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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motivation of additive inverse of a Dedekind cut set

My understanding behind motivation of additive inverse of a cut set is as follows : For example, for the rational number 2 the inverse is -2. Now 2 is represented by the set of rational numbers less ...
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455 views

Motivation behind introduction of measure theory

Is the motivation behind the introduction of measure theory the Lebesgue integral? In order to evaluate such an integral we need the length of each of the horizontal strip of width $h$. I have a ...
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1answer
295 views

Intuition behind the definition of Adjoint functors

I think of adjoint functors as some sort of inverses. So, the first part of the definition looks reasonable that there exists natural transformations $$\epsilon : FG \rightarrow 1_C$$ $$\eta : 1_D ...
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207 views

Visualization of immersed submanifold

I am trying to visualize the difference between immersed submanifold and embedded submanifold. At first, I thought that, for example, if I can embed manifold $M$ in $\mathbb{R}^4$ and if my friend can ...
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1k views

The orthogonal projection onto a plane - explanation

Could somebody explain, why orthogonal projection onto a plane with equation $x_1+x_2+x_3=0$ is given by $$y=(x_1,x_2,x_3)-\bigg( \frac{x_1+x_2+x_3}{3}\bigg)(1,1,1)$$ I don't understand, why we sum ...
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2answers
73 views

Understanding Continuity of Functions

I know that graphically a function $f(x)$ is said to be continuous in $[a,b]$ if there are no breaks in the curve for $f(x)$ in the interval $[a,b]$ I also know that by definition, a function $f(x)$ ...
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221 views

Orthogonal Coordinate Systems Intuition

I'd really love it if you could give some intuition on how to derive the $x$, $y$ & $z$ coordinates from all/any of the orthogonal coordinate systems in this list, how you think about, say, ...
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152 views

Integral and series convergence intuition

I have this problem I ran into during my studies to the upcoming exam: I don't feel I have the intuition of whether a series or an integral converges or not. What are the things I should look for when ...
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300 views

Yet Another Monty Hall Question - Please advise if alternative scenario proves the same principle

Okay, I'm very embarrassed that there are already 71 questions (based on search of "monty hall") and I'm going to post another one. I read the first 5 before succumbing to choice-overload. I'll try to ...
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1answer
164 views

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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1answer
195 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 ...
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388 views

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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290 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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3answers
619 views

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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0answers
96 views

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 $(\lambda-1)\in\mathbb{N}$, ...
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1answer
212 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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4answers
166 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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417 views

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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1answer
153 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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0answers
191 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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11answers
3k views

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
210 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 ...
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8answers
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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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1answer
102 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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66 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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3answers
863 views

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
59 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
1k 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
171 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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203 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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4answers
167 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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448 views

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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1answer
285 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 ...
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836 views

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
481 views

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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3answers
147 views

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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3answers
348 views

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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2answers
145 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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701 views

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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1answer
201 views

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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1answer
114 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 $$ ...
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1answer
105 views

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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1answer
369 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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135 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
283 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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1answer
167 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 ...
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0answers
65 views

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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1answer
492 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
153 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? ...