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Questions tagged [transformation]

Transformation has many meanings in mathematics. If using this tag, add another tag related to the object being transformed. If there is a tag for your specific kind of transformations, use that one instead: e.g., (laplace-transform), (fourier-analysis), (z-transform), (integral-transforms), (rigid-transformations).

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Transformations of RV's Ensuring Absolute Continuity of Quantile Functions

Given a real random variable $X$, suppose $T:\mathbb{R}\to\mathbb{R}$ is non-decreasing. Define $Y=T\left(X\right)$. Let $Q_{X}$, $Q_{Y}$ be the corresponding right-continuous quantile functions. ...
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573 views

Way to Tietze's Transformation Theorem

During our knot-theory lecture we have talking about the following theorem: Given two finite presentations of the same group, one can be obtained from the other by a finite sequence of Tietze ...
6
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50 views

Certain symmetrized product of cosines - can it be transformed into more manageable form

I am interested in the following expression: $$ F_{k_1,\ldots,k_n}(t):=\sum_{\sigma\in S_n}\cos(\sigma(1)k_1t)\cos(\sigma(2)k_2t)\cdots\cos(\sigma(n)k_nt) $$ where $k_1, \ldots, k_n$ are natural ...
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142 views

Trying to find a specific rotation quaternion

I'm looking for a way to find a specific rotation quaternion. Hoping that I'll get the notation right (no mathematician) the basic problem looks as follows Definitions $t \in \{0,...,n\}$ $R(Q, \...
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160 views

Applying PCA on covariance matrix in order to generate a new random variable.

Let $\mathbf{x}$ be a random $n\times1$ real vector, $\mathbf{x}\in\Bbb{R}^n$, which is distributed normally with mean $\bar{\mathbf{x}}$ and covariance matrix $\Sigma_x\in\Bbb{R}^{n\times n}$, i.e. $\...
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428 views

How to use polynomial or conformal transformation

In my research, I came to a transformation problem. The simple version is an initial circle (or sphere) region is advected by some deformational flow. After some time the circle will be deformed into ...
4
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54 views

What area of math develops an overaching theory of transforms?

I am working on some things that require Laplace, Fourier, and Mellin transforms (and a few others lurking in the background). Simply put, if seems a transform is any function $G$ such that $F(g)=F(...
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408 views

Is it possible to obtain a torus by gluing 2 Möbius bands?

Is this transform actually possible? Which is gluing edges of two Mobius bands to make a torus? I tried to do it physically with pieces of paper, but I couldn't complete it. The detailed explanation ...
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79 views

Best sources on complete transforms (classic orthonormal transforms) and overcomplete transforms in signal processing

In the introduction section of a thesis I read a little about classic orthonormal transforms such as Fourier, discrete cosine and wavelet transforms and their application in signal processing. Then ...
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85 views

Transform recurrence relation

Is it possible to transform following recurrence relation $a_n=4a_{n-2}-a_{n-4}$, $a_0=1$, $a_1=0$, $a_2=3$, $a_3=0$ so that it will have nonnegative coefficients? Number of terms, of course, can be ...
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553 views

Hodograph transformation and implicit solution of a non-linear PDE

I am trying to understand how can one apply the Hodograph transformation to a non-linear PDE. I read that this transformation implies the representation of the solution in the implicit form . So, if I ...
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548 views

Geometric intuition for Jordan normal forms (invariant subspaces, shearing, scaling, etc.)

I'm trying to visualize what a linear operator does to a vector space if that operator can be put into Jordan normal form. For concrete motivation, let's take $V = \mathbb{R}^3$, with some linear ...
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33 views

Matrix Transformation for 2D, how do I tell what this matrix does geometrically?

Given a 2x2 matrix, $$\begin{bmatrix}1&-1\\-1&\frac12\end{bmatrix}$$ What geometric effect does it have? So a way I did to solve this was to simply apply it to a unit square that I drew on a ...
3
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281 views

Strong convexity and the Legendre transform

Suppose that I have a strongly convex function $f(\mathbf{x}): \mathbb{R}^m \rightarrow \mathbb{R}$. Is the Legendre transform of this function also strongly convex? As far as I can tell, strict ...
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143 views

Determination of a Joukowski airfoil chord (demonstration)

I'm currently studying Aerodynamics, and one thing that I noticed is that the maximum and minimum $x$ coordinate of the airfoils (which are necessary to compute the chord) on the transformed plane (...
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30 views

Is there a link between the two definitions of homogeneity for differential equations (for a first order equation and diff equations of higher order)?

According to Wikipedia, there are two definitions for homogeneity of a differential equation. The first is for first order differential equations: A first-order ordinary differential equation in ...
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67 views

pdf of transformed random variable $g(X)$ as integral over $X$?

I am not a mathematician, so I am sorry if this question is too easy or some notational detail is not correct. I am trying my best! I have got a random Variable $X$ in $\mathbb{R}^N$ with pdf $p(X)$ ...
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266 views

How to transform (rotate) this hyperbola?

Given this hyperbola $x_1^2-x_2^2=1$, how do I transform it into $y_1y_2=1$? When I factor the first equation I get $(x_1+x_2)(x_1-x_2)=1$, so I thought $y_1=(x_1+x_2)$ and $y_2=(x_1-x_2)$. ...
3
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289 views

Find transformation matrix with respect to another basis

I understand how we can find the transformation matrix $D$ with respect to another basis $B$, by using a transformation matrix that we already know, say $A$: $$D = C^{-1}\cdot A\cdot C$$ Where $C$ is ...
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500 views

Proof that Determinant is Scale Factor

I've seen a lot of supposed properties of linear transformations that're never proven -- just often repeated. These include: The determinant is the scale factor between the volume of region in your ...
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255 views

Invariant functions under integral transforms

We all know Fourier transform has invariants such as $e^{-x^2}$, and another MSE post has shown the non-existence of invariant function under Hilbert transform using Fourier transform. I am wondering ...
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84 views

Heat equation $\frac{\partial \theta}{\partial t}=\kappa \frac{\partial ^2\theta}{\partial x}$ using two transformations to solve

Consider the heat equation $$\frac{\partial \theta}{\partial t}=\kappa \frac{\partial ^2\theta}{\partial x}$$ for an infinite rod. We use the transformation $q_1=\frac{x^2}{kt}$ and $q_2=\frac{\theta \...
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139 views

Changing coordinate system with non standard definitions

The standard coordinate transformation to polar coordinates is $$ \begin{cases} x=r\cos(\varphi)\\ y=r\sin(\varphi) \end{cases} $$ with $r\in[0,\infty), \ \varphi\in[0,2\pi)$ The question is whether I ...
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194 views

Proof for a summation-procedure using the matrix of Eulerian numbers?

I've discussed a procedure for divergent summation using the matrix of Eulerian numbers occasionally in the last years (initially here, and here in MSE and MO but not in that generality and thus(?) ...
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22 views

Definition of arg$(x)$ where $x\in\Bbb R^d$ and further properties.

In a text I am reading occured the reflection in the line $\Bbb R^d z$: For a fixed $z\in\Bbb R ^d $ we define $$T_z : \Bbb R^d \to \Bbb R ^d , T_z (x) := 2\langle x,z_0 \rangle z_0 - x$$ where $z_0 :...
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71 views

3D inverse transformation jacobian

I am working on robotics and I have to deal with transformations. In my case, $\pmb{t} = \left[x \; y \; z \; \psi \; \theta \; \phi \right]^T$ where $x$, $y$ and $z$ is the translation and $\psi$, $\...
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37 views

Does, under iteration, all strongly mixing transformations tend to spread sets out, not only in (ordinary) diameter, but also in harmonic diameter?

In [R. E. Rice, On mixing transformations, Aequationes Math. 17 (1978), 104 – 108; Theorem 2 (motivated by some physical phenomena and offer some clarifications of these phenomena)] it is shown that, ...
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0answers
112 views

how to transform ellipse back to circle

I have a circle in a $xz$ plane with its center at Origin $O$ as shown in the diagram. The circle is being observed by a camera from point $C$ at $yz$ plane at an angle $\beta$ with the $z-$ Axis. ...
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132 views

Continuity and differentiability of the CDF of a transformation of a random variable

I am interested in analyzing the properties of the following function: $$a(z)=Pr\left(g(X) \leq z \right),$$ where r.v. $X$ is absolutely continuous with respect to the Lebesgue measure in $\mathbb{R}...
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125 views

Alternative definition of Legendre transform as an integral

Let $f(x)$ be a convex function. Define $g(y)$ via the integral: $$\mathrm{e}^{-g(y)} = \int_{-\infty}^\infty \mathrm{d}x \, \mathrm{e}^{yx-f(x)}$$ assuming that the integral converges. The domain ...
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39 views

Transform an exponential equation within another exponential equation

My question comes from a statistical problem I am bumping into but I think it is more a math question than a stats question, therefore I post it here. Anyway, I have a Structural Equation Model that ...
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0answers
49 views

How to relate two polar coordinate systems with different origins

Suppose I have a polar coordinate system defined by $\theta$ and $R$. How do I relate this system to a new polar coordinate system $(\nu, r)$ whose origin lies at $\theta=\theta_0$, $R=d$? My first ...
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0answers
75 views

Conformal mapping - known points?

I have a hopefully rather simple question: I want to experiment with different geometries of flowlines and equipotential lines in a 2-Dimensional space in order to fit experimental data. Flow lines ...
2
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0answers
63 views

A transformation of a discrete random vector

I have a vector $X=(X_1,\ldots,X_n)$ of i.i.d. random variables with values in a discrete set $S \subset \mathbb{R}$ and a (continuous but not necessarily linear) function $F:\mathbb{R}^n\mapsto \...
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94 views

How do I draw this picture in squares of discrete $\sqrt{z}$?

From Richard Kenyon's homepage gallery: I want to understand the mathematics of this, and similar/related transformations. ... An explanation in words (1st year uni level maths) would be ideal. I'...
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372 views

Proving non-linear mapping is invertible using partial derivatives only

Given $f : \mathbb{R} \rightarrow \mathbb{R}$, it's possible to show that $f$ is a bijection by considering its derivatives only: if the derivative is always positive or always negative, then the ...
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0answers
251 views

Angle by which tangents to curves at $z_{0}$ are rotated under the mapping $w = z^{2}$

I have to find an angle by which tangents to curves at $z_{0}$ are rotated under the mapping $w = z^{2}$ if (a) $z_{0} = i$, (b) $z_{0} = -1/4$, (c) $z_{0} = 1+i$, and also find the corresponding ...
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0answers
332 views

Perspective correction from 3 points and foreshortening factor

I'm working on creating a homography 3x3 matrix to do a perspective correction of a photograph 2D piece of paper. The paper contains 3 markers (like the 3 corner markers of a QR code) in its corners, ...
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0answers
119 views

How to determine changing scale factors when performing coordinate transfomations?

To explain: I have two coordinate systems. One $(x,y)$ and the other $(x',y')$ as seen in this diagram. Coordinate systems I am trying to convert the coordinate in the $(x,y)$ system to the rotated ...
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0answers
53 views

Distribution of discrete function of continuous random variable?

It has been quite some time that I did statistics, and I am not sure how to figure out the distribution of a function of a random variable if the function itself discretizes (if that is a word) the ...
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0answers
347 views

Fourier Transform of triangle function

I have a question regarding the FT of the triangular function: How does $e^{-j\omega t}$ becomes the cosine function in the first line? What happened to the sine when you go from $e^{j \omega t}$ ...
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0answers
78 views

Change from Fourier Space to Real Space

I have a function in 3D fourier Space $$g(\textbf {k})=\frac{\hat{k}_i}{\hat{k_j}}f(\textbf {k}),$$ where $\hat{\alpha}$ is a fixed vector and $i$ and $j$ are the components of the relevant vector, ...
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0answers
37 views

What is the name of this transformation's property?

I have a transformation $P$ with the following property: $P^n = \mathbb{I}$ (the identity) for some specific $n>1$, and all $P^m \neq 1$ for $m \neq n$. What is the name of the property of $P$? ...
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26 views

What does the s-Transform (exponential transform) mean conceptually? What does it show us?

I don't understand the conceptual idea. If I have PDF, and I calculate its s-transform for some s, what do I know that I did not know before?
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0answers
157 views

Compare between Short Time Fourier Transform and Wavelets

Fourier transform is localised in only frequency domain but Short time Fourier transform(STFT) is localised both in time and frequency domain same as in wavelets. I want to know How are STFT and ...
2
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0answers
147 views

Finding the transformation matrix of a projective transformation in RP^2

So I want to understand how to find the matrix that represents the projective transformation that sends 4 given points to 4 given images, I know that 4 points are necessary to determine it but I can't ...
2
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0answers
57 views

Find a matrix to represent the mapping of a factor module

I have a problem from my past paper I can't figure the logic to, even after seeing the answers. The question goes 【Q】Let $V=\mathbb{R}[X]_{<4}$ be the vector space of real polynomials of degree ...
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0answers
45 views

The Adrian Transformation of a function in $\mathbb{R}^{2}$

Recently I came upon a problem (if you would call it that, more of a thought experiment), which was phrased something like this: Rotate the area formed by $\int_{-1}^12dx$ around the curve $h(x)=-x^...
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193 views

The Composition of Two rotations

So far I rewrote the halfturns of d,c,b,a to halfturn (p,n)(m,l) where n=m because lines c and d are parallel so I can make ambiguous lines n and p parallel too. I also know that lines c,d can be ...
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0answers
68 views

When creating conformal images, how do you change the basis of the input lattice such that spirals result in the transformed image?

I am trying to emulate the results shown in the Wikipedia page on Conformal Images in an attempt to better visualize complex functions (and stare at some trippy images, man). The script I wrote (...