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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Mathematical Source for an algorithm that turns vector to euler angles

Am using the algorithm described here (https://stackoverflow.com/questions/21622956/how-to-convert-direction-vector-to-euler-angles) in the first answer for a thesis in software development. I do ...
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Transformation of RV: Finding PDF

I am trying to work through this example problem in my textbook but I keep getting the wrong final answer. My Notation: PDF X : pX(x) CDF X : FX(x) Question: Consider the transform Y=X2 if pX(x) = o....
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Question about group of automorphism of some $G$-structure.

I'm reading Kobayashi's book Transformation Groups in Differential Geometry and I don't understand a thing at page 15. I don't understand why $U$ consists of transformation $a$ of $M$ that leave each ...
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Pdf of $\sin X$ when $X$ has the pdf $f(x)=\frac{2x}{\pi^2}1_{0<x<\pi}$

I would like to find the PDF of the random variable $Y=\sin(X)$ given the PDF of $X$: $$f(x) = \frac{2x}{\pi^2} \text{ for } 0<x<\pi \text{ and } 0 \text{ otherwise}$$ Following the tips in ...
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Algebraic transformation — where is my mistake?

I tried to find the estimators of $\hat{\beta_1}$ and $\hat{\beta_0}$ via the least-squares method algebraically. Somehow I seem to have messed up. Can you tell me where? My Calculations.
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Continuous automorphism of a lie group in kobayashi's book

I'm reading Kobayashi's book Transformation Groups in Differential Geometry and i dont understand a thing at page 14. My question is why $A_\varphi$ is continuous? $G$ is a subgroup of ...
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Mapping $\Bbb N\to\Bbb Q$

I want to set up a map from $\Bbb N\to\Bbb Q.$ Take $\Phi_S(x)=e^{(S/\ln(1-x))}$ and $M_T(1-x)=\Phi_S(x); S,T\in\Bbb N.$ Set $\Phi_S(x)=M_T(x)$ to obtain algebraic $x$ coordinates. If $x$ happens ...
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How to solve this problem raven matrices problem?

I am doing this free test in http://test.mensa.no/ That, as far as I know, the only problem I can't solve. Basically we shift the first row to the right. From first to second is easy transformation. ...
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Matrix transformations.

I am currently learning matrix transformations and ran in to an exercise that I can't understand. We start with e1 = (1,0) e2 = (0,1) So the first transformation is a shear one where e2 becomes -2e1 + ...
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3D Vector Rotation Matrix with Radians

I have been working on a simple C++ vector library and needed 3D rotation so I found these 3D rotation matrices on Stack Overflow: ...
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Finding the pdf of $f(x,y) = e^{-x-y}$ where $Z = X+Y$

I am having trouble understanding how to find the pdf $f_Z(z)$ when $f_{X,Y}(x,y) = e^{-x-y}, x,y \space \epsilon(0,\infty)$ where $Z = X+Y$ My approach is that $$x = x, y = z-x$$ so using ...
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How does squaring give you a monotonic transformation?

Consider the function: $$ f(x,y) = \sqrt {xy} $$ Is the function $$ f_1(x,y) = x^2 y^2 $$ a monotonic transformation of $ f $? I remember studying earlier that squaring does not give you a monotonic ...
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1answer
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Show $f(n)= \frac{ (1-\alpha) a^{n+1} e^{-a}+\alpha b^{n+1} e^{-b} }{ (1-\alpha) a^{n} e^{-a}+\alpha b^{n} e^{-b} }$ is unique for $(\alpha,a,b)$

Suppose we have the following function \begin{align} f(n)= \frac{ (1-\alpha) a^{n+1} e^{-a}+\alpha b^{n+1} e^{-b} }{ (1-\alpha) a^{n} e^{-a}+\alpha b^{n} e^{-b} }, \text{ where }n=0,1,2,3,4, \ldots \...
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Affine Transformation as Rotation

Im trying to do this textbook question which asks me to "express" a motion T(x) = Ax + b in the form T = Rot(P, $\theta$) (A is the rotation matrix) I know that if I draw the transformation, the ...
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Pinch transform shapes

I'm looking for an algorithm that can pinches a shape in a way to become pointy on both ends. Like this image, transforming the shape on the left to the right. The result is literally similar to ...
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How to transform basis functions

I know how to transform the basis if we have two sets of basis vectors. Now in my situation, I have two basis equation and I want to find out the transform between those basis functions. Specifically, ...
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1answer
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How to find a 2D coordinate field's corners in a 3D Coordinate field if I have 3x 3D points with 3x2D Points?

In order to solve "this" problem, i have to transform my corner-points from a 2D Space to my 3D Space. But my two coordinate fields are only defined by their relation to each other. They have the ...
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Determining partial derivatives after coordinate transformation

I'm currently trying to figure out the partial deriviatives of a function, after there is a space transformation. $$ f(r_1, r_2, w) \rightarrow f(x, y, w)(2\sqrt{xy})^{-1} $$ Where $$ r_1 = x^2 \\ \\...
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Hankel transformation and inverse Hankel transformation of integer order.

Read the definition of Hankel transformation here... https://en.m.wikipedia.org/wiki/Hankel_transform Question: can we define Hankel transformation for any Integer order? Can we define inverse Hankel ...
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Calculate new X & Y coordinate based on compressed or enlarged rectangle

I have two Rectangles as Rect1 ___________ x'', y'' | |dy | | .x',y'| | | | | 0,0------------- Here the value of x''...
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1answer
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Logistic Regression and modelling probabiliy $\pi$

This is a General Linear Models topic but I believe it's just basic failure to remember some more basic math rules that's making it difficult for me. If the link function is $$g(\pi) = \log(\frac{\...
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1answer
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Is $\begin{bmatrix} 0 & A \\ B & 0 \end{bmatrix}$ similar to $\begin{bmatrix} 0 & CAC^{-1} \\ C^{-1}BC & 0 \end{bmatrix}$ by some transformation?

Consider a matrix with two entries being some operator or matrix $$D=\begin{bmatrix} 0 & A \\ B & 0 \end{bmatrix}.$$ I want to construct another $2\times2$ matrix $S$ such that $$SDS^{-1} = \...
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coordinate transformation in differential equation

I'm confused about coordinate transformations. What I understand is, that if we have $$ f(x(t))=g(x(t)), $$ that we can write for each (bijective) $h(x)$ $$ f\circ h(x(t))=g\circ h(x(t)). $$ If we can ...
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1answer
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Transformation Theorem on n-dimensional Sphere

$ n \in \mathbb{N},b >0$ Define $S_n$ as the n-dimensional Sphere in $\mathbb{R}^n$. I cannot figure out the appropiate transformation to use the transformation theorem such that the following ...
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Problem with the basis of a linear transformation

I have this exercise Define and find the expression of the linear transformation Information given: $ T : \Bbb R^5\to \Bbb R^4$ Im T = ⟨(3, -1, 0, ...
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Mathematical expression as LP problem

I am seeking for a linear programming expression for the following expression: enter image description here B(x) is the neighborhod of f(y) (where delta is the size of the neighborhod). It can be an ...
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Affine rectification via vanishing line

I'm trying to understand how to rectify an image given some lines that should actually be parallel in the final image. For example: from the book Multiple View Geometry. I know that the idea is to ...
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3answers
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Existence of a strictly increasing transformation between two functions [closed]

Assume $f$ and $g$ are two differentiable functions defined on a compact interval $X \subseteq \mathbb{R}$ mapping into $\mathbb{R}$ . I want to proof or disproof the following statement $ \forall x \...
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Expansion in power of $\frac{1}{Z}$ and $\frac{ln(Z)}{Z}$

When I read the paper I met the problem in the step expansion in power. We have \begin{align} s(\epsilon)=\frac{A\epsilon^{a}}{bB|\dot\epsilon|}e^{-Be^{b}} \left[1+\frac{a}{bB}\epsilon^{-b}+\frac{a(a-...
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Intuition for a Ingenious Integral Substitution

In the class Measure Theory and Integration, the official solution gave an ingenious solution to the following computation: For $a \in \mathbb{R}^3$ and $r>0$ compute $\int_{\|x \|_2 \leq r} \...
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1answer
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Tranformation of probability density function of a source

i have a source that generates numbers with uniform probability density with in a range. Is there any way by which the data from the source having uniform probability density can be manipulated so ...
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1answer
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Find the pdf of Z=X/(1+Y)

Been trying to solve this problem for some time now. Any help would be appreciated. X and Y are independent R.V's with distribution exp(a) each. I'm asked to find the pdf of Z with: $$Z=\frac{X}{1+...
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CDF after Transformation.

A random variable $Y$ has lognormal ($\mu,\sigma$) distribution if its probability density function is $$ f(y)=\frac{1}{y\sigma\sqrt{2\pi}}exp-\frac{(\ln y-\mu)^2}{2\sigma^2}$$ its CDF will be $$\hat ...
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1answer
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Time transformation of random variable with mass points

Let $B=\left\{b\right\}$ denote the set of atoms of the distribution function G. Define the quantile function $G^{-1}\left( a\right) = inf \left\{ x \in R : G(x) \ge a \right\}$. Let V be independent ...
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2answers
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Corresponding Point for a Glide Reflection

I was wondering if there was an efficient method that could solve these types of problems. Here is the problem: Plot the points K = (0,0), L = (7,-1), M = (9,3), P = (6,7), Q = (10,5), and R = (1,2)...
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Transformation of a scalar function

Given a scalar function, we consider the following transformation: $$\delta f(x) = f'(x') - f(x) $$ But since $f(x)$ is a scalar isn't it true that $ f'(x') = f(x) $ Then the variation is always ...
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Converting a transformation from world frame of reference to local?

I have a matrix $W$ that is the local-to-world transformation (Rotation/Scale/Position) of a point $P$ in 3D space. I also have a rotation quaternion $R$ and a translation vector $T$ that further ...
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Defining Transformations given a set of elements (Apostol Volume 2)

The question is laid out like this: Let $V = \{0,1\}$ . Describe all functions $T: V\longrightarrow V$ . There are four altogether. Label them as $T_1 , T_2 , T_3, T_4$ and make a multiplication table ...
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What hypergeometric transformation rules might I apply to try to simplify a certain expression?

I have (https://mathematica.stackexchange.com/questions/189538/sum-a-certain-hypergeometric-function-based-expression-pertaining-to-an-integrat) a Mathematica expression involving the following six (...
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Cauchy distribution, transformation of univariate random variable

I'm trying to solve a strange exercise. "A cannon is placed on a point P=$(x_0,y_0)$ of the upper side of the Cartesian plane, i.e. $y_0>0$. The cannon shuts projectiles to the ordinate axis in ...
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How do eigenvalues of a matrix X change if we linear transform the matrix X?

I have a matrix $X$ which has eigenvalues $U$. Now create a new matrix $Y = AX$ where $A$ is a nonsingular matrix. How do the eigenvectors and eigenvalues of $Y$ change in relation to the ...
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Perspective Mappings Between Quadrilaterals

Given a quadrilateral represented by $(X_n, Y_n)\;$ I would like to obtain a specific point $\;(A, B)\;$ in the same quadrilateral when it changes perspective knowing only the vertex $\;(X'_j, Y'_j)\;$...
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1answer
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Subset variance order preserving function

Given a finite set with real numbers. X = {x1, x2, x3}. There can be a unique order defined for all the subsets using Variance operator. e.g. X = {1, 2, 4}. $$ {\displaystyle \operatorname {Var} (X)...
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1answer
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Fundamental Theorem of Calculus with Inverse function. Explanation, intution and proof please

I'm an undergraduate student studying for the actuarial exams and was wondering if someone could please walk me through the proof and intuition of this please? I haven't taken an analysis course yet, ...
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Forward and Backward Projections

I have the transform functions (forward and backward projections) such as: $$FP\{f(x,y)\} = \int_{-\infty}^{\infty}f(r\cos(\theta) - z\sin(\theta), r\sin(\theta) + z\cos(\theta))dz$$ $$BP\{g_{\theta}(...
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Standard matrix of a transformation, matrix representation [closed]

I know that the answer is $\left[\begin{matrix} 2 & -1 \\ 1 & 1 \end{matrix}\right]$, but how to get the answer? Let $\mathcal{B} = \{ \mathbf{b}_1 , \mathbf{b}_2 \}$ be the basis for $\...
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Prove $\int_{\mathbb R^{d}}f(|y|)d\lambda^{d}(y)=C_{d}\int_{[0,\infty[}r^{d-1}f(r)d\lambda^{1}(r)$

Let $f:[0,\infty[ \to \bar{\mathbb R}$ measurable, $d \in \mathbb N$ while $E_{d}:=\{x \in \mathbb R^d: |x| \leq 1\}$ Prove $\int_{\mathbb R^{d}}f(|y|)d\lambda^{d}(y)=C_{d}\int_{[0,\infty[}r^{d-1}f(...
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Transformation of Random variable $Y=-2\ln(F(x))$ [closed]

Let $X$ is a continuous Random variable. with strictly increasing function cumulative distribution function $F(x)$. Find and recognise the distribution of random variable $Y=-2\ln(F(x))$. I need some ...
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Canonical (Multiplicative) maps between $m$--dimensional spaces to $n$--dimensional spaces, $n \leq m$.

Let $M$ and $N$ be two smooth manifolds which we may assume are $\mathbb{R}^m$ and $\mathbb{R}^n$ respectively, with $n \leq m$. In fact, assume $M = \mathbb{R}^m - \{ 0 \}$ and $N = \mathbb{R}^n - \{ ...
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1answer
34 views

logarithmic transformation from exponential to linear equation

How to convert this exponential equation to linear equation. $Y =\exp(17.9348)\cdot x^{-2.705}$ what I did is: $Y =\log(17.9348)-2.705\log(x).$ I am confused with this one: $Y=17.9348-2.705\log(x)...