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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affecting the final result of function depending on external factor

Suppose I have a function $f(x) = \frac{x}{x+y}$ whose range is in the interval $[0,1]$ and there is an external factor say $a$, such that $a$ is in the interval $[0,1]$, moreover, a predefined ...
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13 views

Solving a transformation problem in complex (Argand) planes

Below is an example in a text-book section on using complex-number arithmetic to represent transformations in Argand planes Example A transformation $T$ of the $z$-plane to the $w$-plane is ...
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1answer
39 views

What's the formulation of N-point radix-N for NTT

We can write the formulation for the buttlerfly function applied in FFT as \begin{align*}y_0 &= x_0 + x_1,\\ y_1 &= x_0 - x_1. \end{align*} As seen here. For FFT (Fast Fourier Transform) we ...
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1answer
30 views

How to determine if an affine transformation would cause reflection?

I have a list of affine transformation matrices and I want to write a code to delete the transformation matrices that applying them on an image would cause reflection. after seeing this image in ...
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23 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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1answer
24 views

Invariance properties of transformations

In Gentle's Matrix Algebra (2007, p. 175), he presents a table of what features of vectors various transformations preserve. What does it mean to say a transformation T preserves some property of a ...
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1answer
58 views

Bring system in normal form up to the second order

Bring the system $$ x'=y+xz,\quad y'=x^2+y^2+z^2,\quad z'=-2z+xy $$ to a normal form up to the second order (kill all non-resonant quadratic terms). The equilibrium is $(0,0,0)$ and the ...
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1answer
31 views

How to approximate linear relationship between two timeseries? [closed]

I have two time series A and B I would like to solve for the equation in the form $$y_t = m x_t + b$$ that transforms a point $y$ at time $t$ from series $B$ to the corresponding point $x$ at time $...
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120 views

PDFs of Piecewise Transformations: why doesn't it apply in this case?

This is from Casella and Berger's Statistical Inference, although it is more of a probability question than a stats question. Theorem 2.1.8 Let $X$ have pdf $f_{X}$, let $Y = g(X)$. Define the ...
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1answer
34 views

rational integral with a quartic function in the denominator

Let's say I've got the integral: $\int [ 2 Q^4 - 5 Q^2 + 3 ]^{-1} dQ$ This integral evaluates to: $\int [ 2 Q^4 - 5 Q^2 + 3 ]^{-1} dQ = \tanh^{-1}\left( Q \right) + \sqrt{ \frac{2}{3} } \tanh^{-1}\...
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1answer
25 views

The correct order for applying transformations?

I had a line, y=-1/2 x I wanted to reflect everything onto the line so as you can see by my steps in the picture below(sorry, I dont know the coding) I first rotated the line theta degrees then ...
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1answer
36 views

Transform PDE to ODE (3 variable case) with given boundaries

How can I transform the following PDE into an ODE? I tried using three different functions $H(x),G(y)$ and $F(t)$ but that didn't help hence I did not post it here. I really hope someone can help me ...
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1answer
13 views

Three Variable Transformation and Independence

If $X_1$, $X_2$, and $X_3$ are independent, identically distributed random variables and $Y_1$, $Y_2$, and $Y_3$ are functions of them, how do I show that the $Y$ variables are mutually independent? ...
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1answer
20 views

Integral with more-dimensional substitution variables

Good day, In the lecture of partial differential equations we had the following transformation: $$\int_{||\nu||=1} h(x+\nu c t, \tau) d\nu = \frac{1}{c^2 t^2} \int_{||y-x||=ct} h(y,\tau) dy$$ for $...
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41 views

Is there a product integral that preserves zeroes?

The integral essentially takes the arithmetic mean of the range of a function multiplied by the domain, adding together each possible output weighted by the amount of the domain accounted for by that ...
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99 views

(Answered) Transformation between AutoCAD OCS and WCS coordinate systems

This question originates in a problem I am having with transforming between two coordinate systems, that come from AutoCAD. This question is purely about the math of the problem, I just include the ...
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31 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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35 views

Transformation of exponentials

Find the transformation that takes $y=3^x$ to $y=\textit{e}^x$. I have tried: Let $y=3^x$ to $y=e^{x'}$ $$\log_{3}(y)=x\quad\text{hence}\quad\log_{3}(y)=\frac{\log_{e}(y)}{\log_{e}(3)}$$ $$x\...
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26 views

Graph Transformation

This question seems trivial to me but I still cannot figure out how to approach it. For any $x \gt 0$ and $f(x) \le 2^x$, how do I prove the following? $$ f(2x) \le f(x)^2 $$
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1answer
60 views

How can I find the Fourier transform of constant value like $1$.

The textbook told me that $\mathbb F[1] = \delta(f)$ and $\mathbb F[\delta(t)]=1$. It is easy to prove that $\mathbb F[\delta(t)] = 1$. $$ \mathbb F[\delta(t)] = \int_{-\infty}^\infty \delta(t)e^{-...
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0answers
39 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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14 views

Name for affine transform which consists of scale and translate

Is there a concise name for this type of transform? Viewed as an affine transform, the matrix would have the form: k 0 dx 0 k dy 0 0 1 The best I can come ...
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24 views

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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1answer
18 views

Basic inverse $z$ transform

I have trouble finding a (probably) pretty easy inverse of a $z$ transform. $$H(z) = \frac{z-0,5}{z+0,5}$$ I used the polynomial division on it to get a proper fraction and got $$H(z) = 1 - \frac{...
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2answers
22 views

Range of continuous transformation on closes set

let $f$ be a continuous transformation and $F$ closed set. Prove that the range $f(F)$ does not have to be closed.
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3answers
221 views

Find the distribution of the series $Z = X_1+X_2+…+X_N$

"Let $0<p=1-q<1$. Suppose that $X_1,X_2,...$ are independent Ge(q)-distributed R.V.'s and that $N \in Ge(p)$ is independent of $X_1,X_2,...$. Find the distribution of $Z=X_1+X_2+...+X_N$." I ...
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1answer
12 views

Clarification on computing the Moment Generating Function of a Gamma-distribution

In my text book the MGF of a Gamma distributed R.V., $X \in \Gamma(p,a)$, is computed as follows: $$ \psi_X(t)=\int_0^{\infty}e^{tx}\frac 1 {\Gamma(p)}x^{p-1}\frac 1 {a^p}e^{-x/a}dx= $$ $$ =\frac 1 {...
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1answer
12 views

How to compute the Moment Generating Function of R.V.'s $X_1 - X_2$

I want to show that $$Y_1 + Y_2 \overset{d}= X_1 - X_2$$ where $Y_1, Y_2 \sim \text{Laplace}(1)$ and $X_1,X_2 \sim \Gamma(2,1)$, by checking their Moment Generating Functions. For the left-hand ...
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1answer
29 views

What is Fourier transform of $g(y)=e^{-\pi y^2 +2\pi yx}$ for all $y\in \mathbb R.$?

Let $x\in \mathbb R.$ Define $g:\mathbb R\to \mathbb R$ as $g(y)=e^{-\pi y^2 +2\pi yx}$ for all $y\in \mathbb R.$ My Question is: What is the Fourier transform of $g$? In other word, how to ...
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17 views

Z transform to difference equation?

For a z transform to fully describe an equation, you need the z transform itself and the ROC. You can convert the z transform to a difference equation easily if it's rational. How can I covert the ...
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39 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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1answer
34 views

Linear transformation understanding

Let $S: R^2 → R^2$ be the function defined by $S(x, y) = (x − y, y)$ for all $(x, y)$ I've found the matrix for this which was question 1. These are the two questions I am struggling to understand. (...
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1answer
62 views

Calculate the determinant of the matrices $a_{ij}=\frac{1}{i+j-1}$ and $b_{ij}=\frac{1}{i+j}$?

I would like to know if there is any formula for calculating determinants of the following symmetric matrices: $$ A=[a_{ij}]_{n\times n},\qquad a_{ij}=\frac{1}{i+j-1}, $$ and $$ B=[b_{ij}]_{n\times n}...
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22 views

Invariance of stationary wavelet transform

Suppose we are given 64 points $x_1,\ldots ,x_{64}$ and divide them into two groups $x_1,\ldots, x_{32}$ and $x_{33},\ldots , x_{64}$. Then we apply stationary wavelet transform to both these groups ...
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3answers
46 views

Log transformation

Supose I have a series of numbers from 1 to 10. Their mean value is 5.5. Now supose I apply some transformation like $y=2x+1$. Now their mean value is 12. Now, if I want to get back the original mean, ...
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2answers
42 views

Rotation matrix construction

I'm reading a book on how to construct transformation matrices and I'm stuck in a certain point. From the book: Now here's the figure that I don't understand: How come the opposite edge in the ...
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1answer
24 views

Convolution Problem

while working on a signal processing problem i've reached to the following: So my aproach was: Am I doing something wrong? Is it valid Y(f)=[X(f) x H(f)]*W(f)=X(f) x [H(f)*W(f)] If you could ...
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1answer
25 views

Construction of a Transformation of a Random Vector that Preserves Independence

Let $X_1, \dots, X_n$ be $n$ independent random variables, not necessarily normal. Let $Y_1 = \sum_{i=1}^{n}\alpha_i X_i$ a given linear combination of the random variables. Is there a known, "...
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1answer
101 views

optimal monotonic transform: $\min_f (f(x)-y)^2$

Given two vectors of length $N$ denoted by $x_i$ and $y_i$, $1\leq i\leq N$, what is the monotonic transformation $f(x)$ that minimizes the overall distance $D=\sum_{i=1}^{N}{(f(x_i) - y_i)^2}$. Does ...
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1answer
20 views

How do I find the Probability Function of this Transform?

Given the Probability Generating Function for a non-negative, integer-valued, R.V. $X$ as: $$ g_X(t)=\log\left(\frac 1 {1-qt}\right). $$ How do I compute its Probability Function, $P(X=k)$? A step-...
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12 views

Transformation Matrix for particular problem

I have a question regarding transformation matrices. I have two images both showing a table. I have coordinates of the corners of the tables, and now I want to apply a transform to 1 of the images so ...
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1answer
16 views

Finding Equality around Axis of Symmetry

I have a particular function that for even numbers $m$ obeys the following equation: $$f_{m,n}\left(\frac{2}{m}-x\right)=(-1)^nf_{m,n}(x)$$ Now when I put in odd values for $m$ and plot the function,...
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1answer
46 views

Biliniear form to inner product

Let $f:V\times V\rightarrow F$ be a bilinear form in a finite inner product space V. If $F=R$, how can I prove that there exists a single linear transformation $T:V \rightarrow V$ so that for each $v,...
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1answer
159 views

Prove that every triangle is the orthogonal projection of an equilateral one

Prove that every triangle is the orthogonal projection of some equilateral triangle. This problem appears in a book I'm working through in the chapter on transformations in space. There is a rather ...
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1answer
45 views

Probability Theory - Transformation (of two variables) of continuous random variables

Let $X_1$ and $X_2$ be independent and identically distributed continuous random variables, with probability density function $$p(x)=\begin{cases} \exp(-x), & \text{if}\ x>0 \\ ...
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1answer
75 views

Transformation of continuous, independent random variables

Let $X_1$ and $X_2$ be independent and identically distributed continuous random variables, with probability density function $$p(x)=\begin{cases} \exp(-x), & \text{if}\ x>0 \\ ...
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2answers
61 views

Probability Theory - Transformation of independent continuous random variables

Let $X_1$ and $X_2$ be independent and identically distributed continuous random variables, with probability density function $$p(x)=\begin{cases} \exp(-x), & \text{if}\ x>0 \\ ...
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47 views

How do I find the matrix with respect to a different basis?

I tried to solve this question but the answer is totally different, can you explain how to solve it
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2answers
33 views

Transformation Matrix for Derivative

I have figured out how to show Part A by using properties of derivatives. For Part B, we know that $T(f)$ is $\begin{pmatrix} 4a+2b+c\\4a+b\\2a\end{pmatrix}$, so when asked to find a matrix $A$, would ...
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2answers
31 views

How do I scale a triangle given its cartesian cooordinates?

Given the cartesian $(x,y)$ coordinates of three points $a, b$ and $c$ that form an equilateral triangle $ABC$, how do I scale them using its center point so that its position on the cartesian plane ...