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 for visualising dense monotonic discontinuous function

My question is about the function defined in Rudin 4.31, mentioned by this question: Remark 4.31 in Baby Rudin: How to verify these points? I'm having trouble trying to visualise what such a ...
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3answers
78 views

How would you explain Functional Integration to an 8 year old?

I get the definition of the Functional Integral, but what heuristic interpretations are available to better understand the integral? For instance, what motivates the definition? How is it related to ...
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3answers
71 views

How do mathematician make sense of “outcome” and “events” in probability?

One of the biggest challenge for me to understand probability is to make sense of this concept of outcomes and events. To put it plainly, it just doesn't feel like mathematics anymore when we talk ...
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31 views

The Intution Behind Real Symmetric Matrices and Their Real Eigenvectors

I am wondering about the geometric intuition behind real symmetric matrices and their corresponding linear transformations. Is it possible to understand geometrically why real symmetric matrices ...
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1answer
21 views

What is the quickest way to find Nash equilibria in two player bimatrix game?

Suppose the cost/penalty matrix of a game is given as: $$M = \begin{bmatrix} (-5,-5) & (0,0) \\ (0,0) & (-3,-3) \end{bmatrix}$$ Then the game as two equilibria $(u_{11},u_{21})$ and ...
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Is there a way to visualize a group?

Is there a way to picture a group in ones head? I want to "see" the difference between abelian and non-abelian group. And if f is a group homomorphism, is there a way to see that Ker(f)=1<=>f ...
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57 views

Intuition for Burnside's Lemma (aka Cauchy-Frobenius Lemma)

Here is the theorem: Lemma: Let a group $G$ act on a set $S$. Define $\text{Fix}(g)$ as the set of all elements in $S$ fixed by $g$ under this group action. Then the number of distinct orbits of ...
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547 views

Intuitive explanation of the Burnside Lemma

The Burnside Lemma looks like it should have an intuitive explanation. Does anyone have one?
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32 views

Do I have the correct mental map for adjoint operators for inner product spaces?

Let $X$, $Y$ be finite dimensional inner product spaces, let $A: X \to Y$ be a linear operator, let $A^*: Y \to X$ be the adjoint operator to the linear operator, defined using $<y, Ax>_Y = ...
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239 views

Intuition behind $(-\frac{1}{2})! = \sqrt{\pi}$

It can be shown that using the definition of the Gamma function as: $$\Gamma(t) = \int_0^\infty x^{t-1} e^{-x} dx $$ that $$\Gamma(\tfrac{1}{2}) = \sqrt{\pi}$$ or slightly abusing notation, that ...
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35 views

Intuition about an orthogonal projection operator for matrices

Let $A \in \mathbb{R}^{m \times n}$ be of rank $r$, and $A = U\Sigma V^T$ be its SVD with $\Sigma \in \mathbb{R}^{r \times r}$. Let $P_U = UU^T$ and $P_V = VV^T$ be orthogonal projectors onto the ...
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Problems that are largely believed to be true, but are unresolved

Are there unsolved problems in math that are large believed to be true, but for reasons other then statistical justification? It seems that Goldbach should be true, but this is based on heuristic ...
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1answer
39 views

Constant rank theorem: intuition?

Let $f: \mathbb R^n \to \mathbb R^m$ be smooth and let $x_0 \in \mathbb R^n$ be such that $\operatorname{rank}{(J_f(x_0))} = k $. Then there exists a neighboudhood of $x_0$ and diffeomorphisms $\phi, ...
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23 views

What does a functional integral evaluation look like?

I've read the Wikipedia page on functional integration, but it really isn't very easy to understand. There don't seem to be any online videos on the subject either. In addition, when I search online, ...
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2k views

Intuitive explantions for the concepts of divisor and genus

When trying to explain AG-codes to computer scientists, the major points of contention I am faced with are the concepts of divisors, Riemann-Roch space and the genus of a function field. Are there any ...
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6answers
213 views

$\arctan (x) + \arctan(1/x) = \frac{\pi}{2}$ [duplicate]

How can I show that $\arctan (x) + \arctan(1/x) =\frac{\pi}{2}$? I tried to let $x = \tan(u)$. Then $$ \arctan(\tan(u)) + \arctan(\tan(\frac{\pi}{2} - x)) = \frac{\pi}{2}$$ but it does not ...
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34 views

Sheaf cohomology intuition

I am working on understanding specifically what the $n^{th}$ Cech cohomology group $H^n(\mathcal{U}, \mathcal{F})$ measures, where $\mathcal{U}$ is a locally finite open cover on a topological space ...
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38 views

Are these inequalities useless for getting better estimates? If not what is needed?

Are these inequalities useless for getting better estimates? If not what is needed? My motivation for asking this question is to get a glimpse to the mind of masters that can tell if a line of ...
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19 views

Can more pertubations in eigenvalues/vectors lead to smaller changes?

Say i have a $n$ x $n$ matrix $M$, and i change it's smallest eigenvalue from a small negative value $v$ to a small positive value $t$ to obtain $M^*$: $$M^* = VE^*V'$$ $E^*$ is a diagonal matrix of ...
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4answers
3k views

Simple explanation of a monad

I have been learning some functional programming recently and I so I have come across monads. I understand what they are in programming terms, but I would like to understand what they are ...
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1answer
42 views

What is a residue?

I've heard of residues in complex analysis, contour integration, etc. but all I really know it to be is the $c_{-1}$ term in the Laurent series for a function. Is there some sort of intuition on what ...
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1answer
57 views

On the space $L^0$ and $\lim_{p \to 0} \|f\|_p$

For $0 < p < \infty$, the definitions of the spaces $L^p$ are very natural. Then, we of course want $L^\infty$ and $L^0$ to be some kind of limits of $L^p$ spaces. What does the parameter $p$ ...
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1answer
43 views

Limits and nonstandard analysis - is my intuition correct?

Having nonstandard analysis under our belts, would it be wrong to say that $$\lim_{x\rightarrow a^{\pm ^{}}}f(x)$$ is the same thing as $$f(x\pm ^{}{\mathrm{d} x})$$ where ${\mathrm{d} x}$ ...
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Intuition behind accelerated first-order methods

$\newcommand{\prox}{\operatorname{prox}}$ $\newcommand{\argmin}{\operatorname{argmin}}$ Suppose that we want to solve the following convex optimization problem: $\min_{x \in \mathbb{R}^n} g(x) + ...
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Relation between topological denseness and denseness over poset

In the theory of forcing, the notion of dense set is important. Formally, a subset $D$ of a poset $P$ is dense if, for any $p\in P$ we can find some $q\in D$ with $q\le p$. Intuitively, denseness of ...
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387 views

Intuition - The Shortest Curve Between Two Points is a Line

The references below aver that the following is not crudely trivial: The shortest curve between two points is a (straight) line. An elementary school teacher construed it as follows ...
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1answer
36 views

Rank function can change upwards but not downwards: need intuition

Let $f: \mathbb R^n \to \mathbb R^m$ be some smooth map and $J_f$ its Jacobian. Say $x \in \mathbb R^n$ is such that $$ \operatorname{rank}{(J_f (x))} = p$$ Then there exists a neighbourhood of ...
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Intuition for homotopy (co)limits in triangulated categories

The following definition is taken from Daniel Murfet's Triangulated Categories Part I notes. Let $\mathcal T$ be a triangulated category with countable coproducts. Suppose we are given a ...
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1answer
22 views

Expectation of Independent Variables Equals Zero?

Given $n$ independent random variables, $X_1, X_2, ..., X_n$ , each having a normal distribution, why is it that the following expectation holds? $$E[(X_i - \mu)(X_j - \mu)] = 0$$ where $i \neq j$ ...
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Laplace transformations for dummies

Is there a simple explanation of what the Laplace transformations do exactly and how they work? Reading my math book has left me in a foggy haze of proofs that I don't completely understand. I'm ...
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4answers
481 views

Why can't some integral be“found” though they are anti-derivative & exist?

In my book, a list of integrals have been given which the author states ... such anti-derivatives "cannot be found". Some of the members of the list are as under: $\int\dfrac{\sin x}{x} ...
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699 views

Prison problem: locking or unlocking every $n$th door for $ n=1,2,3,…$

I have a problem called "The Prison Problem" that I need to explain to my 9-year-old cousin. I would think that he has just started learning about divisors and perfect squares, and as such, I have a ...
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5answers
2k views

Why isn't the Cantor Set contradictory?

So you start with a 1-dimensional stick, remove the middle third of it, leaving 2 pieces. From each of these 2 pieces, remove the middle third. Etc. Whatever is left at the end of infinitely many ...
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6answers
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Not divisible by $2,3$ or $5$ but divisible by $7$

The question is to determine the number of positive integers up to $2000$ that are not divisible by $2,3$ or $5$ but are divisible by $7$. The answer is supposed to be $76$ but not sure how it was ...
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Intuitive explanation of div(curlF)=0 [duplicate]

If we consider $\mathbf{F}$ as a vector field, then we say that $\mathrm{div}(\mathrm{curl}(\mathbf{F}))=0$. We can prove this in mathematics easily. But I' am not getting an intuitive explanation due ...
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5answers
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The Meaning of the Fundamental Theorem of Calculus

I am currently taking an advanced Calculus class in college, and we are studying generalizations of the FTC. We just started on the version for Line Integrals, and one can see the explicit symmetry ...
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73 views

Geometric intuition for mixed partial derivatives

I'm trying to better understand exactly what $f_{xy}(x,y)$ at a point is geometrically, and possibly understand why $f_{xy}$ and $f_{yx}$ should be equivalent, not just because the math happened to ...
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4answers
119 views

Is there a purpose behind a function?

As I understand it, a function is a relation between two sets of numbers where as for every input value there is only assigned one output. Or for every $x$ there is only one $y$. What I don't ...
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1answer
99 views

Visualize meaning of quotient in quotient map, group - etc?

What are the reasons for the name "Quotient" in Quotient map, group - etc? Overhead picture shows each of the three cosets in $A_4$ is mapped to a single - gray - node. But this isn't division? ...
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1answer
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Can some help me understand Zeidler's intuitive proof of Brouwer Fixed Point theorem

On pg53, Zeidler gives the Brouwer's Fixed Point Theorem The continuous operator $A: M \to M$ has a fixed point provided $M$ is a compact, convex and nonempty set in a finite dimensional normed ...
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1answer
42 views

Why this process is nonergodic?

I am studying a tutorial on stochastic processes and there's an example in it which I don't understand anything of it. First of all there is this criterion for a mean-ergodic random process: For ...
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0answers
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Intuition on $S_4/K_4\cong S_3$ [duplicate]

This is a question I was pondering on my way to class, and may not have an answer. Take the normal copy of $K_4$ in $S_4$ (The group $K_4 \cong\{e,(12)(34),(13)(24),(14)(23)\}$, not the non-normal ...
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1answer
298 views

Characterize normal subgroups - Find all subgroups of $S_3$ conjugate to $\{id, (1,3) \}$ - Fraleigh p. 143 14.29

(27.) A subgroup H is conjugate to a subgroup K of a group G (viz. p. 141 $K \le G$ is a conjugate subgroup of $H$), if $i_g[H] = gHg^{-1} =K$ for some $g \in G$. Show that conjugacy is an ...
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1answer
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Is tossing a die in 10 consequent days an ergodic process?

IT maybe an elementary question but I'm totally new to the concept. In Wikipedia, ergodicity is defined as follows: In statistics, the term describes a random process for which the time ...
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1answer
25 views

what's the difference between variable and process from a statistical point of view?

I'm reading a tutorial stochastic process: ergodicity and temporal averages and I'm totally confused. It is said that: Suppose an IID random process whose marginal PDF is Gaussian with mean ...
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6answers
8k views

In (relatively) simple words: What is an inverse limit?

I am a set theorist in my orientation, and while I did take a few courses that brushed upon categorical and algebraic constructions, one has always eluded me. The inverse limit. I tried to ask one of ...
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4answers
130 views

Definition of Equivalent Norms

Two norms $F,G$ are equivalent when there are constants $a,b$ such that $aF \le G \le bF$. I'm reading about this idea, and so far I've seen that equivalence of norms implies that the underlying ...
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35 views

Hyperplane in a complex vector space

This is my first question on MSE, I'm sorry if there already exists similar questions, I couldn't manage to find it. My friend, who studies Physics, asked me about the meaning of "functional" so I ...
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1answer
55 views

Can someone help me understand this: integrating over a discrete set of points yields 0 under Lebesgue integral?

Suppose I had some linear function $f(x)$ and then I sampled the function over the integers to form $f(n)$, what would be the evaluation of the Lebesgue integral of $\int_\mathbb{Z_+} f(n) d\mu$? For ...
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358 views

If sup A < sup B, there exists an element b ∈ B that's an upper bound for A. (S.A. pp 18 q1.3.8)

My Figure: By definition of $\sup B$, $\sup B$ is an upper bound for $B$. Set $e = \sup B − \sup A > 0$. By Lemma 1.3.7, there exists an element $b ∈ B$ satisfying $\begin{align} & \sup B − ...