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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What is the intuitive meaning of the adjugate matrix?
The definition of the adjugate matrix is easy to understand, but I have never seen it used for anything.
What is the intuitive meaning of this matrix?
Are there examples of applications which may ...
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2answers
94 views
What's the most elegant way of rotating a 3-dimensional co-ordinate system?
For two dimensional rotation of $x$ and $y$ axes anticlockwise by $\varphi$, the equation that transforms $P(x,y) \rightarrow P(x',y')$, $x'=x \cos(\varphi)+y \sin(\varphi)$ and $y'=y \cos(\varphi)- ...
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Does a four-variable analog of the Hall-Witt identity exist?
Lately I have been thinking about commutator formulas, sparked by rereading the following paragraph in Isaacs (p.125):
An amazing commutator formula is the Hall-Witt identity: ...
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147 views
What's the intuition behind non-integer exponents/powers
Consider some $a \in \mathbb{R}$ and $x \in \mathbb{R}\backslash \mathbb{N}$.
Is there some intuition to be had for the number $a^x$?
For example the intuition of $a^2$ is obvious; it's $a*a$ which ...
6
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1answer
211 views
Understanding the three isomorphism theorems
I have learnt the following three isomorphisms for a while but without true understanding:
A group homomorphism $\phi:G\to G'$ can be decomposed into ...
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5answers
189 views
How can it happen to find infinite bases in $\mathbb R^n$ if $\mathbb R^n$ does not admit more than $n$ linearly independent vectors?
How can it happen to find infinite bases in $\mathbb R^n$ if $\mathbb R^n$ does not admit more than $n$ linearly independent vectors?
Also considered that each basis of $\mathbb R^n$ has the same ...
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57 views
A gratifying re-encounter with a piece of math that was out of my mind
A series of real numbers is said to be conditionally convergent if it is convergent but not absolutely convergent.
By rearranging the terms of a conditionally convergent series we can make the ...
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2answers
108 views
Explain convertion algorithm from bytes to Kb, Mb, Gb.
I was trying to convert file size from bytes to human understandable value and found one interesting solution. I will provide it on php with explanation.
...
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166 views
How to 'analyze' problems in analysis; Computing $\int_0^{2\pi}\frac{1}{(a+b\cos(\theta))^2}d\theta$
If $a, b \in \mathbb{R}$ with $a > b > 0$, compute this ungodly thing;
$$\int_0^{2\pi}\frac{1}{(a+b\cos(\theta))^2}d\theta$$
I'm really not a fan of complex analysis... I can't visualize ...
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223 views
Motivating (iso)morphism of varieties
I am reading course notes on algebraic geometry, where a morphism of varieties is defined as follows ($k$ is an algebraically closed field):
Let $X$ be a quasi-affine or quasi-projective ...
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45 views
What is the CONCEPT when we speak of maximum entropy?
What is an intuitive interpretation of the concept of maximum entropy? I want to understand this concept better but what I'm finding is too "advanced" right now. Can anyone simplify it ... imagine I'm ...
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355 views
Intuition on the Orbit-Stabilizer Theorem
The Orbit-Stabilizer says that, given a group $G$ which acts on a set $X$, then there exists a bijection between the orbit of an element $x\in X$ and the set of left cosets of the stabilizer group of ...
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4answers
945 views
Tricks to remember Fatou's lemma
For a sequence of non-negative measurable functions $f_n$, Fatou's lemma is a statement about the inequality
$\int \liminf_{n\rightarrow \infty} f_n \mathrm{d}\mu \leq \liminf_{n\rightarrow ...
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1answer
109 views
Harmonic function.
The function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ given by $f(x) = \|x\|^{2-n}$, where $\|~\|$ denotes the Euclidean norm, is harmonic. This is just a simple computation.
My question is: why ...
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Intuition on group homomorphisms
So I'm studying for finals now, and came across the idea of homomorphisms again. This is not a new idea for me at all, having seen them in groups, rings, fields ect. However, on reevaluating them I ...
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3answers
129 views
Why the principle of counting does not match with our common sense
Principle of counting says that
"the number of odd integers, which is the same as the number of even integers, is also the same as the number of integers overall."
This does not match with my ...
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4answers
297 views
What is the significance of multiplication (as distinct from addition) in algebra & ring theory?
In higher math, operators are defined over a set of objects; and these operators are usually denoted as addition and multiplication with a distribution rule. Assuming multiplication is not repeated ...
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Math Courses involving clever integration techniques
I am a third year undergraduate mathematics student.
I learned some basic techniques for simplifying sums in high school algebra, but I have encountered some of the more interesting techniques in my ...
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4answers
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Variance of binomial distribution
Why for $X\sim B(n,p)$ is $Var(X)=np(1-p)$?
$Var(X)=\sum x_i^2 p_i -(\sum x_i p_i)^2=\sum_{r=0}^n r^2 \binom{n}{r}p^r(1-p)^{n-r}+( \sum_{r=0}^n r \binom{n}{r}p^r(1-p)^{n-r} )^2$
In my ...
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2answers
99 views
Cauchy's Theorem for Groups
Specifically: If $p$ is a prime divisor of the order of a finite group $G$, then there exists an element of order $p$ in $G$
So I'm looking for a little intuition behind this idea. I understand how ...
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283 views
Intuition why the volume and surface area of the unit sphere eventually decrease
The volume formula for a unit sphere, $$\frac{\pi^{n/2}}{\Gamma{(1 + n/2)}},$$
and the surface area formula, $$\frac{2\pi^{n/2}}{\Gamma{(n/2)}},$$
both attain maximum values for finite $n$. We can ...
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27answers
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Too old to start math
I'm sorry if this question goes against the meta for posting questions - I attached all the "beware, this is a soft-question" tags I could.
This is a question I've been asking myself now for some ...
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6answers
173 views
Combinatorial proofs: having a difficult time understanding how to write them out
Can someone explain how combinatorial proofs work? I've included an example questions that's been giving me a hard time. Any insight on the topic would be great.
$$\sum_{k=1}^{n}k{n \choose k} = ...
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1answer
180 views
Multiple choice questions on relations and some of their properties
I'm confused about these 3 selected problems. I have the solutions for each, if necessary, but I'm much more interested in understanding the material. If anyone can offer a clear, concise, and ...
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Intuition for cofibration
The notion of a fibration has a nice geometric intuition of one topological space (a fiber) being parametrized by another topological space (the base) -- this is taken from the Wikipedia entry on ...
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What is the use of $H_s$ for non-integer $s$?
So we have the whole set of theory for Sobolev spaces \begin{equation}
H_s(\mathbb{R}^d)=\{u\in D'(\mathbb{R}^d):(1+|y|^2)^{s/2}\hat{u}\in\mathcal{L}^2(\mathbb{R}^d)\},
\end{equation} and we know that ...
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45 views
Upcoming exam! Any good sources to learn about counting techniques and discrete probability?
If anyone has a free, online source to contribute for a certain topic/topics, please share! I'm not really looking for an intense theoretical grasp of these topics, just an intuitive understanding of ...
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3answers
167 views
Trouble understanding equivalence relations and equivalence classes…anyone care to explain?
What exactly are equivalence relations and equivalence classes? The latter is giving me the most trouble; I've tried to read multiple sources online but it just keeps going over my head.
Example ...
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1answer
67 views
Interpretation of Group Conjugates
So only recently encountering conjugation (in the group-theory sense) in my math adventures/education, and I can't help but ask why? It doesn't seem (at first glance) why its worthwhile defining such ...
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5answers
426 views
Numbers to the Power of Zero
I have been a witness to many a discussion about numbers to the power of zero, but I have never really been sold on any claims or explanations. This is a three part question, the parts are as ...
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What does Frattini length measure?
I have heard derived length, for example, described as a measure of "how non-commutative" the group is. An abelian group will have derived length $1$, whereas a non-solvable group will be so ...
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1answer
118 views
What is a product $\sigma$-algebra?
My question is relatively simple: what is a product $\sigma$-algebra? And why they are important?
Can anyone suggest any links of intuitive (possibly with simple figures) explanations? Or, maybe ...
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1answer
59 views
What is the meaning of detailed balance in Markov Chains?
I know what it means formally to say that a stochastic matrix and a measure are in detailed balance. ($ \lambda_iP_{ij} = \lambda_jP_{ji} \; \forall (i,j)$) but I'm not really sure how to interpret it ...
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1answer
82 views
Suppose you flip a weighted coin…
Suppose you flip a weighted coin that is $3$ times more likely to come up heads. What is the probability that, if you flip the coin $3$ times, you will get an even number of heads?
Can someone help ...
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1answer
165 views
How many strings of six lowercase letters contain the letters a and b in consecutive positions…
How many strings of six lowercase letters contain the letters a and b in consecutive positions with a preceding b, with all letters distinct?
Can someone give me a general method for thinking about ...
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0answers
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intuitive idea of deformations in topology
We know that when we prove that two topological spaces are homeomorphic to each other in fact we are proving that these spaces are in fact equal under deformations.
Why? this question is very ...
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211 views
Intuition behind Cantor-Bernstein-Schroeder
The book I am working from (Introduction to Set Theory, Hrbacek & Jech) gives a proof of this result, which I can follow as a chain of implications, but which does not make natural, intuitive ...
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So $k^2-\Delta: H_{s+2}\to H_{s}$ is a homeomorphism, but what does that tell us?
For each $t\in\mathbb{R}$, we define the Sobolev space \begin{equation}
H_t=\{u\in\mathcal{S}':\int(1+|y|^2)^t|\hat{u}(y)|^2dy<+\infty\},
\end{equation} where $\mathcal{S}'$ is the space of ...
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1answer
424 views
What are some good intuitions for understanding Souslin's operation $\mathcal{A}$?
What are some good intuitions for understanding Souslin's operation $\mathcal{A}$?
Recall the definition: Let $S = \mathbb{N^{<N}} = \bigcup_{n = 1}^\infty \mathbb{N}^n$ be the set of ...
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2answers
66 views
How can we describe isomorphism in a tangible way?
What is the best way of describing isomorphism between two vector spaces? Is there a real life analogy of isomorphism?
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73 views
How does a myopic interpret Wiener's Tauberian?
I just read about this post on the intuition behind convolution. In Terence Tao's answer convolution is interpreted as the blur of image in near-sighted eyes. In Harald Hanche-Olsen's it is made ...
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1answer
80 views
Intuition about the relation of combinations and entropy
It is not difficult to show that
$${n \choose \lambda n} \leq 2^{H(\lambda)n}$$
where $H$ is the binary entropy function:
$$H(\alpha) = -\alpha \lg \alpha - (1-\alpha)\lg (1-\alpha)$$
I was ...
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3answers
58 views
Moment, spheroid, charge redistribution
Let $$I_k:= c \int_{\mathbb R^3} (3x_k'^2-r'^2) \,\,\,d^3 x'$$ where ${r'}^2={x'}_1^2+{x'}_2^2+{x'}_3^2$ and $c$ is a constant = density of charge (uniform) in the body.
Suppose this integral is ...
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202 views
What does an outer automorphism look like?
I am working on a project in my group theory class to find an outer automorphism of $S_6$, which has already been addressed at length on this site and others. I have a prescription for how to go about ...
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Something behind the substitution $h^0=\frac{1}{|G|}\sum_{t\in G}\rho^2_{t^{-1}}h\rho^2_{t}$?
I am quite new to representation theory and I reading Serre's Linear Representation of Finite Groups.
In the first and second chapter, one trick he uses quite often is the substitution ...
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1answer
38 views
How to choose substitution to make the difference equation linear with fixed coefficients?
I am going over some lecture notes and there is the following exercise:
Solve $$(k+1)^{2}y(k+1)-k^{2}y(k)=1$$ with the initial condition
$$y(1)=0$$
where $k$ it for the time, hence not ...
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1answer
164 views
Are Legendre transforms of non-convex functions useful?
Do Legendre transforms have any applications that do not appeal to convexity?
What is the intuitive interpretation of the Legendre transform of a non-convex function?
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How to think about one-point schemes?
As topological spaces, all of $\text{Spec}(k), \text{Spec}(k(x)), \text{Spec}(k[x]/(x^2))$ and $\text{Spec}(k(x_1,\cdots,x_n))$ are all homeomorphic, since they are all one point-spaces.
However, as ...
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Why is the determinant invariant under row and column operations?
I know that we may add any row to any other in a determinant and its value remains the same.
This is clear enough since elementary matrices corresponding to row and column operations have determinant ...
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The Riemann Surface of Logarithmic Function
Who can tell me why the compactification of RS of Log is just Riemann Sphere, please?
