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, ...
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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 ...
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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 $G_1\cup G_2$...
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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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What is the geometric intuition for the $\bar \partial$-Poincare lemma, or for $\bar \partial$ more generally?

The one variable $\bar \partial$-Poincare lemma is proven in Huybrechts and Forster and so on in essentially the same way: one shows that for a local form $f d \bar z$, with $$g(z) := \frac{1}{2\pi i}...
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The axioms of topology [duplicate]

I am just curious about the axioms of topology. In particular with regard to finite intersections. The way that I imagine the axioms of topology is that we give a set $X$ a way of arranging it's ...
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Total order relations on $\mathbb{Z}_{2^n}$

I'm a programmer and not a professional mathematician so I need some abstract-algebra help regarding the following question: Introduction: When performing integer comparison x86 ...
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Interesting explicit convergent subsequence for not converging bounded sequence

To illustrate the (power of) Bolzano-Weierstrass theorem I am searching for an example of a bounded but not convergent sequence and an explicit convergent subsequence. I would like it to be non ...
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If $(x_n)_n$ is an enumeration (sequence) of $\mathbb Q\cap[0,1]$, then is it statistically convergent? Also, what is about almost convergent?

Since, $\mathbb Q\cap[0,1]$ is an countably infinite set, then there exists a bijection (enumeration) from $\mathbb N$ to $\mathbb Q\cap[0,1]$, which gives us a sequence in $\mathbb Q\cap[0,1]$. If $...
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Do I break even in a fair game?

Thanks for reading. The following screenshot I took from the book "The Art of Probability" by Richard Hamming. I'm really confused. How is it that the probability of heads is $\frac{1}{2}$, meaning ...
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Understanding Expected Value and Standard Deviation…

Intro Thanks for reading. Long question incoming - I tried to make it as complete as possible to best explain where I'm coming from conceptually, as I've talked to a lot of people in my ...
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Is there any visual intuition for separation of variables?

Separation of variables is a standard procedure to solve a differential equation of the form $$ u'(t) = g(t) h(u(t)) $$ transforming it to via division and substitution to $$ \int_{u(t_0)}^{u(t)}...
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Definition of the covariant derivative

In Peter Petersen's book Riemannian Geometry (2. Edition) the covariant derivative on a Riemannian manifold is defined by the implicit formula $$2g(\nabla_YX,Z)=(L_Xg)(Y,Z)+(d\theta_X)(Y,Z)$$ where $...
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Negative exponential growth

Thanks for reading! I feel like I have an okay enough understanding of $e$. This is how I understand it (wordy definition incoming, but I'm trying to make it as complete as possible): By ...
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Intuitive understanding of an alternative variance equation

I think I have a fairly okay understanding of what the variance of a random variable is. It's how far we expect the squared distances of the values that the random variable takes to be from the ...
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Modifying Lebesgue Integral using Intuition [closed]

The Lebesgue integral is studied for $F:\mathbb{R}\to\mathbb{R}$ but is it intuitive for $F:[a,b]\cap A\to\mathbb{R}$, where $\left\{A\cap[a,b] \right\}\subseteq[a,b]$? Suppose $A\cap[a,b]$ has a ...
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$P,Q$ is any two subset of $(X,d)$ with $P\subset Q$, then what is the value of $\sup\limits_{x\in P} \inf\limits_{y\in Q} d(x,y)$?

Let $(X,d)$ be a metric space. Then the function $d_H:\mathcal P(X)\times\mathcal P(X)\to\mathcal [0,\infty]$ defined by $$d_H(A,B)= \max \{ \sup\limits_{x\in A} \inf\limits_{y\in B} d(x,y), \ \sup\...
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Visual intuition for $\frac{1}{b - a} \int_{a}^{b} u(t) \ \text{d}t \in \overline{\text{co}}(u([a,b]))$

Let $u: [a,b] \to X$ to be a continuous function and $X$ a Banach space. Then $$ \frac{1}{b - a} \int_{a}^{b} u(t) \ \text{d}t \in \overline{\text{co}}(u([a,b])) $$ holds, where $\overline{\...
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What is the intuitive sense of a closed set on a metric space?

A subset $A$ of $X$ is closed in $(X, d)$ if and only if every convergent sequence of points in $A$ converges to a point in $A$. How is this definition consistent with the open ball definition of ...
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Motivating the multivariable definition of the derivative

When I ask "What is the derivative?" the answer I find I get the most (and the answer I think is most satisfying) is: $$ \text{The derivative at a point is a local, linear approximation of the ...
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Proving that $I+A$ is invertable when $A$ is nilpotent: What intuition leads to a particular approach?

In an answer to this question, it has been suggested to consider the following: $$(I+A)(\sum_{j=0}^n(-A)^j)$$ Through a series of algebraic operations, it can be shown that $\sum_{j=0}^n(-A)^j$ is in ...
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Intuition for Riemannian submersions

A submersion $f:M\to N$ between Riemannian manifolds is called a Riemannian submersion if for each $p\in M$ $d_pf:(\ker d_pf)^\perp\to T_{f(p)}N$ is an isometry. In contrast to Riemannian ...
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What are some intuitive examples of categories which aren't made from structured sets and functions between them? [duplicate]

In my loose reading about category theory, categories are usually introduced in terms of sets, often with some structure (such as algebraic or topological), and functions between them that preserve ...
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Looking for a particular lecture/lecturer on the fact that intuition may is not always enough

Last night I was about to close off my phone and saw the beginning of a very interesting lecture on how intuition is not always enough in Mathematics... The lectures (so far unknown to me) started ...
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Followup question: How to come up with this counterexample?

Given: Original question:Alternative Proof to "Prove that it cannot be proven that "The United States had more fallow acreage than planted acreage" My Question: A ten year ...
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Intuitive understanding of Euler's Formula

Thanks for reading! In this link... https://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ ...Euler's formula is explained pretty well. I understand that multiplying a ...
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What does the fraction of correctly identified edges mean in community detection / clustering?

In some papers in community detection or network clustering, they use the following measure of quality for the detected communities (see for example https://arxiv.org/abs/0906.0612, section XVB, 3rd ...
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Wrong intuition about counting number of surjections

Consider the sets $A:= \{1, 2, 3, 4\}$ and $B:= \{1,2,3\}$. It is not hard to count the number of surjections $A \to B$, namely $36$, by subtracting the number of non-surjections from $81$. But I'm ...
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Sorting out what's true in the generic model in the classifying topos of a theory

I'm interested in trying to understand the generic model in the classifying topos of a particular coherent theory$^1$, and more specifically trying to sort out what non-coherent formulae hold in said ...
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$f(x+h)=f(x)+\int_0^1(Df(x+r h)h dr)$?

I saw this in a book explaining the Infeasible Newton Method. It used this to prove an inequality which is then used to prove the convergence of the method. Can anyone explain how it is derived and ...
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Confused with a system of equations with three variables that has infinitely many solutions

I'm studying High School Algebra and it had this question: Solve the system by equations: \begin{align*} x + y - z &= \,0 \\ 2x + 4y - 2z &= 6 \\ 3x + 6y - 3z &= \,9 \end{...
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Unique Equivalence Classes

Not sure if this is a suitable question here, but I'm having trouble understanding the intuition behind a theorem I've read in a textbook. So it says the following: "If $\mathscr R$ is an ...
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Intuitively, why are the two limit definitions of $e^x$ equivalent?

Thanks for reading! Intuitively, why does... $$\lim_{n \rightarrow \infty} \left(1+\frac{1}{n}\right)^{xn}=\lim_{n \rightarrow \infty} \left(1+\frac{x}{n}\right)^{n}=e^x$$ Note, I'm not asking why $...
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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 $\ell$....
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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 ...
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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 ...
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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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Definition of a reducing subspace

Gerald Teschl in his book "Mathematical Methods in Quantum Mechanics" defines a reducing subspace of an unbounded operator the following way: Let $T$ be an unbounded operator on a complex Hilbert $...
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Intuitive proof for the second directional derivative

I understand that the second directional derivative can be found by taking the gradient vector of the first directional derivative and multiplying it by the given unit vector, as follows: $$\nabla^2_{...
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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 ...