# Questions tagged [intuition]

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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### Intuition behind vector field along a local parametrization of a manifold.

In our course on several variable calculus,the following notion was defined: Vector Field along a parametrization: Definition: Suppose $M$ is a $k$-manifold in $\mathbb R^n$.Let $q$ be a point on the ...
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### Condition for $\int_{1}^{+\infty} x^a(x-1)^b dx$ be convergent?

I did some full tests $\int_{1}^{+\infty} x^a(x-1)^b dx$ and apparently it is always convergent if $a<-1$, $b>-1$ and $a+b<-1$. Intuitively I think that b must be greater than $-1$ because ...
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### Why pyramid with sharp corners and edges is still a manifold intuitively

I'm new to manifolds and in my Computer Graphics class, we briefly explored the topic superficially through visual examples. From my intuitive understanding, we should be able to place a Euclidean ...
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### intuitively Understanding meaning of a Combinatorics problem to reach solution

I have been recently taking course on Combinatorics and landed on following problem, Here is the formal statement of problem: A room contains a single bulb and $(2^{2^{10}}+2^{2^9})$ identical ...
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### Category of LCAG's "measures" the difference between the integers and the reals: what does this mean?

The Wikipedia article on Locally compact abelian Groups (https://en.m.wikipedia.org/wiki/Locally_compact_abelian_group) has the following excerpt in the Categorical properties section: Clausen (2017) ...
1 vote
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### The special property of the principal part at a pole

I'm going through Ahlfors, and in the section about the Residue theorem, and he says that about the poles, a function $f$ can be expanded to $$f(z)=B_h(z-a)^{-h}+\cdots+B_1(z-a)^{-1}+\varphi(z)$$ he ...
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### Intuition behind the the definition of homomorphisms and isomorphisms [duplicate]

I'm familiar with the definition (inverses and bijections, preserving operations) in the context of groups and vector spaces and do know the formal definitions of morphisms as a whole. However, what I'...
1 vote
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### Why or when can we ignore specific requirements in probability problems due to symmetry?

The Red Sox play the Yankees in a best-of-seven series that ends as soon as one team wins four games. Suppose that the probability that the Red Sox win Game $n$ is $\frac{n-1}{6}$. What is the ...
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### Intuition behind quadratic form of SPD matrices $S\Sigma S$

Is there an intuition behind the quadratic form of SPD (symmetric and positive definite) matrices: $$(S, \Sigma) \in \mathcal{S}_{++}\times\mathcal{S}_{++} \mapsto S\Sigma S$$ where $S$ and $\Sigma$ ...
1 vote
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### Finding a Kähler manifold from a given exponential family. Intuition?

I'm wondering how to go from: $$\mathrm{Exponential~ families} \implies \mathrm{Kähler~manifolds}$$ I read that the tangent bundle of an exponential family naturally forms a Kähler manifold. I also ...
1 vote
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### One of the Specific Intuitions behind Central Limit Theorem

This is a relatively vague question since it occurred during the class couple of semesters back when the Professor tried to explain the "intuition" of Gaussian distribution. The information ...
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### differential dynamic programming : Intuition, key idea and difference from dynamic programming

I cam across 'Differential Dynamic Programming' in a course on Optimal Control. In this course , we were introduced to Dynamic Programming prior to DDP. I went through the Wikipedia Post on ...
1 vote
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### Nature of the set of deltas in limit definition

I was proving the fact that if f is differentiable at a, then (cf)’(a)=cf(a)’ After showing it, I started to play with some ideas and wondered from intuition how to see that depending on the value of ...
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### Why is optimal transport theory so relevant?

I see plenty of papers published with "optimal transport" in their title and I know that at least 2 Fields medal in the last 10 years were assigned for something related to optimal transport ...
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### How to understand change of variables intuitively?

I've been trying to prove or have an intuitive understanding of the change of variables, and I tried it for the function $f(x)=x^2$ using $u(x)=x^2$, the transformed function then becomes $g(u)=u$. ...
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### Algebraically why must a single square root be done on all terms rather than individually?

Let's assume we know that $x+9=10$. I understand this is illegal: $$\sqrt[]{x} + \sqrt[]{9} = \sqrt[]{10}.$$ And this is correct: $$\sqrt[]{x + 9} = \sqrt[]{10}.$$ Is there an intuitive way to ...
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### Intuition behind $Q_t=\sum \langle M^{\alpha},M^{\alpha}\rangle_t+\sum |A^{\alpha}|^3_t+|A^{\alpha}|_t+t$

Consider the semimartingale $Z$, which by Doob decomposition can be written as $Z=M+A$, where M is a martingale and A is the process of total bounded variation. I am trying to make sense of the ...
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### Definition of nested partition of the circle

Below is an excerpt from the paper Boundary torsion and convex caps of locally convex surfaces, in which the author defines a so-called nested partition of the circle. I am having a hard time ... 51 views

### Explanation of $\{A_i\bigcup B_j\}_{i,j\in I\times J}$

It is in my understanding that a family, $\{A_i\}_{i\in I}$ is a collection of sets $A_i$ with $i$ taking on all possible values within $I$. Of course this is the less formal definition; the most ...
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### Why is it "obvious" that the expected value of a continuous uniform distribution is (a+b)/2?

For a continuous uniform random variable X with support on an interval [a,b], where a<b, one can always calculate the expected value, by integrating, to arrive at the value of (a+b)/2. However, ...
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### What is the intuition behind the method of characteristics for a second order PDE?

I understand the idea behind the method of characteristics as applied to first-order PDEs: watching how $u(x,y)$ changes along special curves $(x(s),y(s))$ simplifies the problem to a set of coupled ...
In this question I'm looking for an intuitive explanation, which could provide me some "a-ha!" moments. As we all know, the derivative is just a $\tan$ of an angle in the triangle, whose ...