem states that the only normed division algebras are $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$. What is some good pedagogical motivation why we should think ...
23
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7answers
2k views

Can you use both sides of an equation to prove equality?

For example: $\color{red}{\text{Show that}}$$$\color{red}{\frac{4\cos(2x)}{1+\cos(2x)}=4-2\sec^2(x)}$$ In high school my maths teacher told me To prove ...
6
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0answers
102 views

Abstraction and Genaralization

This is a (bit funny) ultra-soft question regarded to a type of thinking that is puzzling me. Suppose $a(p_{1}, p_{2}, p_{3}, p_{4}, ... , p_{n}, Q)$ denote: from the items $p_{i}$, find a common ...
0
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1answer
68 views

What is the name of people who do algebra? [closed]

People who do topology is called topologists, people who do analysis is called analysts, people who do geometry is called geometers, then how about algebra?
7
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1answer
162 views

Any math competitions dedicated to calculations by hand (on a college level)?

Most of the people consider hand calculations the thing of the past. However, I recently started thinking about it and there are many interesting ways to do basic arithmetics on large numbers, ...
2
votes
1answer
25 views

Intuition of the distribution of the minimum of exponential random variables

Let $X,Y$ be two independent random variables with exponential distribution with parameters $a$ and $b$ respectively. It is known (see e.g. here) that $Z:=\min\{X,Y\}$ is exponential distributed with ...
1
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2answers
90 views

Product of negative numbers [duplicate]

Why is a negative number multiplied by a negative number a positive number? I'm trying to know what does multiplying by a negative number mean. If you think of multiplication as a "groups of" ($3 \...
1
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0answers
28 views

Is there a mathematical treatment of continuously (or smoothly) deforming surfaces, or, in general, continuously deforming manifolds?

I'm wondering if there's been any attempts to construct formal mathematical descriptions of deforming surfaces, whether evolving to some steady state configuration or reacting to some pulse, force, or ...
40
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5answers
4k views

Why do we study real numbers?

I apologize if this is a somewhat naive question, but is there any particular reason mathematicians disproportionately study the field $\mathbb{R}$ and its subsets (as opposed to any other algebraic ...
2
votes
1answer
36 views

Is there an agreed upon convention for naming ZFC+Large Cardinal Axioms?

Is there an agreed upon convention in general for what to name ZFC+[Large Cardinal Axiom]? Or would one have to state explicitly which axiom was being added? To explain what I mean, note that anyone ...
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0answers
20 views

Limit Terminology

From the $\epsilon-\delta$ definition of a limit, we can see that any limit can be broken up into two "one-sided limits". These "one-sided" limits are simple cases that arise as a consequence of the ...
2
votes
2answers
108 views

Good books on integrals [duplicate]

I'm a math student at the sixth semester and I've had my courses in calculus and complex analysis. I'm able to solve integrals with the usual techniques, e.g. with substitution. However, whenever I am ...
-2
votes
3answers
43 views

ordinary differential equation project suggestion [closed]

My professor asked to write a project on ODE just to experience on how to write projects. It need not be a research project. Being in second rate school from third world country, we never did those ...
0
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0answers
33 views

Necessity of Differential Forms

All the undergraduate and graduate texts on analysis introduce Differential and integral calculus (I will assume this introduction of basic calculus/analysis). Among them, some books also introduce ...
8
votes
1answer
95 views

Why is complex analysis so nice? And how is it connected/motivating for algebraic topology?

This is very much a soft question, but after seeing Cauchy's integral formula in lecture today I was really struck by how neat complex analysis is. I don't understand how all of these amazing analytic ...
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0answers
21 views

A illuminating theorem on general modules over a ring

I'm going to do a one hour presentation on modules, free modules and projective modules over a ring. While most of my motivation for studying them comes from my interest in starting homological ...
0
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1answer
53 views

Infinite Lie algebra

It is well known fact that the finite dimensional Lie algebra will always be closed under commutation relation. But I have doubt about infinite Lie algebra. In some of the case it is closed under ...
1
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1answer
76 views

What exactly is a “chaotic” sequence?

In response to this question, I was told that a certain sequence $a_n$ is "chaotic" and "wandering." The particular sequence in question is defined by $a_0 = z, z \in \mathbb{C}$ and $a_{n+1} = {a_0}^{...
1
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0answers
57 views

Finding a problem book in algebraic topology

I simply need book with problems solved with greatest explanation possible. I know about Hatcher and have a great lecturer, so I do not need theory. I need problems solved in detail.
2
votes
1answer
28 views

Reference request: examples where probabilistic methods make theoretical contributions to deterministic fields?

This idea came up during a discussion on whether you need randomness for the concept of probability to be valid, and if not, what is needed. I argued that having a stationary, recurrent sequence, is ...
0
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2answers
58 views

Why are there no “continuous maps” in algebra. [closed]

Or maps that behave similarly? Sorry if this is a strange question.
1
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0answers
20 views

Hirota's Bilinear Form

My query is related to conversion of PDE into bilinear form using some transformation. The detailed description about bilinear forms can be seen in book of Yoshimasa Matsuno . The bilinear operator ...
2
votes
0answers
47 views

When is it appropriate to write “Then it follows”

I am reading a proof, and before the proof fully finishes, the author writes "Then it follows [the statement we are trying to prove] is true" I have been spending the last three hours justifying the ...
1
vote
1answer
105 views

create interest in Maths to my son [closed]

I hope this is the right place to ask this question : my children are presently studying in Class 8 & Class 4. Elder child studying in Class 8 says he likes Physics, Chemistry, Astronomy but does ...
2
votes
1answer
39 views

What Similarity of Matrices really mean?

If matrices are similar then what in layman language it would imply? Are there some properties of matrices which we would expect to be similar?
0
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1answer
39 views

What is a good criterion to quickly show a map is or is not closed?

Definition: Let $X$ and $Y$ be topological spaces. A function $f: X \rightarrow Y$ is closed (respectively, open) if, whenever $A \subseteq X$ is closed (resp. open) in $X$, then $f(A)$ is closed (...
0
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0answers
20 views

$Y$ is a difference of a factor of $X$

To say $Y$ differs by a factor of $X$ means $dY = X$, or equivalently $Y_2 - Y_1 = X$ I believe. But this does not make sense to me, as a factor is what multiplies another number, e.g. $2$ is a ...
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2answers
34 views

General results on the change of the parity of a number by repeatedly dividing by 2 [closed]

I know this question may seem open, but I'm a bit interested in figuring out, getting some ideas, or at least getting some sources on how the parity of a number is affected by repeatedly diving it by ...
2
votes
4answers
215 views

Why do we use degrees? [closed]

I see a lot of people who ask why we use radians instead of degrees. But why do we use degrees instead of radians. In the cases we use degrees instead of radians, what convenience does it bring? The ...
0
votes
1answer
73 views

Interpretation help: Showing that Riemann Hypothesis holds “almost surely”

I was perusing this textbook on algorithmic number theory, where I came across this page where they appear to prove that the Riemann Hypothesis holds almost surely. This seems like an odd statement ...
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1answer
27 views

Functional analysis as a prerequisite [closed]

Can someone give me an example of a mathematical field in which functional analysis is a prerequisite?!
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0answers
45 views

Is PA the most common foundation for arithmetic?

Are Peano Axioms the most common and widely accepted axiomatization of arithmetic, just as ZFC is the most common and widely accepted foundation for all of mathematics?
1
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1answer
37 views

Why are alternating divergent series generally easier to evaluate? [closed]

Why is it that alternating divergent series tend to be easier to evaluate or that there are more ways to evaluate them? Is there a particular reason for the difficulty to evaluate series that don't ...