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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Intitutive explanation of why probability of getting an even integer is more when random numbers are multiplied.

If n(>1) random integers are selected, then the probability of their product being odd is $1/2^n$, which is less than that of the product being even. But this doesn't sound intuitive to me. If ...
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Why is $a\hat{\imath} + b\hat{\jmath } + c\hat{k}$ meaningful when $\hat{\imath}$, $\hat{\jmath }$, $\hat{k}$ are not 'alike' quantities?

For the standard form: $a\hat{\imath} + b\hat{\jmath } + c\hat{k}$. Since the $\hat{\imath}$, $\hat{\jmath }$, and $\hat{k}$ directions are different, why are we 'allowed' to write them this way? Isn'...
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Intuition for this definition of "almost orthogonal" in Banach space

I'm looking into an intuitive understanding of a following lemma in functional analysis about "almost orthogonal elements": Let $X$ be a Banach space. $\dim X=\infty$. $U\subseteq X$ is a ...
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Is the Axiom of choice intuitive? How was it first introduced?

I will refer to the Axiom of choice as ($AC$). As far as I know, the widely accepted axioms of set theory is the Zermelo-Fraenkel axioms together with the $AC$. But the last axiom seems to be the most ...
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Any good explanation why $\sigma(\tau f)=(\sigma\tau)f$ holds.

I am reading "Linear Algebra" by Ichiro Satake. The following proposition and its proof are in this book: Let $f(x_1,\dots,x_n)$ be a mulitivariable polynomial. We define $\sigma f$ as $$(\...
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Intuitively, why does P(switch then win) = P(DON'T switch then win) × $\dfrac{\text{total doors} - 1}{\text{after host opened h, doors still open}}?$

I already know — NOT asking — about the algebra. Please focus solely this generalized Monty Hall problem, with c cars, g goats. After the contestant picks his door, the host will open $h$ doors ...
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The Intuition and Logic Behind Adjusting Pearson's Contingency Coefficient So That It Can Reach 1

One of the measures of association for the nominal data based on Chi-Squared test is the Pearson's contingency coefficient: $$ C' = \sqrt{\frac{\chi^2}{\chi^2 . N} } $$ This measure suffers however ...
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1answer
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Is there some practical intuition when working with a cooperad given by cogenerators and corelations?

In the case of algebras and operads, a description by generators and relations is common practice and I have a good understanding of this. A non-symmetric operad $\mathcal{P}$ given by a linear space ...
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1answer
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Gronwall's Lemma intuition

Ive recently had this real analysis exercise which is apparently a special case of Gronwall's lemma. The exercise is to show that if $f,g:[a,b]\to[0,+\infty[$ are continuous functions and $\exists\...
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Intuitive explanation of the transience of symmetric random walks

We know that the symmetric random walk on $\mathbb{Z}^d, \, d \geq 3$ is transient. However, if we looked at each coordinate separately, we would see a lazy symmetric random walks on $\mathbb{Z}$, ...
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Intuition for a uniform space?

Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and uniform convergence. According to Wikipedia, ...
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Intuition behind Principal Ideal Domain

In a group, normal subgroups are most likely not cyclic and hence not generated by a single element. Ideals, as kernels of homomorphisms, is a similar concept to normal subgroups. Why would we expect ...
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The Intuition of the Construction of a Non-Measurable Set (Vitali Set) on the Real Line

I suppose the question could be stated another way: if you were asked to construct a non-measurable set the first time in the history, what would motivate the construction to a Vitali set? For a ...
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Given a coin's bias, how can two flips be conditionally independent?

I am seeking merely (to correct my) intuition, not formal proofs. Andrew Chinery answered A clearer example paraphrased from Norman Fenton's website: if Alice (A) and Bob (B) both flip the same coin, ...
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Proving $\forall A \subset X: f(A)\cap f_{\text{ess}}(X)=\emptyset \implies \mu(A)=0$ & How to think about proofs with essential range.

I have had a very quick introduction to measure theory and haven't built up much intuition to guide the proofs I attempt. I have been struggling for a while now to prove the following result which ...
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1answer
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The Laplace transform resolves a function into its moments.

The Laplace transform is related to the Fourier transform, but whereas the Fourier transform expresses a function or signal as a series of modes of vibration (frequencies), the Laplace transform ...
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Intuition behind using random variables in Monte Carlo methods / localization

I am trying to get a better intuitive understanding of why Monte Carlo works so well in approximating a solution to complex problems, such as calculating irrational numbers or the Particle Filter / ...
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What does finding a "free local ring" have to do with finding the spectrum of a ring?

In Tierney's 1976 paper On the Spectrum of a Ringed Topos (which you can find here) at the top of section 2 we read Let $A$ be a commutative ring in [a topos] $\mathbf{E}$. We look at the problem of ...
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Understanding Analysis by Abott (Prerequisites)

I just bought „Understanding Analysis“ by Stephan Abott. I am a CS Student who already passed his math lectures in Germany but I want a better understanding since I memorized a lot. I want to build a ...
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What is the intuition behind pseudogroups and why can they capture local symmetry?

What is the intuition behind pseudogroups and why can they capture local symmetry? I read on Wikipedia that they can capture the notion of local symmetry. What does this mean exactly and how can they ...
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What is the intuition behind using the fixed point of this function to show the convergence of this sequence?

In https://ssmr.ro/gazeta/gma/2021/gma3-4-2021-continut.pdf on page 33 we have the solution of a problem from a contest for university students. I will write here the statement and the solution of the ...
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Spectral representation intuitive explanation

I am following a course in functional analysis and in the lectures we encountered the following theorem: Theorem: Let H be a Hilbert space and $T:H\to H$ a self-adjoint and compact operator. Then: $T ...
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SVD confusion for matrices $m\times n$ with $m>n$

According to my understanding, the SVD decomposition consists of decomposing a matrix into a rotation, scaling, then another rotation. This suggests to me one can use that to create arbitrary linear ...
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1answer
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Maximum number of edges one can add to a tree without making it non-planar

I actually have a question regarding the accepted answer to this question by the OP themself: How many edges can you add to a tree with $n$ vertices so it stays planar? The answer employs induction ...
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How to interpret the fact that the Fourier transform of $e^{-x} \operatorname{sinc}(x)$ is a constant - and what, if any, is its significance?

Recently, I was fiddling around with computing Fourier transforms of different functions. At a certain point, I found out that $$\mathcal{F}_{x} [ \operatorname{sinc}(x)e^{-x} ] (t) = \int_{-\infty}^{\...
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Intuition for when a problem may be amenable to the "umbral calculus"?

I've always been interested in situations where we can apply "illegal" operations to objects and still solve problems (as seen here, say), and a common justification for these techniques is ...
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Intuition for orientation of a simplex (in 3 dimensions)

In trying to begin to learn basic homological algebra, i am confronted with orientation of simplices. The definition seems unmotivated and unintuitive: for $n$-simplices with $n \in \{-1,0,1,2\}$, it ...
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1answer
58 views

Intuition for the Cauchy-Schwarz inequality that does not rely on a geometric interpretation of vectors

I know that the Cauchy-Schwarz inequality can be understood by interpreting vectors as arrows in the coordinate plane, but since the inequality should hold for any inner product space, I'm looking for ...
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1answer
117 views

Duhamel's Principle Intuition

I am trying to understand Duhamel's Principle by applying it to some simple problems. I am thinking of $P(t)$ as expressing a bank account balance at time $t$, to try to gain an intuition for Duhamel'...
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The intuition behind the formula for counting distinct permutations with repetitions

I'm going through a book right now where the author is going over some combinatorics equations. I apologize if the following is an easy question. I'm trying to gain an intuitive understanding of the ...
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333 views

What's the intuition for Cross Multiplication?

I already know, I'm NOT asking about, the algebra. It's NOT intuitive why 3 pears x 4 tangelos = 6 quinces x 2 riberries $\iff$ 3 pears/6 quinces = 2 riberries/4 tangelos. I stumbled the picture ...
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What is an intuition for subdirect irreducibility?

I am reading "A Course in Universal Algebra" by Burris and H.P. Sankappanavar and they provide this definition: "An algebra A is subdirectly irreducible if for every subdirect embedding ...
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1answer
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How to show a sequence of measure converges weakly?

The doubt I have is that in all equivalent definitions of weak convergence of finite measures via the Portmanteau theorems, some knowledge of $\mu$ is required to check those conditions, for example ...
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152 views

Expected Geometric Growth Rate (Kelly's Criterion)

The Wikipedia article for Kelly Criterion establishes its main formula using the expected geometric growth rate $r = (1 + fb)^p * (1 - fa)^q$, where $f$ is the fraction of an account (that starts with ...
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Seemingly conflicting notions of a function

Throughout my mathematical education, I have seen a few, seemingly, different and conflicting notions of what a function is: A function is a a type of mathematical object that maps every element of a ...
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Subdifferential, two different definitions.

The subdifferential of the function $W$ at $x$ was defined as $$\partial W(x):=\{\kappa\in\mathbb{R}^d:W(y)−W(x)≥\kappa\cdot(y − x),\text{ for all }y ∈ \mathbb{R}^d\}.$$ I understand what this means ...
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Intuition for derivatives of trigonometric functions

I’m trying to understand the geometric/infinitesimal ‘derivations’ of the formulae for derivatives of trigonometric functions here. What I don’t understand is how the similarity of the triangles gets ...
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1answer
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What are broad tips for creating pushdown automatas (PDAs) for context free languages (CFLs)?

What are tips for creating PDAs for context free languages? I know there is typically no bit by bit answer for producing these PDAs, however there should be some broad tips to direct one to make them. ...
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Continuity equation with measures / Vizualise the divergence of a measure?

I kind of understand the meaning of the continuity equation when $\rho_t(x)$ is a density $$\partial_t\rho+div_x(\rho_t v)=0$$ Which means that the time variation of the density in a point is the ...
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Intuitively, why does growth proportional to the population size not diverge but growth proportional to pairs diverges to infinity in finite time?

It's interesting that growth which is proportional to the current population doesn't diverge to infinity, while growth that is proportional to higher powers of the current population. That is, the ...
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1answer
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Why not define boundary to be $\partial S= S-S^o$?

The standard definition of boundary is $\partial S= \bar{S}- S$. Intuitvely in geometric shapes it seems to be the same as $S-S^o$, is there a reason we use the closure definition instead of the ...
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Why is the number of heads and the number of tails independent in $n$ coin flips?

In probability, in different manifestations, I have noticed that the sum of random variables $X$ and $Y$ may be given as $n$, and yet $X$ and $Y$ may still be independent. I find this very counter ...
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Operations with the $\mathcal O$ notation

I'm quite new to the big $\mathcal O$ notation, so my question might sound ridiculous. I was told we wouldn't learn almost anything about $\mathcal O$ and $\mathcal o$ notations during our bachelor ...
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Example of an Affine space

let $f_1$ and $f_2$ be some fairly simple polynomial functions. I let $F_1$ and $F_2$ be some elements of the set of their respective antiderivatives. Now, can I say that the set of ordered pairs $\...
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1answer
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What are general tips for generating grammars for context free languages?

Perhaps, this does not have a "correct" answer, but what are general tips for creating context free grammars for context free languages? I know there is usually no step by step solution to ...
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1answer
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Non-singularity of $Df$ implies that $f$ is locally one-to-one

I'm on the following lemma, in the Inverse Function Theorem section of Munkres's Analysis Book: Let $A$ be a open in $\mathbb{R}^{n}$. Let $f: A \to \mathbb{R}^{n}$ be of class $\mathscr{C}^{1}$. If ...
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How can a positive factor change the sign of an expression? Need intuitive explanation.

From this answer it follows that $\operatorname{sign} X \ne \operatorname{sign} a X$ for any $a>0$ and nilpotent $X$. For instance, in dual numbers, $\operatorname{sign} \varepsilon \ne \...
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1answer
25 views

Equivalent definitions of congruences (in context of universal algebra)

I am familiar with this (informal) definition of congruence relation (or simply congruence): (1) A congruence is an equivalence relation on an algebraic structure that is compatible with the structure ...
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Purely algebraic context in which homology arises naturally?

I know that homology is most commonly introduced from a topological context (and/or Stokes theorem related context), but in say a homological algebra course/text, you are just given the definitions (...
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62 views

Doubt in derivative computation

I quote Life Insurance Mathematics (Gerber, 1997). Let us consider a person aged $x$ years, denoted by $(x)$. We denote his/her future lifetime by $T$ or, more explicitly, by $T(x)$. Thus $x+T$ will ...

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