Mathematical intuition is the instinctive impression regarding mathematical ideas which originate naturally without regard to formal mathematical proofs. It may or may not stem from a cognitive rational process.

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Real number construction : additive inverse of a cutset

For a given cut set $\alpha$, what is the intuition behind considering the set of all such $p$'s such that some number less than $-p$ does not belong $\alpha$ as the inverse of $\alpha$ ? i.e. $\...
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331 views

Intuitive interpretation of the Fourier transform of the derivatives of a function

Let $f$ be a smooth function, $f(t)\in C^\infty(\mathbb R)$, and $F$ be its Fourier transform $$F(\omega):=\mathcal F f\,(\omega)\,=\,\int_{-\infty}^\infty \mathrm e^{-\mathrm i\omega t}f(t)\mathrm{d}...
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Why should $|2^\mathbb{N}|>|\mathbb{N}^2|$ be true?

I've been thinking a bit about infinite things lately, and this question I had wondered about came back to me. One of the classic expository demonstrations of Cantor's work is the two equally ...
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479 views

Taking a Fourier transform of Taylor series

My (naive) question is whether it is possible to take the Fourier transform of a Taylor series? Could one use multiplication with $\delta$ to get the function sampled at the point of expansion and ...
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Understanding Discrete and Fast Fourier Transform intuitively

I have done both of these in my math courses, but without understanding what they actually are intuitively. I would be very much grateful if you could give me an intuitive explanation of them.
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What is the intuition behind $P(X={\lambda})\ {\equiv}\ P(X={\lambda}-1)$ for a Poisson distribution

Where $X_\lambda$ is a Poisson random variable with mean $\lambda$,$$P(X_\lambda=k\in\mathbb{N}) = \frac{\lambda^{k}e^{-\lambda}}{k!} $$ When it happens that $(\lambda-1)\in\mathbb{N}$, $P(X={\lambda})...
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309 views

Real number construction : prove that $Q$ is a subfield of $R$

I am slightly confused about the proof presented in Rudin. It says that the ordered field $Q$ is isomorphic to the ordered field $Q*$ whose elements are the rational cuts. It is this identification of ...
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181 views

A Basic question on intuition of rational cut set in the construction of real numbers

The intuition for cuts presumably comes from the standard experience of approximation by terminating decimals. For example, we can approximate $\sqrt{2}$ by the sequence $1,1.4,1.41,1.414,1.4142, \...
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Motivation behind Dedekind's cut set

I want to know the motivation behind Dedekind's real number construction. The motivation of such properties of the cut sets is not clear to me. BTW, I am new to real analysis and just have started ...
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293 views

cauchy schwarz equality: difference in proving style for linear algebra and expectation version

I am interested in proving the following sub version of Cauchy Schawrz equality. 1) LA version : If $x$ and $y$ are two real vectors and the following holds $$<x,y> = ||x||.||y||$$ then $x$ ...
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297 views

What is the intuition of conjugacy classes?

How can I fully understand what are conjugacy classes are in groups? I know the definition, that $a$ and $b$ are conjugate if $gag^{-1}=b$ for some $g\in G$. But what is the intuition? Using a ...
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Why is $\frac{1}{\frac{1}{X}}=X$?

Can someone help me understand in basic terms why $$\frac{1}{\frac{1}{X}} = X$$ And my book says that "to simplify the reciprocal of a fraction, invert the fraction"...I don't get this because isn't ...
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2answers
230 views

If $G$ is a finite group, $H$ is a subgroup of $G$, and $H\cong Z(G)$, can we conclude that $H=Z(G) $?

If $G$ is a finite group, $H$ is a subgroup of $G$, and $H\cong Z(G)$, can we conclude that $H=Z(G) $? I learnt that if two subgroups are isomorphic then it's not true that they act in the same way ...
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Intuitive meaning of Exact Sequence

I'm currently learning about exact sequences in grad sch Algebra I course, but I really can't get the intuitive picture of the concept and why it is important at all. Can anyone explain them for me? ...
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166 views

smooth approximate parameterization to polygonal boundary

I can "almost" parameterize the boundary of a square using $${\bf r}(t) = (\cos t)^{1/p} {\bf i} + (\sin t)^{1/p} {\bf j},$$ $0\leq t\leq 2 \pi$, and $p$ is odd. This parameterization is smooth (or at ...
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83 views

How much can we “reduce” this polynomial division?

Let's say we start with a univariate polynomial, $p(x)$: $$p(x) = x^n-1$$ We can then divide by another polynomial; for instance $q(x)$. For example, if $p(x) = x^6-1$ and $q(x) = x^2-x+1$ we have: ...
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Can this pattern of denominators be expanded?

We can take a polynomial, $p(x)=\left(c_0 + c_1 x^1 + c_2 x^2\right)$, and get a repeating pattern of it by: $$\frac{p(x)}{1-x}=c_0+(c_0+c_1)x+(c_0+c_1+c_2)x^2 + \dots + (c_0+c_1+c_2)x^m + \dots$$ .....
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About NOT elimination/introduction and RAA rules on Natural Deduction

Can somebody explain the $\neg$-elimination rule in natural deduction?. Searching on books and the web, I found different definitions for it. For example, in my logic I course, the rule is: $A, \...
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2answers
79 views

Constraint satisfaction problem - Arc consistency

The Constraint satisfaction problem (CSP) is roughly speaking a formalism that defines a finite set of relations over a domain. The relations are simply defined by enlisting elements in certain ...
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4answers
2k views

An intuitive approach to the Jordan Normal form.

I want to understand the meaning behind the Jordan Normal form, as I think this is crucial for a Mathematician. As far as I understand this the idea is to get the closest representation of an ...
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4answers
209 views

What is the use of, and intuition behind, writing $\frac{d^2}{dx^2}$ for the second derivative?

Is it possible to take a second derivative without taking the first derivative before? Why do we multiply the $d$ and $dx$ operators? Like, does $\dfrac{d^2}{dx^2}$ really mean $\dfrac{d}{dx} \cdot \...
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220 views

What happens after the cardinality $\mathfrak{c}$?

While having measure theory this year the following came in my mind: When we go from finite objects to infinite we "lose" a lot of properties. For example the summation isn't well defined ...
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255 views

A basic question on the derivative of a continuous function

Is the following condition necessary for the existence of derivative of a continuous function at point $x$: $$\lim_{h \to 0^+}\frac{f(x+h)-f(x)}{h} = \lim_{h \to 0^-}\frac{f(x)-f(x+h)}{h}$$
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A basic intuition on a probability problem

Two players take turns shooting at a target, with each shot by player $i$ hitting the target with probability $p_i$, $i=1,2$. Shooting ends when two consecutive shots hit the target. Let $\mu_i$ ...
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538 views

Intuition of law of iterated logarithm

Let $X_i$ be iid random variables with $EX_i = 0$ and $Var X_i=1$ and $S_n=X_1+\cdots+X_n$. Then the law of the iterated logarithm says almost everywhere we have $$\limsup_{n\to\infty}\frac{S_n}{\...
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Intuition of Addition Formula for Sine and Cosine

The proof of two angles for sine function is derived using $$\sin(A+B)=\sin A\cos B+\sin B\cos A$$ and $$\cos(A+B)=\cos A\cos B-\sin A\sin B$$ for cosine function. I know how to derive both of the ...
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3answers
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What is the relation between vectors in physics and algebra?

Vector math is something I find very interesting. However, we have never been told the link between vectors in physics (usually represented as arrows, e.g. a force vector) and in algebra (e.g. ...
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Why the $\nabla f(x)$ in the direction orthogonal to $f(x)$?

Why the $\nabla f$ in the direction orthogonal to $f$? I can't understand it intuitively. Is it because of the definition of gradient?
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Compact subsets of function spaces, geometry

The subset is called compact when every open cover contains a finite subcover. In Euclidean spaces, it is easy to visualize this by imagining some open ball that contains this set, thinking about the ...
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How to write $\pi$ as a set in ZF?

I know that from ZF we can construct some sets in a beautiful form obtaining the desired properties that we expect to have these sets. In ZF all is a set (including numbers, elements, functions, ...
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158 views

Does $P(A\cap B) + P(A\cap B^c) = P(A)$?

Based purely on intuition, it would seem that the following statement is true, when thinking of the events as sets: $$P(A\cap B) + P(A\cap B^c) = P(A)$$ However, I am not sure if this is true, and ...
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Why is the Derangement Probability so Close to $\frac{1}{e}$?

A derangement is a permutation of $\sigma$ of $\{1,2,3,\dots,n\}$ such that $\sigma(i) \neq i$ for every $i$. A common application of inclusion/exclusion in undergraduate combinatorics and ...
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What is the physical meaning of fractional calculus?

What is the physical meaning of the fractional integral and fractional derivative? And many researchers deal with the fractional boundary value problems, and what is the physical background? What ...
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161 views

Best way to write the characteristic polynomial

In linear algebra I've seen that there are two major different ways to write the characteristic polynomial of a mapping $f$: As $$ p_{f}=\pm\left(T-\lambda_{1}\right)^{m_{1}}\cdot\ldots\cdot\left(T-\...
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Geometric intuition behind the Uniform Boundedness Principle

Is there a way to visualize why the Uniform Boundedness Principle should be true? I understand the statement of the theorem but I'm having a hard time seeing a picture of it in my head.
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582 views

Prove that any circuit contains a cycle

This is a practice question (not HW) Prove that any circuit in a graph must contain a cycle AND that any circuit that is not a cycle contains at least two cycles. Note : This is for a first course ...
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167 views

What exactly is the complex plane, and how is it useful?

A lot of functions are defined on the complex plane, like the Gamma function: the Lambert W function, etc. But I have no idea about what the complex plane means and how it's useful, or just ...
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1answer
435 views

Geometric representation of product rule?

At time 1:06 of this video by minutephysics, there is a geometric representation of the product rule: However, I don't understand how the sums of the areas of those thin strips represent $d(u\cdot ...
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200 views

How to understand $\frac{d}{dt}\{(\exp(tX))_*(Y)\}|_{t=0}=[X,Y]$?

Let $G$ be a Lie group on which $X$ and $Y$ are two vector fields. Let $G\xrightarrow{\exp(tX)} G$ be the (Lie theory) exponential map corresponding to $X$. Then of fundamental importance is \begin{...
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Intuition behind criterion for an irreducible Markov chain to be transient

I have been looking over my notes for Markov chains, and I have come across the following: Theorem: An irreducible Markov chain is transient iff for some state $i$ there exists a nonzero vector $y$ ...
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868 views

How to improve mathematical creativity?

To introduce myself: I'm an undergraduate mathematics student in Germany. Currently I'm studying in the second semester and until now I'm doing well, but I still got the feeling that my ability to ...
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1answer
207 views

Volume of a hypersphere

We know that the area of a circle (2-D) =$\pi r^{2}$ and the volume of a sphere (3-D)= $\dfrac{4}{3}\pi r^{3}$. Question:What is the "volume"(or whatever that is called) of a n-dimensional sphere? ...
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$\pi$ from the unit circle, $\sqrt 2$ from the unit square but what about $e$? [duplicate]

If one wants to introduce $\pi$ to a not mathematically savvy person, the unit circle would be a good choice. The unit square would be the way to go for $\sqrt 2$. But what about $e$? I've reviewed ...
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627 views

It is possible to define our intuitive notion for probability in subsets of $[0,1]$

I've always heard and read the sentence: If you pick a real number $x\in[0,1]$ at random, the probability to obtain a rational number is $0$. What is the meaning for that? Is this the "real" ...
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60 views

What to take from representation of $S_d$?

I am reading about group representations, and books I read all contain the representation theory for symmetric groups $S_d$. However none of them presents the material in a friendly way. After reading ...
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951 views

the role of logic in math and education

My question is somewhat related to this discussion: Is Mathematics one big tautology? I have a computer science background and I have always approached math from the logic point of view (formalism?)....
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1answer
116 views

How information works?

I am really confused after reading wikipedia... What I don't get is how can something "bring" information, and in mathematics, how a mathematical object (like a set) can "have" information. For ...
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2answers
239 views

Logical explanation of Euler's formula

This question is a about (if not proving) at least guessing the Euler's formula. I don't want the proof using the infinite sums. We can guess by logic that for example that the equation $x^2+1=\sqrt{...
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847 views

Algebraic geometry in representation theory?

I heard that today algebraic geometry plays some significant role in representation theory, which is a little surprising because when I learnt representation theory it is basically algebra, topology, ...
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The thought process of derivatives explained (intermediate calculus) “derivatives with respect to what”

My intention here is to contribute, if there is a problem with my solution or explanation--if it is wrong--please add a comment and don't just down vote. My answer represents my understanding and I ...