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), ...

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Linear Transformations, Linear Algebra

Let T:P3→P3 be the linear transformation such that $T(−2x^2)= −2x^2 − 2x$, $T(0.5x + 2)= 3x^2 + 4x−2$, and $T(2x^2 − 1)= 2x + 1$. Find $T(1), T(x), T(x^2)$, and $T(ax^2 + bx + c)$, where a, b, and c ...
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36 views

Transforming a sawtooth into a sinus with one parameter

Can you help me in finding the analytical expression of a function $f_\alpha(\theta)$, with one parameter $\alpha=(0,1)$ by which one can continously transform a sawtooth curve into a sinus? With ...
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41 views

Where should I learn the theory of transformation?

I am trying to learn the dihedral group in group theory and I feel a bit confused about the composition of rotation and reflection, here are my questions: In what area of mathematics will I learn ...
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Rotating the Spherical Harmonics around the x axis

I was trying to rotate the Spherical harmonics around the x axis by an angle of $\pi/2$ radians. At first, I thought that adding a simple substitution like $\theta \rightarrow \theta+\eta$ and $\phi ...
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27 views

Cosine of a triangular random variate

Good morning, I want to calculate the probability density function of a random variate $Z=cos(Y)$, where $Y=Φ_1−Φ_2$ and $Φ_{1,2}∼U(0,2π)$, that is both variables are uniformly distributed in ...
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25 views

For $X$ exponential with mean $\frac{1}{\lambda}$, find pdf of $X^2, X^3,$ and $e^{-\lambda X}$

Supposed to express the answers in terms of $\lambda$. I tried X^2 and did $F_Y(x)=P[Y \leqslant X]=P[X^2 \leqslant x]=P[-\sqrt{x} \leqslant \sqrt{x}]$ Then this equals $F_X(\sqrt{x})-F_X(-\sqrt{x})$ ...
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50 views

How is double integral variable substitution different from one variable trigonometric substitution?

I'm studying variable change in double integrals and I understood the reasoning behind the formulas as described really well here. However, geometric arguments for analysis don't convince, as well as ...
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12 views

Determining the distribution of univariate transformation

If Y is uniformly distributed on the interval $(0, 1)$ and if $Z = –a * ln(1 – Y)$ for some $a > 0$, then to which of the following families of distributions does Z belong? Lognormal ...
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17 views

Probability of no heads in terms of a moment generating function

Define a R.V. $N \geq 0$, and let $M_N(s)$ be the associated transform. Assume that we have an unfair coin which lands heads with probability p and we toss it N times independently. Show that ...
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40 views

Can the equation $\cfrac{n^2(s-2)-n(s-4)}{2}=2^p-1$ be transformed into an equation of the form $x^2 + D=AB^y$, for fixed $s$, $s \ge 3$

Can the equation $\cfrac{n^2(s-2)-n(s-4)}{2}=2^p-1$ be transformed into an equation of the form $x^2 + D=AB^y$ (where $D,A,B$ are fixed and $x,y$ are variables) when the variable $s$ is fixed to any ...
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Linear Transformation: Orthogonal Projections

Define $\mathbf{u_1} =$ $\begin{align} \begin{bmatrix} 0 \\ 0 \\ 1 \\1\\ \end{bmatrix}\end{align}$ and $\mathbf{u_2} =$ $\begin{align} \begin{bmatrix} ...
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32 views

Steps of transformation

Given the function $y=-5-3 \sqrt{-2x-4}$ and base function y= $\sqrt{x}$ describe the transformations that have been applied to obtain the function from the base function. I tried, horizontal ...
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32 views

matrix transformation : shear, composition

Suppose T_m is the shear in the x direction with factor k, and suppose T_p is the rotation by angle \theta use matrix multiplication to find the image of the vector v=(1;2) under the composition T_p ...
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31 views

Does translating a function change its domain?

In Spivak's Calculus chapter 3, there is a part which essentially states: $\textrm{if} ~~~ r(x)=x^2\ \textrm{such that} \ -17\leq x\leq \frac{\pi}{3}\\ \textrm{then} ~~~ r(x+1)=x^2+2x+1=r(x)+2x+1\ ...
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34 views

A question concerning Jacobians of coordinate transformation

Apologies for perhaps a very trivial question, but I'm slightly doubting my understanding of Jacobians after explaining the concept of coordinate transformations to a colleague. Basically, as I ...
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13 views

Extracting sign of scaling from modelView matrix

I want to retrieve the sign of scaling for each axis from modelview matrix. Right now I am able to extract the sign only if all 3 signs are same but it fails when one of them is different. Here is the ...
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Transformation of functions: proof for the time period

Given the standard form of a trigonometric function: $a \times \cos(b(t+c)) + d$, what is the proof that the period $p = \frac{2 \times \pi}{b}$. We don't have the proof in our syllabus. I'm just ...
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17 views

Determine a change of variables to transform one DE to another

Given two ODE's, is it possible to determine if one can be obtained from the other via a change of variables? In particular, I have the two ODEs: $$ \begin{split} ...
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18 views

Find coordinates of intersection

The question says "The line with equation $y = - \sqrt{3}$ intersects the graph at points A and B, find coordinates of point B." I worked out that the graph formula is $y = 2\cos(2x)$ and I think I'm ...
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11 views

Can a boolean value concerning changing the sign or not of a scalar value considered a 1D rotation?

So going descending order on dimensions: 3D: 3 scaling scalars, 3 rotation scalars, 3 translation scalars 2D: 2 scaling scalars, 1 rotation scalars, 2 translation scalars 1D: 1 scaling scalar, 1 ...
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37 views

How to transform a 2-d space on a circle to a higher dimension space

I have 2 points $A=(x_A,y_A), B=(x_B,y_B)$ on a unit circle $O$. The distance between $A$ and $B$ goes through the perimeter of the circle. How can I transform this space to a space with higher ...
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Exponential to Weibull Distribution transformation

Let X~Exp($\lambda$) and let Y=$\lambda$$X^{1/\gamma}$. Find and name the distribution Y. So considering the CDF of Y, I have that $F_Y(y)=1-e^{-\lambda(y/\lambda)^\gamma}$. This is looking very ...
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Exponential Distribution Transformaion

Let X~Exp($\lambda$) and let Y=$\lambda$$X^{1/\gamma}$. Find and name the distribution Y. So considering the CDF of Y, I have that $F_Y(y)=1-e^{-\lambda(y/\lambda)^\gamma}$. Now I think the next ...
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44 views

Mapping from the z-plane to the w-plane

I'm struggling with this question Show that the transformation w=z-1/z maps |z-1|=1 in the z-plane to |w|=|w-1| in the w-plane. Any help would be much appreciated. Thanks
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A Lipschitz transform maps measurable set to measurable

Prove that a Lipschitz transform $T: \mathbb{R}^n \to \mathbb{R}^n$ maps measurable set to measurable. Assume the only thing that we know about Lipschitz transform is that we can find $M>0$ ...
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33 views

How do I solve this complex numbers problem: transformation from the z plane to the w plane?

The point $P$ represents a variable point $z = x + iy$ in an Argand diagram. The point $Q$ represents a variable point $w = u + iv$ in a second Argand diagram and $x$, $y$, $u$ and $v$ are real ...
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Linear transformations that preserve permutations of a vector.

Forgive me if this question sounds silly. Let $v$ be a $m \times 1$ vector and $P$ be a $m \times m$ permutation matrix. Can there be a transformation $T$ such that $\min_\limits{P\neq I}\|Tv-TPv\|$ ...
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How to solve this graphical function transfromation problem?

How to solve problems like this. I always face problem in solving function transformation related problems. Is there any good way to solve problems like this..
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Triple integration, Spherical coordinates

How do we get limit such as $0\le\theta\le\pi$, $0\le\phi\le2\pi$ in spherical coordinate system where $$x=r \sin\theta\cos\phi, y=r \sin\theta\sin\phi, z=r \cos\theta$$ Why is the $\theta$-limit ...
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Similarity between two 2-D Transforms

How can I measure the similarity between two 2-D transforms? For instance, I would like to find how much similar is the 4x4 Hadamard Transform (H) with the 4x4 integer DCT (D), as used in H.264/AVC (a ...
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134 views

A linear transform of a closed set is closed

A linear transform of a closed set $E\subset \mathbb{R}^d \to \mathbb{R}^d$ is closed. I have seen a lot of similar questions here, but none of them exactly addresses the issue. Please if you ...
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Can the z-transform/laplace transform be generalized?

The fourier transform is a special case of z/laplace where the countour being projected on is the unit circle. The ztransform/laplace transform then generalizes this to a circle of any radius. Is ...
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Does the amount of translation for depends on whether it goes before or after dilation?

One question in my practice book, asks me to describe $y=f(-ax+b)$ based on y=f(x). According to the book, the order of transformation must be listed in this order: Reflection Dilation (scaling) ...
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transformation of differential equation

1.$$ x^2(1+u)dx + x^3(1-u)du = 0 $$ 2.$$ \frac {1-u}{1+u}du + \frac {dx}{x} = 0 $$ I know from 1 to 2 the equation is divided by $x^3(1+u)$, but i don't understand how that happens,was reading ...
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Corresponding matrix field basis

Hi people, I'm reviewing my notes for an exams and this is a question which I was unable to wrap my head around for many months. It should be fairly simple but I might be lacking a crucial piece of ...
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Is there a concise, specific name for a transform that consists of rotate, scale and translate?

I'm working on software that involves transforming between different mapping coordinate systems. In one part of the maths/logic, I have to derive, then apply a transform between two cartesian ...
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30 views

Tranformation of random variables

Let $X$ have the p.d.f $f(x)= e^{-x}$, $ x > 0$, $0 $ otherwise,find the pdf of $Y = X^2$ and space range $Y$. I use the change of variable formula The inverse is $${x} = \sqrt{{y}} $$ The ...
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What is this method of scaling called? Can it be generalised?

Consider the problem of finding the values of $\alpha_1, \alpha_2, ..., \alpha_k$, subject to constraints, such that the following equation is satisfied \begin{equation} \alpha_1 x_1 + \alpha_2 x_2 + ...
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fourier transform of a function that is only partially known

I'm interested in periodicity of a function that is not known completely, one way to think about it: our function $x(t)$ is defined at discrete points $t_1, t_2, ...$ but the values at these points ...
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Find me a sigmoid function with fixed point at the point of inflection in the unit interval

I am interested in finding sigmoid (S-shaped) functions $f$ on the unit interval $[0,1]$ such that For some $a \in (0, 1)$, $f(a) = a$ and $f'(a^-) > 0$ and $f'(a^+) < 0$. $f(0) = 0$ and $f(1) ...
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Before and after a transformation apply another transformation and its inverse?

So I have something like $y(x) = x\cdot5$, so whatever $x$ is, it's being scaled up by $5$. But I could center that scaling to a different point reference other than $0$. For example $10$. So for ...
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Method for transforming one curve around another?

I'm working with a complex problem involving waveforms. Essentially I want to bend a given waveform around a circle. At it's most basic, I want to take one curve on a linear graph and map it onto a ...
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Z transformation with $k$ from non-zero

we known that $Z$ transformation of $f_k$ is defined as $$F(z)=\sum_{k=0}^{\infty}f_k z^k$$ My problem is if $k$ starts from $m$, where $m >0$, then $\sum_{k=m}^{\infty}f_{k+m} z^{k+m}$ is still ...
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How to get the transformation matrix from 3D normal Vector

I have one plane object with a 3D normal vector $(X,Y,Z)$. How to get the transformation matrix from this object to my 3D origin frame $(x_0,y_0,z_0)$. The normal vector should be aligned with $z_0$. ...
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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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23 views

Transformations of discrete random variable

I understand how to do transformations. Well, I thought I did. But I can't seem to comprehend how to do a discrete to discrete. Here is the question I am working with. (I am currently studying for an ...
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f(2x + 1) transformation

I was working on a problem that asked: If there is a vertical asymptote at $x = 5$ for $f(x)$, where is the vertical asymptote for $f(2x + 1)$? The correct answer is at $x = 2$, but this ...
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Linear Transformations finding matrix in respect to a basis and coordinate change matrix.

Define $T: Poly_2 \ to\ Poly_2$ by $$T(at^2+bt+c)=3ct^2 +2at-b$$ 1) Show that T is a linear transformation and give a matrix A for T with respect to the basis $B=\{t^2,t,1\}$. 2) Give ...
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Find Fourier transform of triangular function based on a Fourier results of rectangular

I have a triangular pulse given by $$x\left(\frac{t}T\right) = \begin{cases} 1-\frac {|t|}T, & \text{if $T\ge t$} \\ 0, & \text{otherwise} \end{cases}$$ Given that ...
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Order of Operations for Horizontal Transformations

We know that when we want to combine two horizontal transformations, specifically that of translating and stretching a function, we have to translate $f(x)$ first, and then afterwards stretch it. ...