# 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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### Visual Interpretation for the Sum of a Finite Geometric Series

Is there an intuitive visual explanations for the sum of a finite geometric series? I know there are some pretty intuitive ones out there (and on this site), but I haven't seen any visual ...
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### Combinatorial argument for the mean of Beta distribution

I know how to prove via the formula that $\mathbb E[\beta(a,b)]=\frac{\alpha}{\alpha+\beta}$ but at least for the case that $\alpha,\beta \in \mathbb N$ there should be some nice intuitive prove for ...
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### Understanding multivariate orthonormal polynomial bases

I have recently started to delve into the construction and parameterization of orthogonal polynomial bases (e.g., for polynomial chaos expansion) and there are some things that aren't yet clear to me, ...
1answer
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### Relative contribution of parallel resistors to final resistance

If you have two resistors A and B in parallel, how could you represent their individual percent contribution to the equivalent resistance? I thought of this question while trying to map the parallel ...
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### The Motivation and Intuition Behind Matrix Tri-factorization. [closed]

What is the motivation and intuition Behind Matrix Tri-factorization? $𝐵=𝑃^{−1}𝐴𝑃$
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### Intuition on why does $P(-1) = 0$ leads to identify one factor of a 3rd Degree Polynomial?

Given $P(x) = x^3 + 3x^2 -13x -15$ In order to detect occasions in which $P(x)=0$, the factors of the remainder $15$ i.e. $\{-1,-3,-5,1,3,5\}$ were used as values of $x$ These were: $P(-1)$, $P(3)$ ...
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### Intuitive example for Identical random variables

I intuitively understood the idea of independence between two random variables. But hard in getting sense of identical random variables on the same sample space. I saw many examples for identical ...
0answers
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### Union of graphs is a sort of matrix concatenation

Just an intuition for discussion: If we consider a set $G=\lbrace G_1, G_2\rbrace$ of two connected simple graphs, where $G_1$ and $G_2$ have no vertex in common then the the graph union $G_1\cup G_2$...
3answers
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### Bounding points by lines

Take a set of N points where no group of points with more than two points can be co-linear. The points also lie in the plane. What is the minimum amount of straight lines it takes to bound each point ...
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### If $(x_n)$ is almost convergent to $\ell$, then $\lim\limits_{p\to\infty}\frac{x_n+x_{n+1}+x_{n+2}+\dots+x_{n+p-1}}{p}=\ell$ holds uniformly in $n$.

Let $(x_n)$ be a real sequencce. I have showed that if $\lim\limits_{p\to\infty}\frac{x_n+x_{n+1}+x_{n+2}+\dots+x_{n+p-1}}{p}=\ell$ holds uniformly in $n$, then $(x_n)$ is almost convergent to $\ell$....
1answer
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### Determinants through dot products

Thanks for reading! So, in Paulo Buchsbaum's answer to this question on Quora... https://www.quora.com/What-is-the-mathematical-intuition-behind-the-determinant-of-a-matrix-How-was-its-definition-...
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### Intuition/Significance of adjoint representation of a Lie group

I understand the definition of the adjoint representation of a Lie group. But why is that important? In particular, why is it a natural choice of group representation?
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### Cross Products and Determinants Geometrically

In this question... Geometric interpretation of the cofactor expansion theorem ...Grigory explained (beautifully, in my opinion) why the cofactor expansion for calculating determinants worked by ...
1answer
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### What information does the operator norm provide?

I am trying to improve my intuition for the operator norm (of bounded linear transformations between normed spaces). The definition $\sup_{\|x\| = 1} \|Tx\|$ is tells me that $\|T\|$ bounds the ...
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### Intuition behind One-sided Lipschitz

In my current lecture Numerical Analysis of Ordinary Differential Equations we introduced the concept of One-sided Lipschitz functions. A function $f: D \rightarrow \mathbb{C}^d$ satisfies a one-...
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### examples of simple insight using category theory about Set or Vect?

i'm looking for a beginner's example to a somewhat non-obvious insight one can get by formalising things using category theory. such things clearly exist among many areas of pure math but i am looking ...
0answers
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### How to understand Non-positive Definite Metric Tensors intuitively?

My motivation for this comes from Minkowski space and general relativity more broadly, but I don't want to focus on the real world details currently because I want to understand what a positive vs ...
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### Why do we divide by standard deviation when standardizing a normal distribution?

We have this random variable $Y= \frac{x - μ}{\sigma}$ to convert a normal distribution $N(\mu, \sigma)$ to a $N(0, 1)$. It is quite intuitive to subtract $\mu$, since you move all the values ​​in the ...
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### Can strong duality for linear programming be viewed intuitively?

Is there a somewhat intuitive way of understanding strong duality in linear programming? I do understand weak duality quite well since it pretty much follows from how the dual problem is defined but I ...