# Tagged Questions

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-...

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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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### Is there a similarity solution for this PDE? (with discussion, kindly check)

I have a PDE for $h(x,t)$ of this form $$h_t+Ah^{-1}+(h^3h_x)_x+Bh_{xx}+(h^3h_{xxx})_x=0,$$ where the subscripts denote the partial derivatives, and $A$ and $B$ are all constants. I'm wondering ...
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### 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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### 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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### Decomposition of 4x4 or larger affine transformation matrix to individual variables per degree of freedom.

There are a couple of problems and solutions where affine matrices are decomposed into their seperate tranformations. However they are all for the 2D case and I`m finding it difficult to generalise it ...
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### 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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### Get 2D coordinate transformation matrix based on points in a system and their angles in the other?

I'd like to get the parameters (rotation angle,$\Theta$, and translation coefficients, $x_0$ and $y_0$)) of a transformation for translating and rotating points in a coordinate system to another. As ...
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### Transformed Laplace “solution space”

From my own knowledge I can tell that when we take the Laplace transformation of a function we are in essence transforming our f(t) into a F(s). I've looked at several Q/A here asking for the ...
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### Stabilize Variance for Statistics (Transformation)

Problem: When $Y (> 0)$ has mean and variance equal to $\mu$ and $\mu/n$ respectively, it is shown in the textbook that the appropriate transformation of Y to stabilize variance is the square root ...
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### Inverting a discrete linear transformation

Consider the transformation from the set $\{a_n\}_{n=0}^N$ to the set $\{p_j\}_{j=0}^N$: $$p_j = \sum_{n = 0}^Na_n\phi_n(x_j)$$ where $\{\phi_n(x)\}_{n=0}^N$ is a set of basis functions (linearly ...
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### Perhaps an easy algebra problem, but it still evades me

I need help spotting a corresponding transformation Let $x,y$ be some variables and $$z=z(x,y)$$. We have a transformation $X(\lambda):(x,y,z)\to (x',y',z')$, such that x'= x\exp(a\lambda)\\ ...
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### Following a polyline along the surface of a polygon that is twisted

I have (hopefully) an interesting problem regarding geometry. I will also search online and in literature but I thought to pose the question here as a third resource. For my problem I need to get the ...
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### Identity for reducing the power in the parameters of Hypergeometric functions

Is there any identity/formula for reducing/increasing the power in the parameters of the Gauss Hypergeometric function $_2F_1(a,b;c;z^d)$ (d is a real) let's say to z? Is there also any identity for ...
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### 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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### Is there an intuitive understanding of what a walsh coefficient is?

I am working with Walsh coefficients. I know the intuitive understanding is almost that that they are the degree of connectivity, but it is there a better way of thinking about it? What is the ...
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### bound on Hilbert transform

Consider $\widehat{Tf(\xi)}=m(\xi)\hat{f}(\xi)$, where $m(\xi)=(1-\vert\xi\vert)1_{[-1,1]}$, i.e. $T$ is the operation of taking Fourier transform and multiplying with the function $m(\xi)$. I am ...
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### Understanding Discrete Cosine Transformation

I'm currently working on some software and a key component is 2D DCT. But my question is more general, as I'm trying to understand the DCT in general, let's say from engineers point of view. For start,...
I've got 6 points in 3D space: $A,B,C,D,E,F$, that represent 4 vectors. $AB$ is perpendicular to $AC$ and $DE$ is perpendicular to $DF$. I need to find a transformation matrix M, that transforms $AB$ ...