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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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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75 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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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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54 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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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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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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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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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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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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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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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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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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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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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 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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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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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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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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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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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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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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39 views

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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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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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
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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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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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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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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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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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Show that $\int x\mathrm{e}^{-\alpha x^2}\mathrm dx =\dfrac{-1}{2\alpha} \mathrm e^{-\alpha x^2}$ + Constant

I tried to do this integration by parts and got $\int x\mathrm{e}^{-\alpha x^2}\mathrm dx =\dfrac{-1}{2\alpha} \mathrm e^{-\alpha x^2} +\alpha\int x^3\mathrm{e}^{-\alpha x^2}\mathrm dx$ + constant. ...
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Disk in analytic topology vs. the spectrum of a Henselian DVR in etale topology

In this informative and concise set of notes on vanishing cycles by Donu Arapura, it is stated that the theory of vanishing cycles ports nicely to the etale world if the role of the disk is replaced ...
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71 views

A matrix as a point in $\mathbb{R}^{nm}$

I just had a really quick question to ask. I was reading a book on linear algebra and have just been trying to wrap my head around what exactly a matrix represents. At one point, the book said "In a ...
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1answer
26 views

Geometric interpretation of the derivative of a Bezier curve

For a given set of control points $b_0, b_1, \ldots, b_n$, the Bezier curve is defined as $$b^n(t) := \sum_{j=0}^n b_j B_j^n(t),$$ where $B_j^n(t):=\binom{n}{j}t^i(1-t)^{n-i}$ are Bernstein ...
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58 views

Intuition on the Representable Functor

Given a locally small category C, and an object $C$, the functor: \begin{equation} \mbox{Hom}_\textbf{C}(C,-):\textbf{C} \longrightarrow \textbf{Sets} \end{equation} that sends objects to hom-sets ...
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Elements of order $3$ in $\text{Aut}\left(\mathbb{Z}/91\mathbb{Z}\right)$

It looks like someone has already been here, but the question I have goes farther. To summarize my work, as well as the work in the above post, we know that ...
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1answer
60 views

What are the Results of the First and Second Axioms of Countability?

What are the consequences of a space being first or second countable? What was the motivation for these axioms in the first place?
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404 views

What is the intuition behind generating functions? What makes them valuable?

I'm sorry if this question makes no sense. I have been reading generatingfunctionology and I have been able to solve the problems in the first chapters and I understand the mechanism I have to follow ...
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1answer
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Can I have a logical explanation for why this number is so ridiculously close to a whole number? [duplicate]

$$e^ {\pi\sqrt{163}}=262537412640768743.9999999999992\cdots$$ Why does this number run so incredibly close to a whole number? Can I have a logical explanation for why this finding? I know how to ...
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54 views

Is there a way to visualize, like a picture in mind, the $n$-th derivative?

Is there a way to visualize (like a picture in mind) the $n$-th derivative ? For $n=1$ is the tangent line and we can visualize it quite well. More abstractly is it possible to see the geometric ...
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60 views

Why does the division algorithm work for converting between number bases?

I know and have observed that the the division algorithm can be used to convert any number in the decimal system to the binary system. However, I have tried searching for an intuition of why this ...
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What exactly is a differential? [duplicate]

I've seen the formula for differentials alot, namely $$dy=f'(x)dx$$ but what I think when I see this is that someone is manipulating the "formula" $$f'(x)=\frac{dy}{dx}$$ When I think of ...
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1answer
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Problem with Justifying the Formula for First Order Seperable Differential Equations

I am reading this text http://www.math-cs.gordon.edu/courses/ma225/handouts/sepvar.pdf to justify the method to solve first order seperable differentiable equations, where we are told first told that: ...
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1answer
89 views

$A^tA-AA^t$ in Mathematical Physics

In very different contexts of mathematical physics (rigid body mechanics, fluidodynamics, general relativity, quantum field theory,...) I have come across the following expression: $$ A^tA-AA^t, $$ ...
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102 views

Explaining the meaning of equality

I've been tasked with explaining to a group of people what the notion of equality means in mathematics, I've come up with a working explanation, but would appreciate some input, suggestions etc. ...
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How do I visualize this quotient space?

If $V = [0,1] \times [0,1] \subset \mathbb{R}^2$. We define the equivalence relation $\sim$ on $V$ as follows: every element $(x,y) \in V$ is equivalent with itself and besides that the three ...
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Verify $\frac {\partial B} {\partial T} =$ $\frac{c}{(e^\frac{hf}{kT}-1)^2}\frac{hf}{kT^2}e^\frac{hf}{kT}$

Find an expression for $\frac {\partial B} {\partial T}$ applied to the Black-Body radiation law by Planck: $$B(f,T)=\frac{2hf^3}{c^2\left(e^\frac{hf}{kT}-1\right)}$$ The correct answer (I believe) ...