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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13 views

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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1answer
13 views

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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0answers
30 views

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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73 views

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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0answers
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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5answers
33 views

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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1answer
93 views

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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1answer
32 views

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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1answer
49 views

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. ...
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1answer
37 views

PDF for $\frac1a Uniform(−a,a)$

Problem: Let $X$ be $Uniform(−a,a)$ distrubuted. Calculate the PDF for $Z = \frac1a abs(X)$. Attempt: I think graphically here. $X$ is $U(-a,a)$ with $PDF = \frac1{2a}$ so $abs(X)$ is $U(0,a)$ with ...
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1answer
108 views

Show random variables are mutually independent

Let $X_1$,$X_2$, $X_3$ denote a random sample from the distribution having p.d.f $$f(x) = e^{-x},\,\, 0<x <\infty,$$ zero elsewhere. Show that $$y_1 = \frac{x_1}{x_1 + x_2} $$ $$y_2 = ...
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7 views

Variance Stablizing Transformations on Kernel Density Estimates

Suppose if I had a data set and performed a non-parametric kernel density estimate and generated a probability distribution function - using the delta method/taylor expansion (as seen in: ...
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1answer
67 views

Mellin transform of Gumbel distribution

The probability density function (PDF) of Gumbel distribution is given as: $$f\left(x\right)=\frac{\exp \left(-\left(\exp \left(-\frac{x-\mu}{\beta }\right)+\frac{x-\mu}{\beta }\right)\right)}{\beta ...
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2answers
38 views

Using the dimension formula to prove isomorphism

Let V be a finite-dimensional vector space and $T: V\rightarrow V $. T is a linear transformation. Use the dimension formula to prove that if T is injective, it must also be surjective; if T is ...
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0answers
60 views

Normalize a diagonal matrix such that each element belongs to $[0,1]$.

Let $\mathbf{x}\in\Bbb{R}^n$ be an $n$-dimensional real vector and $C\in\Bbb{R}^{n\times n}$ be a diagonal real matrix. Suppose that two vectors $\mathbf{r}_\min,\mathbf{r}_\max\in\Bbb{R}^n$ ...
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0answers
17 views

Deriving the formula for Fixed-rate mortgage using z-transform

So, using z-transform, one could easily derive the formula for fixed-rate mortgages. However, when I tries this, I noticed one thing I'm very uncertain of. The relationship for fixed-rate morgages ...
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2answers
17 views

how to map series of coordinates onto a series of coordinates with different resolution

I have a set of target coordinates and a set of actually clicked coordinates which should be approximately the same, but not identical. The y coordinates are equal, however, the x-coordinates differ, ...
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1answer
26 views

Find inverse $z$-transform of $\frac{5}{z^{2}-z-6}$

How can I find inverse z transform of $$X(z)=\frac{5}{z^{2}-z-6}$$ What I did: first i factored denominator and i got (z+2)(z-3), now we get A(-2^{n}) + b(3^{n}). To get A and B i used Partial ...
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1answer
42 views

Find inverse $z$-transform of $\dfrac{(z-1)^2}{z^3}$

How can I find inverse z transform of $$X(z)=\frac{(z-1)^{2}}{z^{3}}$$ What I did: I am thinking to do Partial Fraction Decomposition or long division. Is there another method ?
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1answer
35 views

Help tranformation of random variables?

Let $X$ have the p.d.f $f(x)= \frac{x^2}9$ , $0 < x < 3$, $0$ otherwise, find the pdf of $Y = X^3$ I have this exercise, but I do not know how to start, how do I know if it is a one to one ...
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2answers
27 views

Image of $A \subset \mathbb{R}^2$ under general transformation

If I have a transformation $\varphi: \mathbb{R}^2 \to \mathbb{R}^2$ which doesn't have any particular property, for example, which is not linear, how do I know what is the image of a subset $A \subset ...
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3answers
145 views

Calculate centers of circles from their ellipse perspective.

Originally there are 4 circles in a plane and after perspective transform we get four conics. Now I know the equation of those ellipses. How could I get the origin of those four circles ? I know ...
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2answers
25 views

Show the solution for $\mathcal{F}(e^{-\left | t \right |})$ [closed]

I'm trying to show that $$\mathcal{F}(e^{-\left | t \right |}) = \frac{2}{\sqrt{2\pi}(1 + w^2)}$$ Knowing that the Fourier transform is in the form $$\mathcal{F}(f(x)) = ...
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0answers
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Where do $H(3,1)$ and $H(3,2)$ come from when building Homography?

So you're given 4 pairs of points, including the original and where it maps to, i.e. $(x_1, y_1)\Rightarrow (x_1', y_1')$, $(x_2, y_2)\Rightarrow (x_2', y_2')$, $(x_3, y_3)\Rightarrow (x_3', y_3')$, ...
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1answer
59 views

Determine shift between scaled rotated object and additional scale step

I am trying to find the amount to move an object so that, when it rotates and resizes, the resize would be smooth. I have a Qt program, where I have to rotate objects around center, and resize based ...
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0answers
47 views

Prove something is not a linear transformation

As I understand it, the two things that define a transformation as linear are: 1) $T(u+v) = T(u)+T(v)$ 2) $T(cu) = cT(u)$ I want to prove that $T(x,y) = x+y+1$ (where $T: \mathbb{R}^2\to ...
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0answers
17 views

Inverse z-transform of $z^4+1.827z^3+2.338z^2+1.827z+1$

I need to transform the following $H(z)$ back to time domain: $$ H(z)=(z-e^{j\frac{8}{15}\pi})(z-e^{-j\frac{8}{15}\pi})(z-e^{j\frac{12}{15}\pi})(z-e^{-j\frac{12}{15}\pi}) $$ I did the following steps ...
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1answer
37 views

Transformations of the form $f(ax+b)$

Suppose I wanted to sketch the graph $y=\sqrt{5x-10}$ for $5x-10 \ge0$ Is there a direct method? I know that I can define first $g(x)=\sqrt{x}$ and consider $y=g(5x)$ or $y=\sqrt{5x}$ This is a ...
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0answers
19 views

Is a signal summable when given a z-transform?

The output of a system is given as z-transform: $$ Y(z)=\frac{1+z^{-2}}{(1+\frac{1}{4}z^{-1})(1-\frac{1}{2}z^{-1})} $$ I want to know if the signal in the time domain is summable, meaning that the ...
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2answers
211 views

Del operator in Cylindrical coordinates (problem in partial differentiation)

I am currently reviewing basic vector analysis and trying to understand every single detail, however, I got stuck in some derivation. What I want to show is the following: Given the del operator ...
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1answer
32 views

If $f:\mathbb{R}^{2}\to \mathbb{R}$ continuous, strictly increasing with $f(x(y),y)=0$ for unique $x(y)$ for each $y$

A claim: If $f:\mathbb{R}^{2}\to \mathbb{R}$ is a continuous function, $f(x,y)$, strictly increasing in each of its arguments, with $f(x(y),y)=0$ for a unique $x(y)$ for each $y$, then $f$ must be a ...
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0answers
16 views

Fourier transformation: $t\cdot\sigma(t)e^{-t}$

$$f(t) = t\cdot\sigma(t)e^{-t}, \text{where}\,\sigma(t)\text{ is the unit step function}$$ I thought about using the convolution theorem because we know that $F[\sigma(t)e^{-t}](\omega) = ...
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2answers
708 views

Should I trust Mathematica or numerous other sources on this Fourier transform

Assume $a>0$ So Mathematica claims $$F\{e^{-a|t|}\}(\omega) = \frac{a\sqrt{\frac{2}{\pi}}}{a^2+\omega^2}$$ However, I've read about another transform pair (page 3): $$F\{e^{-a|t|}\}(\omega) = ...
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1answer
23 views

Proof of Ptolemy's inequality using $z \mapsto \frac1z$

Ptolemy's inequality: if $A,B,C,D$ are the vertices of a convex quadrilateral $ABCD$, then: $AC \times BD \le AD \times BC + AB \times DC$, with equality only if this quadrilateral is inscribed in a ...
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0answers
20 views

Transformation of a function; different methods, different values?

Given the function $ g(x) = 2f(2x + 2) - 3 $ for the point $ (1, 2) $. Now, we take 2 common and we get $ f(x) = 2f( 2(x + 1) ) - 3 $; the Horizontal stretch is $ \frac{1}{2} $. Solving for x: ...
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4 views

Binomial Logistic Regression with variables transformation

I'm a high school student studying accelerated maths. I understand that variables can be transformed for better linear line fit. This one is easy to picture an image. But for an dependent variable ...
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1answer
8 views

How can I know the points within contours will stay within the mapped contours?

I don't know if this is true of conformal mapping or just mapping in general but I want to be completely sure that if I know how the contour of a region transforms then the points within the original ...
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0answers
15 views

How coordinate lines transform under $e^z=\frac{a-w}{a+w}$

I was asked this question as part of a homework assignment. I'm assuming $w$ is my transformation function such that $w=f(z)$ and that the coordinate lines in question are the cartesian coordinate ...
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2answers
59 views

Using DTFT to find the sum of $\sum_{n=-\infty }^{\infty }\text{sinc}(n\alpha_1)\text{sinc}(n\alpha_2)$

I am trying to use DTFT (as asked in a problem) to find the following sum $$\sum_{n=-\infty }^{\infty }\text{sinc}(n\alpha_1)\text{sinc}(n\alpha_2)$$ for real $\alpha_1>0$ and $\alpha_2<1$. I ...
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1answer
29 views

homogeneous transformation matrix - How to use it?

I am trying to understand the homogeneous transformation matrix, for which i don't understand what kind of input it requires. What bothering me is the subscript (new) used at the "Location of old ...
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1answer
32 views

Expectation of odd one-to-one transformation of a random variable.

Let $X,Y$ be random variables taking values in $[-1,1]$. Suppose we know that $\mathbb{E}[X] = c \cdot \mathbb{E}[Y]$ where $c \in \mathbb{R}^-$, and that neither expectation is $0$. Suppose ...
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1answer
25 views

Convert a density function $\rho(r)$ of sphere to ellipsoid.

Note: This is not a homework problem. As the title somewhat eludes to, I have a density function for a sphere as a function of radius $\rho(r)$. I would like to then flatten the sphere slightly into ...
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1answer
55 views

Calculate the inverse Laplace transform of $\frac{1}{1-e^{-s}}$

During my signals and systems class i came across this and i have to find it's inverse Laplace transform. I don't know how. $$\mathcal{L}^{-1} \Big\{ \frac{1}{1-e^{-s}} \Big\} = \ ?$$ Any help ...
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1answer
65 views

Point between two points given time? [closed]

Let's say there are two separate points in a 3 dimensional space. An object at point A can move to point B at any speed given (let 'S' be units per second). If I start moving the object from point A ...
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1answer
32 views

Which functions of a normally distributed random variable are also normally distributed

I know that if $X$ is normal then $Y$ = $f(X)$ = $aX + b$ is normal, and this is covered in other questions. Are there any other cases?
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1answer
33 views

Inverse of the transformation $X \mapsto Y = X\cdot X^t$ [duplicate]

I have a Matrix Y of Kind : $n\cdot I$ where $I$ is the identity Matrix of size $n\times n$ , I need to find Matrix $X$ if it exists such that all elements are either 1 or -1 and satisfies $XX^t$ = ...
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33 views

Zero padding property of FFT

I wonder if the following basic property I thought up is a real property of FFT (or more specifically the discrete version of Fourier series), and if so what is it officially called? The property ...
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1answer
43 views

Creating a sparse matrix from a dense matrix

I would like to know whether there is a general method (and, if so, which one) to create a sparse matrix from a dense matrix. I know a sparse matrix simply does not include the zero entries, but since ...
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1answer
29 views

Difficulty calculating velocity after lorentz transformation

I'm working on understanding Lorentz transformations via a text by Garrity, "Electricity and Magnetism for Mathematicians: A Guided Path from Maxwell's Equations to Yang-Mills". On pages 43 and 44 he ...
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2answers
94 views

Transformation of second order ODE

I am a beginner for a ODE theory. There is a ODE: $$ x^{2\beta+2} \frac{d^2 V}{dx^2}+x\frac{dV}{dx}+bV=0. $$ In the paper, the authors introduce some transformation $$ ...