# Tagged Questions

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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### Intuitive understanding of the “Multiplication Rule”?

I apologize in advance that this question has a long set-up. In the set up I am presenting how I currently understand the material, and the actual question is if my understanding is correct and ...
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### Dividing a plane with lines

A while back, one of my friends challenged me to find out how many regions I can divide a plane into given $n$ lines. For instance: He also told me that the formula to find the maximum number of ...
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### Fourier Analysis and its applications [duplicate]

My question has two parts: $1)$ Could anyone explain in simple terms what a Fourier Transform is? $2)$ What are some of the applications of Fourier Analysis in the field of high school mathematics?
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### How does the multiplication theorem correspond to the concept of intersection?

Given two events A and B defined on a sample space S. S : Rolling a six-sided dice A : Getting an even number B : Getting a number ≥ 4 In an elementary sense (the experiment being carried out once), ...
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### Integral of a function defined on dense subset

[Edited] For a real-valued continuous function $f$ defined on a Lebesgue measurable dense subset of $[0,2]$, consider an integral $$\int_{[0,1]}\frac{f(s)}{\sqrt{1-s}}ds.$$ My question is whether ...
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### What does the derivative of a function at a point describe? [duplicate]

I understand that the derivative of a function $f$ at a point $x=x_{0}$ is defined as the limit $$f'(x_{0})=\lim_{\Delta x\rightarrow 0}\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}$$ where $\Delta x$ ...
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### Intuition behind Fourier and Hilbert transform

In these days, I am studying a little bit of Fourier analysis and in particular Fourier series and Fourier/Hilbert transforms. Now, I am confident with the mathematical definitions and all the ...
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### Homotopy Equivalence intuition

Can somebody tell me intuitively what does it mean geometrically when we say two spaces are homotopy equivalent ? I understand the technincal definition.
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### Why should I expect the product of sum of four squares to be a sum of four squares? How did Euler come up with it?

Euler discovered the lovely identity shown here: https://en.wikipedia.org/wiki/Euler%27s_four-square_identity Is there a natural reason to assume a solution can be found? Any intuition? I saw that ...
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### Extension theorem from Guillemin-Pollack, motivated sketch of proof?

Let $W$ be a compact, connected, oriented $k + 1$ dimensional manifold with boundary, and let $f: \partial W \to S^k$ be a smooth map. Could anybody sketch with good motivation that $f$ extends to a ...
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### Log-concave function changes when scalar is added

A function $f$ is log-concave if $\log(f)$ is concave. Intuitively, one might guess that adding a scalar to a function would not affect properties like concavity, log-concavity, quasi-concavity etc., ...
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### Basic probability. Is the textbook wrong?

A and B are independent events such that P(A)=0.7, P(B)=k, P(A U B)=0.8 Find the value of k. Solution given: P(A U B) = P(A) + P(B) - P(A ∩ B).....(i) [Addition Rule] P(A ∩ B) = P(A).P(B).....(...