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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How Fourier transform(opeartor) behaves with the operation composition?

It is well-known that Fourier transform takes convolution into a point-wise multiplication, that is, $\widehat{f\ast g}= \hat{f}\cdot \hat{g}$, for $f, g \in L^{1}(\mathbb R).$ That is, bit roughly, ...
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11 views

Math Bases of Comparison and Association [on hold]

My question is about the cognitive phenomenon of intuitive pattern-matching or association by similarity / dissimilarity. Imagine a situation where a person has a particular experience, which might ...
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55 views

Interpretation of partial derivatives of vertical coordinate with respect to $x$ and time

My question is from my lecturers notes, this is what he wrote and I don't know what he is on about : What is a physical meaning of partial derivatives of $y(x,t)$? $y_x(x,t)$ is the rate of change of ...
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50 views

$1$ is not congruent because of Fermat's Last Theorem?

I would like someone to explain something I did not understand. I was reading a page called "nuking the mosquito" where they give very complex proofs for very simple results. The proof I want to talk ...
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1answer
21 views

Elements in the same coset and the Cayley Diagram

A question from Visual group theory, by Nathan Carter. In a Cayley diagram, if $aH$ is a coset of a subgroup $H$ of a group $G$ and $b$ belongs to $aH$, why is it that every node that can be reached ...
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65 views

Geometric idea behind equations of the form $|x-a|\pm|x-b|=c$

So let's say I want to solve $$|x-a|\pm|x-b|=c$$ Using the classic multiple cases approach, one can show that the solutions are given by $$x=\frac{a+b\pm c}2 $$ But how can one make sense of this ...
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Why does solving $\int \frac{v}{9.8-0.0025v^2}\mathrm{d}v=\int1{d}x$ for $v^2$ in terms of $x$ produce 2 completely different answers?

In this question $g=9.8$ (acceleration of free fall). You are also given that when $x=0$ $v=0$. My answer is $v^2=400g(1-e^\frac{x}{200})$. I obtained it by integrating both sides so that ...
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55 views

Intuition for the compactness of real projective space $\mathbb{R}\mathbb{P}^n$.

I want to have an intuition for why the $n$-dimensional real projective space defined as $$\mathbb{R}\mathbb{P}^n:=\mbox{set of 1-dimensional subspaces of }\mathbb{R}^{n+1}$$ is compact. I don't see ...
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1answer
37 views

What does local space of a given Banach space says intuitively?

We put, $\mathcal{D}(\mathbb R)=$ The space of $C^{\infty}-$ functions on $\mathbb R$ with compact support Example: For instance bump function is in $\mathcal{D}(\mathbb R)$ Let $E$ is a Banach ...
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26 views

The meaning of the connection between power spectral density and auto correlation

I know that if we have a signal $x(t)$, then its Fourier transform would be the signal in the frequency space, which I understand to be how much of each frequency exists in the x(t) signal. $ ...
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100 views

Show that $q^4+2pq^2 +p^2 = 2pq -(pq)^2 -1$ becomes $p^3+q^3+3pq-1=0$.

I know that these two are exactly the same equation but I can't seem to prove it. You are also given that $p+q=1$. This is a follow up from a similar question.
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64 views

By plugging $p=1-q$, into the $3$ equations show that $x=y=z$

By plugging $p=1-q$, into the 3 equations: $$\begin{cases} z=py+qx \\ x=pz+qy \\ y=px+qz \end{cases}$$ show that $\boxed{x=y=z}$ This is from the final part of question 7 in this STEP paper, and is ...
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1answer
31 views

Computing of the Gamma Function

I have stumbled upon Gamma functions when dealing with Gamma distributions on my studies with basic statistics. However, I have not understood how its computation expands factorials to real and ...
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46 views

How to determine value from willingness to pay?

I use the British pounds symbol instead of dollars because $ conflicts with Mathjax. Source: p 296, The Legal Analyst, Ward Farnsworth "... one time in a thousand we do lose the film; if you’re ...
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1answer
58 views

Series problem:

Parts i) & ii) I can solve. For part iii) I get $z=py+qx$ [For $n=0$] $x=pz+qy$ [For $n=1$] $y=px+qz$ [For $n=2$] leading to $(1-pq)x=(q^2+p)z$ [1] $(1-pq)z=(q^2+p)y$ [2] ...
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65 views

Sequential Algebraic Problem:

The first part i) I can do. For part ii) this is how far I can get: If n is odd then $y=px+qy$ If n is even then $x=py+qx$ After some rearranging i end up with $y(1-q)=px$ & $x(1-q)=py$ and ...
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53 views

Explaining multiplication of fractions

The best way I've been able to describe multiplication is as $$ a\times b = \sum^a_{i=1} b$$ But my definition does not account for things such as $2.99792458\times8.987551787$ and ...
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1answer
33 views

Everyday life examples of hyperbolic rotations

I'm trying to find some intuition in Lorentz transformations. I understand that they are basically rotations by imaginary angle of vector of the form $\{ict,x\}$ (for $1+1$ space-time dimensions), and ...
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1answer
18 views

Understanding Inequalities.

If $(\displaystyle\frac{2(a-1)}{a+1}\lt\ln a\lt\frac{2(a-1)}{2+\ln a})$ equals $(\displaystyle\frac{2(a-1)}{a+1}\lt\ln a \lt -1 + \sqrt{2a-1})$. Does that mean that $-1 + \sqrt{2a-1}$ = ...
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1answer
22 views

Show that $ \frac{2+\ln a }{2}\lt\frac{a-1}{\ln a} \lt \frac{1+a}{2}$ becomes $ \frac{2(a-1)}{a+1}\lt\ln a \lt -1 + \sqrt{2a-1}$

Show that $\displaystyle \frac{2+\ln a }{2}\lt\frac{a-1}{\ln a} \lt \frac{1+a}{2}$ becomes $\displaystyle \frac{2(a-1)}{a+1}\lt\ln a \lt -1 + \sqrt{2a-1}$ The closest I can get is $$ ...
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31 views

Trigonometric series problem: finding a second valid solution.

Given that I can do part of this question so here goes: Substituting $\theta=\frac{1\pi}{11}$ into LHS of given expression gives $$\cos\frac{1\pi}{11} + \cos\frac{2\pi}{11} + \cos\frac{3\pi}{11} ...
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28 views

Problem involving summing exponential series:

I can show the first part (i) (a), but the second part (b) i think it should be $S=\infty$ since the denominator is zero with that value of $\theta$. However, this is not the answer, any ideas? ...
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1answer
47 views

Using De-moivres to solve the following problem:

Part (i) I can solve and understand that the solutions are $Z=e^\frac{2ki\pi}{5}$ for $k = 0,1,2,3,4$ Its the part (ii) I cannot understand. Could someone kindly give me a ...
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1answer
35 views

Using De Moivre's theorem with relation to the argument of a complex number

Given that $Z^4 = 64(\cos\pi+ i\sin\pi)= 64(-1+0i) = -64$ I understand that the argument [$arg(Z^4)$] is $\pi$, now if instead given the form $Z^4 =64(-1+0i)$ and I desired to find the argument ...
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73 views

Distinction between nowhere monotone and nowhere differentiable

It is known that all functions that are continuous and nowhere differentiable are also nowhere monotone but that there is a function that is everywhere differentiable but nowhere monotone. I have ...
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1answer
57 views

Intuition concerning Riemann Sums

I have just started learning integrals, and I want to know the following: In the definition of a riemann integral, it states that the interval that the integral is to be evaluated, is partitioned ...
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77 views

The semidirect product as a deformation of the direct product

The way I think of the semidirect product is as a "deformation" of the direct product. Is there a way of making this intuition precise? Perhaps using some certain (co-) homology theory of groups?
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21 views

What nontrivial operations exhibit $\text{op}(f(x)) + \text{op}(f(x+1)) = \text{op}(f(x) + f(x+1))$?

What nontrivial operations exhibit $\text{op}(f(x)) + \text{op}(f(x+1)) = \text{op}(f(x) + f(x+1))$? For example, I know that summation, integration, and their inverses all exhibit this property. To ...
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160 views

Why is a raised to the power of Zero is 1? [duplicate]

Why is $a^0=1$ $\forall a \in Z, a\neq0$. I understand $2^4=2\cdot2\cdot2\cdot2$ How can I express $a^0$. I am serious about the practical proof of this
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37 views

Show elementarily that $\lim_{R\to\infty}\int_{\Gamma_1} \frac{e^{iz}}{z} = 0$

Context: I am trying to show that $\int_0^\infty x^{-1}\sin x dx = \frac{\pi}{2}$ using complex analysis, by first integrating $\oint_{\Gamma} z^{-1}e^{iz}$, where $\Gamma$ is a closed contour ...
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Why is it called a primitive root?

I am looking for a paper or reference that explains why primitive roots are called primitive roots. I know what they are but was wondering if there was a reason?
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4answers
158 views

What if we removed all the irrational numbers from the real number line? [closed]

(1) Imagine drawing the real number line and tippexing out all the irrational numbers. What would the resulting shape look like- would there even be a line? (2) And what about if we ...
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94 views

Geometric Intuition for Dihedral Group Automorphisms

I noticed the other day that the automorphism group of the dihedral group $D_{2n}$ (of order $2n$) is $\operatorname{Aff}(\mathbb Z/n\mathbb Z)$, the group of affine transformations of the $\mathbb ...
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Gambling interpretation of conditional probability

In Billingsley, when defining conditional probability the following property has been given a gambling interpretation : $$ \int_G P[A||\mathscr{G}]dP = P(A \cap G), G \in \mathscr{G} $$ where at ...
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Single variable justification for the multivariate chain rule.

I $\def\d{\mathrm d}\def\p{\partial}$am going to ask everyone to switch their paradigms to that of the real line. I am looking for a "lowbrow" explanation of the following phenomena. I am talking ...
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Why isn't the Cantor Set contradictory?

So you start with a 1-dimensional stick, remove the middle third of it, leaving 2 pieces. From each of these 2 pieces, remove the middle third. Etc. Whatever is left at the end of infinitely many ...
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3answers
210 views

Expressing the probability density function of $Ax$ in terms of the pdf of $x$

I understand that, for example, you might have a density function which measures the probability of observing an outcome in a certain interval measured in feet, but someone wishes to use meters ...
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113 views

Getting an intuitive feel for induced representations

I'm reading about induced representations for research. Particularly, I'm trying to get a firm grasp on the finite group case before venturing on to the locally compact case. I've been looking at ...
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37 views

Derive $f(Ax)=f(x)g(A)$: the property for scale invariance.

This is part of a proof for the following question: Show that if a probability density function $f(x)$ with $x>0$ is scale invariant then $f(Ax)=\frac{1}Af(x)$ where $A$ is a real constant. ...
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35 views

Differentiable curves that are not smooth

We call a curve admitting a parameterization $t\to z(t)$, $t\in[0,1]$ differentiable if the vector function $z$ is differentiable. We call the curve smooth if it is differentiable and its derivative ...
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Intuitive interpretation of $\frac{\partial S(a,t)}{\partial t} = -\frac{\partial S(a,t)}{\partial a}$

I'm trying to visualize what the following equation is saying: $$\frac{\partial S(a,t)}{\partial t} = -\frac{\partial S(a,t)}{\partial a}$$ where $S$ is a probability-density, but I think you can ...
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1answer
26 views

Understanding arguments to functions in $\mathbb R^n$

Example of two theorems I have problems with: Mean value theorem: $U\subseteq\mathbb R^n$ open, $f:U\to\mathbb R^m$continuously differentiable, $x\in U$, $\xi\in\mathbb R^n$ such that $x+t\xi\in U$ ...
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98 views

$2^{1/4} \times 4^{1/8} \times 8^{1/16} \times 16^{1/32} \times \ldots\to2$

$2^{1/4} \times 4^{1/8} \times 8^{1/16} \times 16^{1/32} \times \ldots\to2$ How can I explain this to a school student who doesn't know what a limit is?
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104 views

Why do we denote $S^1$ for the the unit circle and $S^2$ for unit sphere?

Maybe a quite easy question. Why is $S^1$ the unit circle and $S^2$ is the unit sphere? Also why is $S^1\times S^1$ a torus? It does not seem that they have anything in common, do they?
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1answer
29 views

Intuition behind prism operators to prove homotopy invariance of homology

I'm trying to understand the proof of homotopy invariance of induced maps on homology. However, I do not really understand the intuition behind this proof and especially what the prism operators (as ...
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90 views

Can a statement in FOL be equivalent to two separate independent statements?

This may seem like a dumb question, and it certainly seems dumb to me asking it, but I'm running into a contradiction. I'm looking at the problem of finding a statement $\phi$ such that $\psi$ and ...
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1answer
26 views

Meaning of “Identify a set with another set” in group theory

There is a exercise problem that asks "Identify a set with another set ". I don't understand what I should do. Do I need to establish a bijection between them? Thanks EDIT-I: Actual question: G is a ...
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1answer
86 views

Arrows-only implication & disjunction in $\mathbf{Set}.$

Just before the truth-arrows in a topos subsection of Goldblatt's "Topoi: A Categorial Analysis of logic," descriptions of the truth functions $\Rightarrow$ and $\smallsmile$ are given in ...
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193 views

The physical meaning of ${n \choose k} = {n \choose n-k}$.

They say that $${n \choose k}={n \choose n-k}.$$ Can someone explain its physical meaning? Among many problems that use this proof, here is an example: The english alphabet has $26$ letters ...
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715 views

What are the most important functions every mathematician should know? [closed]

I am an undergrad in math and was wondering, what are for you the most important functions every mathematician should know? At the moment I think ...