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 explanation of variance and moment in Probability

While I understand the intuition behind expectation, I don't really understand the meaning of variance and moment. What is a good way to think of those two terms?
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Relationship between torsion modules and topology

I was reviewing my class notes and found the following: "The name 'torsion' comes from topology and refers to spaces that are twisted, ex. Möbius band" In our notes we used the following definition ...
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Motivation behind standard deviation?

Let's take the numbers 0-10. Their mean is 5, and the individual deviations from 5 are -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 And so the average (magnitude of) ...
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Example: Function sequence uniformly converges, its derivatives don't

Could anyone give an example of a sequence of differentiable (real) functions that uniformly converges to a differentiable function, but the derivatives of which don't converge to the derivative of ...
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A “geometrical” representation for Ramsey's theorem

The [infinite] Ramsey theorem states that Let $n$ and $k$ be natural numbers. Every partition $\{X_1,\ldots ,X_k\}$ of $[\omega]^n$ into $k$ pieces has an infinite homogeneous set. Equivalently, ...
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The intuition behind generalized eigenvectors

An ordinary eigenvector can be viewed as a vector on which the operator acts by only stretching (without rotating) it. Is there a similar intuition behind generalized eigenvectors? EDIT: By ...
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359 views

Is this interpretation of Stieltjes integration correct?

If $f$ is a positive function, the intuitive interpretation of the Riemann integral $\int_a^b f(x) dx$ is the area under the curve $f$ between $a$ and $b$. Suppose $f$ and $g$ are smooth positive ...
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Why do we require a topological space to be closed under finite intersection?

In the definition of topological space, we require the intersection of a finite number of open sets to be open while we require the arbitrary union of open sets to be open. why is this? I'm assuming ...
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Intuitive explanation of Cauchy's Integral Formula in Complex Analysis

There is a theorem that states that if $f$ is analytic in a domain $D$, and the closed disc {$ z:|z-\alpha|\leq r$} contained in $D$, and $C$ denotes the disc's boundary followed in the positive ...
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Finding all normal subgroups of a group

On my homework today, we had to find all the normal subgroups of $D_{n}$, the dihedral group of order 2n. I solved the problem by looking at how the conjugacy classes change based on whether n is even ...
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Why do we care about dual spaces?

When I first took linear algebra, we never learned about dual spaces. Today in lecture we discussed them and I understand what they are and everything, but I don't really understand why we want to ...
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What is the Direction of a Zero (Null) Vector?

To be more precise, I am interested in knowing if the intuition that a Euclidean zero vector does not have a particular direction is actually correct, and if there is a rigorous formulation that would ...
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Intuition for the definition of the Gamma function?

In these notes by Terence Tao is a proof of Stirling's formula. I really like most of it, but at a crucial step he uses the integral identity $$n! = \int_{0}^{\infty} t^n e^{-t} dt$$ coming from ...
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Could you explain why $\frac{d}{dx} e^x = e^x$ “intuitively”?

As the title implies, It is seems that $e^x$ is the only function whoes derivative is the same as itself. thanks.
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Can this standard calculus result be explained “intuitively”

Recently I stumbled upon someone who said he wanted to understand why $\arctan x = \int\dfrac{dx}{1+x^2}$ At first I was confused. This is an easy result in any integral calculus course. But then he ...
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Best intuitive metaphors for math concepts (of any level)

Frequently, we introduce a new concept with a formal definition, then immediately say "Intuitively, what this means is..." What are the absolute best metaphors you've seen (for concepts of any level)? ...
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Elementary proof of the Prime Number Theorem - Need?

Although i am very much new to "Analytic Number Theory", there are some non mathematical questions which puzzle me. First of all, why was G.H.Hardy so much keen to have an elementary proof of the ...
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Is it misleading to think of rank-2 tensors as matrices?

Having picked up a rudimentary understanding of tensors from reading mechanics papers and Wikipedia, I tend to think of rank-2 tensors simply as square matrices (along with appropriate transformation ...
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Bugs walking in a plane

There are $N$ bugs in a plane. All bugs are moving at the same constant (nonzero) speed, but no two bugs are moving in the same direction (velocity vectors are of the same speed, but no two are ...
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An example of a scheme in the language of schemes

Somewhat related to this question, but almost infinitely more basic. A Confession I am, should classification prove essential, a differential geometer and a topologist by inclination and by ...
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intuitive explantions for the concepts of divisor and genus

when trying to explain AG-codes to computer scientists, the major points of contention i am faced with are the concepts of divisors, Riemann-Roch space and the genus of a function field. are there any ...
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Visualising $\mathbb CP^2$: a problem of attaching cells with a dimension gap >1

For the uninitiated Morse theory, as many other early alebraic-topology widgets, leads to a picture of smooth manifolds as being built up from 'cells', copies of $\mathbb{D}^n$ for varying $n$, ...
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Surprising Generalizations

I just learned (thanks to Harry Gindi's answer on MO and to Qiaochu Yuan's blog post on AoPS) that the chinese remainder theorem and Lagrange interpolation are really just two instances of the same ...
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How can I understand and prove the “sum and difference formulas” in trigonometry? (cos(a ± b) = …, etc.)?

The "sum and difference" formulas often come in handy, but it's not immediately obvious that they would be true. \begin{align} \sin(\alpha \pm \beta) &= \sin \alpha \cos \beta \pm \cos \alpha ...
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Intuitive explanation of the Burnside Lemma

The Burnside Lemma looks like it should have an intuitive explanation. Does anyone have one?
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Intuitive Way To Understand Principal Component Analysis

I know that this is meant to explain variance butthe description on Wikpiedia stinks and it is not clear how you can explain variance using this technique Can anyone explain it in a simple way?
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Taylor expansion to show that for Stratonovich stochastic calculus the chain rule takes the form of the classical one

How can I show with a heuristic argument based on a Taylor expansion that for Stratonovich stochastic calculus the chain rule takes the form of the classical (Newtonian) one? Concerning Ito calculus ...
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Is there a geometrical interpretation to the notion of eigenvector and eigenvalues?

The wiki article on eigenvectors offers the following geometrical interpretation: Each application of the matrix to an arbitrary vector yields a result which will have rotated towards the ...
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Intuitive explanation of covariant, contravariant and Lie derivatives

I would be glad if someone could explain in intuitive terms what these different derivatives are, and possibly give some practical, understandable examples of how they would produce different results. ...
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What's the difference between open and closed sets?

What's the difference between open and closed sets? Especially with relation to topology - rigorous definitions are appreciated, but just as important is the intuition!
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Meaning of closed points of a scheme

This is a question in Liu's book. Let $X$ be a quasi-compact scheme. Show that $X$ contains a closed point. Well I'm unable to do this question, so any help would be appreciated. This question also ...
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Why is it so hard to find the roots of polynomial equations?

The question that follows was inspired by this question: When trying to solve for the roots of a polynomial equation, the quadratic formula is much more simple than the cubic formula and the cubic ...
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What does it mean to be going 40 mph (or 64 kph, etc.) at a given moment?

I was coming back from my Driver's Education class, and something mathsy really stuck out to me. One of the essential properties of a car is its current speed. Or speed at a current time. For ...
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How to visualize a rank-2 tensor?

The notion (rank-2) "tensor" appears in many different parts of physics, e.g. stress tensor, moment of inertia tensor, etc. I know mathematically a tensor can be represented by a 3x3 matrix. But I ...
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What's an intuitive way to think about the determinant?

In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t ...
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Stacks are just sheaves up to Isomorphism

I have heard that one can think of stacks on a site as taking sheaves but instead of the restrictions being equal, we just loosen it to isomorphic, and treat the sheaf conditions with the "obvious" ...
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1answer
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Why does the log-log scale on my Slide Rule work?

For a long time I've eschewed bulky and inelegant calculators for the use of my trusty trig/log-log slide rule. For those unfamiliar, here is a simple slide rule simulator using Javascript. To ...
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Is there a relationship between $e$ and the sum of $n$-simplexes volumes?

When I look at the Taylor series for $e^x$ and the volume formula for oriented simplexes, it makes $e^x$ look like it is, at least almost, the sum of simplexes volumes from $n$ to $\infty$. Does ...
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genericness and the Zariski topology

What does it mean (in a mathematically rigorous way) to claim something is "generic?" How does this coincide with the Zariski topology?
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What is a Markov Chain?

What is a intuitive explanation of a Markov Chain, and how they work? Please provide at least one practical example.
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Simple explanation of a monad

I have been learning some functional programming recently and I so I have come across monads. I understand what they are in programming terms, but I would like to understand what they are ...
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Different kinds of infinities?

Can someone explain to me how there can be different kinds of infinities? I was reading "the man who loved only numbers" and came across the concept of countable and uncountable infinities, but ...