Questions about understanding a problem geometrically.

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Interpretation of this linear application

If $r$ is a unit vector, the reflection with respect to the hyperplane of normal $r$ corresponds to the matrix $I-2rr^\top$ (known as a Householder matrix). Now, let $r_1,r_2$ be two unit (column) ...
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34 views

Is $T^2 \# \mathbb RP^2\cong \mathbb RP^2\# \mathbb RP^2\# \mathbb RP^2$?

I was reading a little about how to imagine the projective plane and I have some weird intuition that says $T^2 \# \mathbb RP^2\cong \mathbb RP^2\# \mathbb RP^2\# \mathbb RP^2$. Is this true, and if ...
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9 views

Adding two functions represented by a table of values with a different step size?

Let $f(t)$ be some numerically obtained $T$-periodic function represented by a table of values over one period or a set of points $(t, y)$ with a time step $\Delta t.$ Now let's change the frequency/...
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1answer
43 views

How to find the closest point to three vector lines?

So this is the question here I know the angles $A$ and $B$ for each individual, and their positions in longitude and latitude (assuming height of person $z =0$), am I correct in thinking that for any ...
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2answers
42 views

What is the interpretation of homogeneous line intersection?

I understand homogeneous coordinate systems. I read the intersection of lines in homogeneous coordinate can be computed by taking a cross products of lines $l_1(a_1,b_1,c_1)$ and $l_2(a_2,b_2,c_2)$. ...
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19 views

Covering maps as bundles

One geometric way to see a continuous map (or any set function really) is as a "fiber bundle" with the usual picture of a comb - the base space indexes the fibers of the map and there's a nice picture ...
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30 views

Pulling back along surjective étale maps vs being “locally in $\mathcal M$” vs being “locally in $\Sigma \mathcal M$”

(Closely related) This question centers around section 6.5 of Borceux and Janelidze's Galois Theories. Definition 1. Let $\mathcal M$ be a class of arrows in a category (in our case $\mathsf{Top}$). ...
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41 views

What is geometrical interptetation of a set being measurable.

What is geometrical interptetation of a set being measurable. I mean what does it mean geometrically by a set is measurable...
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89 views

Is there a geometrical interpretation of this equality $2\cdot 4\cdot 6\cdot\ldots\cdot(2n)=2^nn!$?

$$2\cdot 4\cdot 6\cdot\ldots\cdot(2n)=2^nn!$$ How it can be seen in a plane? I have found many proofs with by induction but I wish to understand it geometrically.
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69 views

Relationship between gradient and velocity?

I recently learned that when you have a real-valued function $f(x,y) = c$, you create a plane parallel to the $xy$-plane. Inside that plane we have a level set, such that the derivative of its ...
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3answers
70 views

Geometrical representation of speed?

I've been learning about position vectors, and how their derivatives show the velocity (first derivative), and acceleration (second derivative) of a moving body. From Mechanics I learned that, the ...
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48 views

Geometric interpretation of $ \frac{x^2+y^2}{y}=\text{constant} $.

I would love help in interpreting the following expression geometrically $$ \frac{x^2+y^2}{y}=\text{constant} $$ for simplicity, let $ c = \text{constant} $, and then through rearrangement we have $...
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17 views

Geometric intuition for conjunctive spaces

A topological space $S$ will be called conjunctive if for each open set $A$ containing a point $p$, there's a point $q\in S$ satisfying $\overline{\left\{q \right\}}\subset A\cap \overline{\left\{p \...
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1answer
68 views

Question about geometric interpretation of modules

I would like to understand the accepted answer to this MO question about the geometric interpretation of modules. In particular, I would like clarification on the following excerpt. Let $R$ be the ...
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66 views

Geometric meaning of vanishing of higher cohomology of quasi-coherent modules over affine schemes

One of the basic vanishing results about quasicoherent (sheaves of) modules over affine schemes is that their non-zero cohomology vanishes. My only geometric intuition for sheaf cohomology is via ...
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56 views

What is the geometric implication of subtracting two Matrices representing linear transformations?

If we have two linear transformations denoted by matrices $A, B$ operating on an arbitrary vector $v \in \mathbb R^n$, then how does $Av$ and $Bv$ differ geometrically from $(A-B)v$ ? Does the ...
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28 views

Whats the terminology for defining a point on a triangle?

There are 2 common methods of representing a point on a 2d/3d triangle. 2 numbers (often called "UV coordinates" in 3D graphics):Where 2 edges of the triangle are axes, which the point is translated ...
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26 views

How to resample translated grid to ensure consistent interpolation?

I have a grid of values, sampled at certain locations. I'd like to translate grid by some offset and resample it. The questions is: what should be the resampling and interpolation formulas such that ...
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2answers
50 views

Interpretation of $a+b \ | \ a^n + b^n$ for odd $n$

It is not hard to show that $a+b \ | \ a^n + b^n$ for odd $n$. (because $f(x) = x^n - b^n = (x-b)h(x)$ we have $a - b \ | \ a^n - b^n$, so $a - (-b) \ | \ a^n - (-1)^n b^n$) Is there a nice ...
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19 views

How to mathematically model a realistic aperture illumination?

I want to know a mathematical expression that i can use to model a realistic aperture illumination to produce the primary beam of an antenna so that the radial distribution of this aperture ...
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28 views

Intersection of 4D curves, one is 3-sphere other is the determinant of the four dimensions

Let the curve with 3-sphere be represented by equation, $$a_{11}^2 + a_{12}^2 + a_{21}^2+ a_{22}^2 = C_1$$ And the curve with the determinant of matrix, $A = \begin{bmatrix} a_{11} & ...
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3answers
143 views

Geometric interpretation of inverse complex function?

Function $f\colon\mathbb{R}\to\mathbb{R}$ and its inverse $f^{-1}$ are symmetric over line $y=x$. It's easy to imagine inverse of real function, we just have to "flip" the plot over $y=x$. But what ...
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1answer
103 views

What can/do mixed second-order partial derivatives represent?

What I know from basic calculus: The first derivative of a function $f:\mathbb{R}\to\mathbb{R}$ gives an indication of the function's behavior (namely, how the value of $f(x)$ changes as $x$ changes). ...
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102 views

What does Dini continuity mean?

What does Dini continuity (the integral condition) mean visually? Description of Dini contuity: https://en.wikipedia.org/wiki/Dini_continuity
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Why does the Hilbert transform produce the Hilbert envelope?

The Hilbert transform is often described as an improper integral. It can be also characterized as a convolution between two functions. But why is it these descriptions that produce the so called ...
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1answer
105 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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70 views

What is the Geometric interpretation of $i^i$?

We know that $i^i$ is real. But how to explain it geometrically maybe in terms of rotation. like we can explain geometrically multiplication of two complex numbers and so on. Can someone show me a ...
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1answer
75 views

Geometrical condition for the intersection of two circles

On the plane, one can give a simple condition for the intersection of two circles $\mathcal{C}_1(O_1,r_1),\mathcal{C}_2(O_2,r_2)$ : they intersect iff $|r_1-r_2|\le|O_1O_2|\le r_1+r_2$. Likewise, can ...
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43 views

Open sets in $\mathbb C$ and open sets in $\hat{\mathbb C}$

I usually have a lot of trouble with complex variable when it comes to the geometric representation of $\hat{\mathbb C}$ and what happens in there. I have the next exercise and from quick look at it I ...
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1answer
125 views

Is there a nice meaning to the geometric triple product?

Using geometric algebra, I can easily find the geometric tripleproduct of three vectors $a,b,c \in \mathbb{R}^3$ to be $$abc = a \left(b \cdot c \right) - b \left( c \cdot a \right) + c \left( a \...
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97 views

Geometric Interpretation of Fractional Derivatives

I was looking for a geometrical interpretations of fractional derivatives and fractional integrals. I would be glad to see any kind of intuitive and preferably visual interpretation of the objects ...
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206 views

Geometric Interpretation of the Basel Problem?

I'm probably asking a question no one knows the answer to, but everyone must mentally ask at some point. Does the Pi in the solution to the Basel problem have any geometric significance? Every time ...
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25 views

How do I put this into a linear equation?

I really don't know what I would do first. I moved $x$ to the other side of the equal sign to make it $2y = 12 + x$ and then I was going to divide it by $2,$ but i realized you can't divide $x$ by $2,$...
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95 views

Geometric interpretation of 2D-Translation's Matrix representation

I just learned the trick of writing a translation of a 2-dimensional real vector as a matrix multiplication in a 3-dimensional space - wikipedia explains it here. Basically it shows: $$ \begin{...
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85 views

Geometric interpretation of determinant of a system of homogenous linear equations

What is the geometric interpretation of the determinant of a matrix representing a system of homogenous linear equations? We know that iff the determinant is equal to zero the system has a non-...
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76 views

Are there any formulas or identities that involves $\pi$ but have no obvious trigonometric interpretation?

Every formula that involves $\pi$ has an underlying trigonometric interpretation but it is not usually obvious. I wonder if there are any formula like the Gaussian integral $f_{n}(x)=\sqrt{\frac{1}{...
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59 views

Find the geometric interpretation of general solution

Here are the linear equation: $$2x+z=0$$ $$-x+3y+z=0$$ $$-x+y+z=0$$ I have found that the general solution is, $$t \begin{bmatrix} \frac{-1}2\\ \frac{-1}2 \\ 1 \\ ...
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264 views

physical interpretation or practical meaning of a distance

How can I find/understand the physical explanation or practical meaning of a distance? For example Euclidean distance is the shortest path between two points, how about for other distances?
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459 views

Why is it that $\int_a^b \int_c^d f(x)g(y)\,dy\,dx=\int_a^b f(x)\,dx \int_c^d g(y)\,dy$?

The title sums it up. It's simple to prove, but I'm wondering if there is a geometric interpretation?
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51 views

Interpretation of the complement $S_5 \setminus D_6'$.

On account of a homework question $$ \text{does there exist an injective homomorphism } D_6 \to S_5 \text{ ?}$$ which I solved positively, I raised the question $$ \text{what does the complement }...
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166 views

Rotation of Matrices and their interpretation

Given are now two matrices and I have to discuss what the given functions are doing (geometrically). Maybe you can revise/add the following: given are the matrices $ A = \begin{pmatrix} \cos(a) &...
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108 views

Interpretation of vector line integrals

I recently came across two types of line integrals which I had not previously come across before and am unsure as to what their interpretation would be: Integrating a scalar field with vector line ...
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1answer
28 views

Showing for $f,g:\mathbb{R}\rightarrow\mathbb{R}$ that for $M>0, |f(x)|\le M|g(x)|$ for $x>x_0$

Showing for $f,g:\mathbb{R}\rightarrow\mathbb{R}$ that for $M>0, |f(x)|\le M|g(x)|$ for $x>x_0$. This is a repost of a question that is probably too long to ever get an answer (I feel compelled ...
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Odly phrased big-oh question, have I done what is required? (because it's not really big-oh, it's more graph sketching)

I've encountered these before, but never phrased or defined as follows, I'd like to know if I've done whatever the question wants to draw attention to (if it didn't want to draw attention to something,...
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3answers
362 views

Geometry of the dual numbers

A dual number is a number of the form $a+b\varepsilon$, where $a,b \in \mathbb{R}$ and $\varepsilon$ is a nonreal number with the property $\varepsilon^2=0$. Dual numbers are in some ways similar to ...
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1answer
102 views

Can something like $\text{Hom}(V,K)$ be visualised?

I have no trouble visualising vector spaces like $\Bbb R^3$ and (e.g.) a subspace of dimension $2$, which would just be a plane through the origin of a $3$-D space, but I'm having trouble visualising ...
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Interpretation of Linear Algebra and superposition

I have been told and have seen that knowing Linear Algebra is foundational in pursuing advanced mathematics. After dealing with it enough I have gotten "used to it" (I still need a lot more practice!) ...
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1answer
2k views

Do the BAC-CAB identity for triple vector product have some intepretation?

As in the title, I was wondering if the formula: $$a\times (b\times c)=b(a\cdot c)-c(a \cdot b)$$ for $\mathbb R ^3$ cross product has some geometrical interpretation. I've recently seen a proof (from ...
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2answers
80 views

expected value of random variables

Take two random variables $X=a+bX_0$ and $Y=c+dY_0$, and define $T=X-Y=\mu+\sigma Z$ where $\mu$ is the mean of $T$, $\sigma$ its standard deviation and $Z$ is a standardized random variable with mean ...
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183 views

$i^{th}$ root(s) of unity

If we define $S:=\{z\in\mathbb{C}:z^n=1\}$ (i.e. the $n^{th}$ roots of unity), then $|S|=n$ (i.e. we have $n$ of them). We can even go as far as to say: $$S=\{z_k:k\in\mathbb{N}\cap[1,n]\}=\{e^{2\pi i/...