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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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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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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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0answers
38 views

Finding the the partial derivatives of a transformed surface, in the original space

I am trying to calculate the from the surface described by a function $P(x, y, z, w): \mathbb{R}^4 \to \mathbb{R}^4$. The surface is really only dependent on $x$ and $z$ but for computational reasons, ...
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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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1answer
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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
18 views

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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1answer
26 views

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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1answer
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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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2answers
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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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1answer
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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
23 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)...
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1answer
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Constant lengths (dimensional constants) and scaling

Suppose in the $xy$-plane we have defined the constant length $L$. This can be a fixed radius of a circle; or a boundary condition or any condition such, that $L$ has dimension of "meters" and is ...
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0answers
25 views

Rotation matrix to yaw, pitch, roll matrices

I need to decompose a rotation matrix $R$ to yaw, pitch, roll matrices so that: $R=Yaw*Pitch*Roll$
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0answers
43 views

Jacobian and area differential

A transformation T (u, v) is said to be a conformal transformation if its Jacobian matrix preserves angles between tangent vectors. Consider that the vector $\langle 1,0\rangle$ is parallel to the ...
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1answer
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Which direction is clockwise when rotating around x-axis in 3D?

Picture 1 shows a demonstration that rotations around an axis is positive for clockwise directions. An example later on, picture 2, applies a rotation matrix for 60 degrees in the x-axis for a ...
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1answer
24 views

Affine transformations satisfying conditions

I'm asked to find ALL affine transformations from $\mathbb{R^3}$ to itself which satisfy that the point $(-1,2,2)$ is fixed and that the lines $$\textbf{a}: y-z=y+z-2=0$$ and $$\textbf{b}: z-1=x-z=...
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1answer
21 views

Calculate rectangle position after scaling from his center

I have a rectangle with these bounds height: 31.010414123535156 width: 215.8330078125 x: 10.95849609375 y: 90.49478912353516 I want to scale up the rectangle by ...
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1answer
51 views

Circle transformation

Let $K$ be a circle $(x-1)^2+(y-4)^2=5$ and $P$ a line with the equation $y=-3x$. We transform the circle $K$ with a transformation, defined with the next conditions: points are tranformed, so that ...
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3answers
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Extract data of 3D graph rendered in 2D

I have a 2D pdf of some 3D graphs from which I would like to extract the data (I have the authors permission, original data has since been lost). The graphs in question look like: Focusing in on the ...
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1answer
32 views

Setting up the standard matrix for orthogonal projection

For a subspace $V$ of $\mathbb{R}^4$, you are given these three ordered bases: $A= (\mathbf{a}=(1,-2,-1,3), \mathbf{b}=(1,3,-2,-4))$ $B= (\mathbf{a}=(1,-2,-1,3), \mathbf{c}=(2,1,-3,-1))$ ...
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0answers
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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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1answer
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Difference in rotation matrix

I have two objects A and B, with a start transformation matrices MA1 and MB1 (they include translation, rotation and scale). End matrix of the first object is MA2. How do I apply the same rotation as ...
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0answers
31 views

Riesz transform representation on the torus

Does anyone knows how to do the representation of the Riesz transform on the torus? I know that on the space $\mathbb{R}^d$, is it given by $${\displaystyle R_{j}f(x)=c_{d}\lim _{\epsilon \to 0}\int ...
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0answers
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Efficient method for looking for the closest point

I have $N$ points ${P_1, P_2, P_3, ..., P_N}$ with dimension $K$, where $K\gg N$. Given another point $Q$ with the same dimension, I want to get which of is the point $P_n$ which is closest to $Q$. ...
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1answer
33 views

Why when it comes to translating functions (x-b) negative? [closed]

Quick question: when in the formula:(x-b), if it was (x-2) moves to the right but when it is (x-b)+c, you have (x-2)+3 then it moves up 3. I do not understand why it is a minus sign instead of a plus ...
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1answer
33 views

Universal formula to calculate rotating by angle

I am generating roads and buildings that belong to them and since I want the streets to be rotated and then connected with each other, I need to rotate both them and their respective buildings. Is ...
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2answers
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Linear operator find function

Let $T(x,y)=(x,3x-y)$ be a linear operator on $R^2$ and let $f(x)=x^2-4$. Find $f(T)$ I'm very new to this and I'm stuck on what it's asking for. I'm guessing I make a matrix from the ...
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0answers
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Are tensors always invariant?

I'm new to tensors and what I've learned about them is that they are objects that are 'invariant' under transformations, is this true for all ranks of tensors? If I got $n$ functions $A^i$ where $i = ...
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2answers
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Transformation matrix with respect to an orthonormal basis

I have this question here... Let $V$ be the span of $v_{1}=(0,1,2)$, $v_{2}=(-1,0,1)$ and $v_{3}=(-1,1,3)$. $(a)$Construct an orthonormal basis $B'$ for $V$ (usual dot product). $(b)$ ...
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1answer
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Finding the transformation that preserves the values of a polynomial.

I have a polynomial in multiple variables, and would like to have a transformation act on them so that the value of the polynomial is preserved. For example, the polynomial $x^2+y^2$ is preserved in ...
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1answer
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Finding the transform function of two histograms [closed]

May you please help me how can I find the transform function of the following histograms? Thank you
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35 views

Quaternion that transforms a point like a 2D angle

Am looking for a way to transpose a 2D solution of a problem to a 3D solution of the same problem. The algorithm I've implemented in 2D works as follows: Given the points $A (A_x, A_y)$ and $B (B_x, ...
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0answers
19 views

Non-monotonic transformation of a specific function

I'm working on my home project about probability right now but stuck at some point and can't figure out where. The problem and my (probably a wrong)answer is: SET UP: Let $X$ be an RV which follows ...
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1answer
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Find the density function of the kinetic energy, $E = \frac 12 mV^2$

A particle of mass m has a random velocity, $V$ , which is normally distributed with parameters $μ = 0$ and $σ$ . Find the density function of the kinetic energy, $E = \frac 12 mV^2$. (Rice, 2.64). ...
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1answer
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The Jacobian of $(x,y)\mapsto (x+y^2,y+x^2)$ under the substitution $u=x+y^2$ and $v=y+x^2$.

I am given the map $(x,y)\mapsto (x+y^2,y+x^2)$. I am unable to find the Jacobian by making the substitution $u=x+y^2$ and $v=y+x^2$. Any hints would be appreciated. (I am trying to find whether the ...
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2answers
67 views

Where can I find a proof, that $s(n+1,k+1) = \sum_{i = 0}^{n} \binom{i}{k}s(n,i)$?

In our Combinatorics Script it is written, that $$s_{n+1,k+1} = \sum_{i = 0}^{n} \binom{i}{k}s_{n,i}$$ for $n,k \in \mathbb{N}$. The problem is that I can't find a combinatorial proof for that, not ...
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0answers
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Deriving the Transformation of the Coefficients of Surfaces of Second Degree as a Tensor. Per Einstein

See pages 12 and 13 of Einstein's https://en.wikisource.org/wiki/The_Meaning_of_Relativity/Lecture_1 This question involves rectangular Cartesian coordinates in $\mathbb{R}^{3}$ related by orthogonal ...
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0answers
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Distances under kelvin transformation

a part of my task is to show that you can express $\int\limits_{S_{1/2}(1/2 e_1)} \dfrac{1}{\vert x-z \vert ^n} \ dH(z)$ as $\int\limits_{S_{1/2}(1/2 e_1)} \dfrac{1}{\vert x \vert ^n \vert z \vert ...
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2answers
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Help with simple linear transformation

can you please help me with this simple question? I want to know if this is linear or not $f: \Bbb R^2\to\Bbb R^3$ $f(0,0)=(1,0,0)$ $f(0,1)=(0,0,0)$ If you could explain why it is or it isn’t ...
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1answer
30 views

What conditions are required for transformations of limits to be valid?

Consider a continuous single variable real valued function $f(t)$ defined on some non empty finite interval $D$. Say we wish to find the limit of $f(t)$ as $t$ approach some point $t_{d} \in D$ (...
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2answers
256 views

If All the Points Lie On a Plane, Then Why Does the Linear Mapping Reduce to …?

I previously asked a question with regards to what the matrix $\mathrm{H}_{3 \times 3}$ is/represents in the following textbook excerpt: In applying projective geometry to the imaging process, it ...
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1answer
32 views

Question about inversion images

"Let $W$ be a circle with center $O$ and radius $r$. Let $S$ be a point outside the circle. Let $l_1$ and $l_2$ be two tangent lines to the circle $W$ passing through the point $S$ . Let $T_1$ and $...
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2answers
20 views

How to calculate a transformed conic curve?

The origin question came from I want to move from A to B with a conic curve, and its model was like this: Say We have a curve transformed from $y=x^2$ (just with rotation and movement) And is there ...