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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Distinguish Normal Distribution, Gaussian Distribution and Normalised Gaussian Distribution?

As I understood it, the 'normal distribution' is $$\frac{1}{\sqrt{2\pi}}\exp\left(\frac{-(x-\mu)^2}{2{\sigma}^2}\right)$$ Now according to this the 'normal probability density function' is ...
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21 views

Intuition of transversality equation

I am studying Differential topology from Guillemin / Pollack and unfortunately i cannot understand ıntuition of Transversality equation Let $f$ be a smooth map between smooth manifolds $X$ and $Y$ ...
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2answers
72 views

Partition of Unity for defining Riemannian metric.

Why do we need Partition of Unity for defining a Riemannian metric on a manifold ? What role does it play ?
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35 views

Logarithmic growth and integrability

I was helping a graduate student with a problem, and I realized that I didn't have a good intuitive explanation for the following phenomenon: $$ \int_e^\infty \frac{dx}{x\ln^p(x)}<\infty \quad ...
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66 views

Geometrical interpretations of SVD

I'm a bit confused by the various geometrical/visual interpretations of SVD or better I'm wondering how to reconcile them. Transformations : As explained here, the 3 matrices produced by the SVD can ...
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2answers
79 views

2 Cards are drawn from a deck of cards.

I use this site for learning mathematics. I have come across this question. and the explanations are very much clear. Suppose the question is 2 Cards are drawn from a deck of cards. What is ...
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1answer
101 views

surface area of cylindrical spiral

consider $$\{(r\cos(\phi),r\sin(\phi),\phi)|r \in (0,1), \phi \in (0,2\pi)\}.$$ is the surface area the same as that of the unit circle, that is $\pi$? Intuitively yes, maybe not :S
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44 views

fourier series of unknown functions

I am confused in understanding use of fourier expansions of functions. This answer, for example says that we can write voice as a sum of sines and cosines of different frequencies and amplitudes, but ...
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170 views

Intuition for the GP formula.

Okay, so I wanted to ask, whether the formula for a geometric progression also has an intuition. There is an intuition for the AP formula: $$\frac n2(2a+(n-1)d)=\frac n2(a+(a+(n-1)d))$$ So what we ...
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1answer
49 views

Fractional Sobolev spaces definition

Fractional Sobolev space $H^s_p(\mathbb R), s>0, 1<p<\infty$ is a space of tempered distributions $f$ that satisfy $F^{-1}((1+|\xi|^2)^{s/2} F(f)) \in L^p(\mathbb R)$. Here, $F$ denotes the ...
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119 views

Initial and Final Objects in a Category

I understand the definition of initial and final objects in a category: that an object $\frak{I}$ is initial in a category $\frak{C}$ if for every other object in $\frak{C}$ (we'll just call it ...
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1answer
60 views

Intuition for orthogonality in infinite dimensions

I'm trying to explain orthogonality in inner product function spaces (e.g. Hilbert spaces) intuitively. As main expample, take the $L^2$ inner product given by $$<f,g>_{L^2(I)}:=\int_I ...
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84 views

How do I acquire an intuitive understanding of the distributive law over a disjunction?

The distributive law over a disjunction is given to be: $ P \vee (Q \wedge R) \equiv (P \vee Q) \wedge (P \vee R) $ I want an intuitive understanding of this statement, in order to do I tend to write ...
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1answer
67 views

Are there any connections between the probability of poisson distribution when x=k and x=k+1?

We know in poisson distribution, we have: $$P(x=k)=\frac{\lambda^k}{k!}e^{-\lambda}$$ so $$\frac{P(x=k+1)}{P(x=k)}=\frac{\lambda}{k+1}$$ I want to know are there any intuitive explanation of this ...
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1answer
43 views

Degree of an algebraic set

I am learning about degree of algebraic sets. I know the definition from Wikipedia https://en.wikipedia.org/wiki/Degree_of_an_algebraic_variety , but it is not too clear to me what it is. Could ...
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2answers
56 views

Clarification needed regarding DeMorgan's Law

I'm currently going through Daniel Velleman's "How to Prove It". He states DeMorgan's Law as follows: 1. $\neg (P\wedge Q)$ is equivalent to $\neg P \vee \neg Q$ 2. $\neg (P \vee Q)$ is ...
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66 views

Intersection of 2 planes?

The question asks to find the parametric equations of the line of intersection between the planes $3x+2y-z=28$ $x-4y+2z=0$ I think I know how to do it and I think I got the right answer, but I ...
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128 views

Why non-regular curves have cusp(s)

I'm very confused about the author's explanation of regular curves in my calculus book. The author says that a regular curve $\gamma:[a,b]\to\mathbb{R}^3$ is a curve such that $\gamma'(t)\neq 0$ for ...
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78 views

What is the connection between a metric and a manifold?

I am in process of reading a paper which contains something called a "Shahshahani Metric" which has uses in mathematical biology ...
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1answer
68 views

Does the equality $ \dfrac{10!}{6!} = 7! $ hold any special (geometric) meaning?

I've come across the following simple, but unexpected equality numerous times accidentally. $$ \frac{10!}{6!} = 7! $$ which is the same as $$1*2*3*4*5*6*7 = 7*8*9*10$$ Does it hold any specific ...
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64 views

Intuition on why there can't be a continuous bijection between $(a,b)$ and $[c,d]$?

Intuition on why there can't be a continuous bijection between $(a,b)$ and $[c,d]$? I'm not (necessarily) looking for a proof for this, I want to understand why does this happen, intuitively: if I ...
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4answers
53 views

Clarification on meaning of “Gaussian random variable”

When my lecturer uses the word Gaussian random variable, he always writes the pdf of the Gaussian instead of the random variable itself. For example, given a random variable $X$ Gaussian, $f_X(x) ...
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1answer
26 views

Subtle difference between convergence in measure and almost uniform convergence?

I am reading Terence Tao's post on convergence types. Consider the following two types: We say a sequence $f_n : \mathbb R \to \mathbb R$ converges in measure to a function $f$ if and only if ...
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1answer
34 views

Invariance of the number of holes

Suppose that $\Omega$ is an open subset of $\mathbb{R}^2$. What conditions on $\Omega$ ensure that $\Omega_k$ homeomorphic to $\Omega_l$ implies $k=l$, where $\Omega_k$ denotes $\Omega$ with any $k$ ...
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71 views

Some clarification needed on the Relation between Total Derivative and Directional Derivative

I will consider here functions of several variables only. If both directional derivative $D_{v}f(x)$ at $x$ along $v$ and total derivative $D f(x)$ at $x$ exist then $$D_{v}f(x)=Df(x)(v).$$ ...
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2answers
68 views

What is the difference between $-1^2$ and $(-1)^2$? [duplicate]

Intuitively, I though that $-1^2$ and $(-1)^2$ were exactly the same thing; however, it seems I was wrong as Wolfram Alpha (and any other calculator) returns $-1$ for the first case and $+1$ for the ...
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1answer
55 views

Sine as derivative of cosine and vice versa. [closed]

The derivative of the sine of angle is the cosine of the same angle. Depending on which book you look at or which teacher you ask or which website you're looking at, I've seen it as derivative of sine ...
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1answer
95 views

How to visualize differential forms geometrically

I've been attempting to teach myself differential geometry and I have heard that one can visualise them geometrically and that this can sometimes be helpful for an intuitive understanding of them. For ...
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41 views

How to understand the geometry of bilinear forms that are not positive-definite?

I simply cannot find a good resource that explains intuitively how to understand the geometry that is induced on a vector space when the bilinear form is not positive-definite. In the ordinary ...
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1answer
52 views

Definition of adjoint functor similar to the definition of homotopy equivalence?

I am new at category theory and I haven't get the definition of adjoint functor. I have seen the another definition of natural transformation much similar to the definition of homotopy. Other ...
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86 views

Connecting a vector space to its dual - why?

Can someone explain to me - intuitively - why embedding a vector space into its dual should naturally fix its geometry? I mean, I can run the usual statements through my mind - "The injection into the ...
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893 views

Why countable unions, intersections etc.?

I was just wondering why one always insists on countability when it comes to the definition of a $\sigma$-algebra in measure theory. I mean, measure theory works as it does, but is there a deeper ...
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38 views

Examples of Lp spaces in Applied Math

I was wondering if there are examples of exotic Lp spaces being used in applied mathematics. I know that the "special" p's (1,2 , infinity ) are of use, for example in statistics, L1 is mean, L2 is ...
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3answers
43 views

A more formal but intuitive understanding (on a definition) of a group action

We know that a symmetric group $S_n$ acts on the set $\{1, 2,\ldots, n\}$. The definition of an action of a group $G$ on a set $S$ is a function $G\times S\to S$ such that: 1) $e\ast s=s$ 2) ...
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69 views

What is the good way to remember the signs of the rotational matrix?

Recall rotational matrix in (x,y) is given by: $R = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}$ For the life of me I cannot remember if the ...
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23 views

What is the purpose of continuous and differentiable dependence

In learning Gronwall's inequality you also get to learn about continuous an differentiable dependence. I know the theorems but I have no idea about their application. What is the big idea of ...
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1answer
142 views

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? The function is defined as $$f(x) \colon= \sum_{x_n < ...
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64 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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3answers
109 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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1answer
62 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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0answers
109 views

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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1answer
97 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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3answers
89 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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42 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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1answer
66 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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1answer
68 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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39 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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3answers
322 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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59 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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242 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 ...