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), ...

learn more… | top users | synonyms

0
votes
1answer
15 views

Finding the eigenvalues and eigen vectors of linear transformation

$$T:Z_5^{2x2}\rightarrow Z_5^{2x2}$$ $$T \left( \begin{array}{ccc} a & b\\ c & d \\ \end{array} \right) = \left( \begin{array}{ccc} a & a+b\\ b+c & c+d \\ \end{array} \right)$$ This ...
0
votes
0answers
22 views

Log-like transform function which is almost linear for small numbers

I am regular user of stackoverflow.com, but I think that my question can be better answered here. Please let me know if my inputs are not clear or if should move this question to some other forum. I ...
1
vote
1answer
27 views

Why does this hyperboloid change into a surface? [duplicate]

Given this equation $x^2+y^2+z^2+2xy+2xz+2yz-x-y-z=6$ and the corresponding quadric: If I rearrange the equation to $(x+y+z-3)(x+y+z+2)=0$ (which is equivalent), I get: So, which is the right ...
3
votes
0answers
36 views

How to transform (rotate) this hyperbola?

Given this hyperbola $x_1^2-x_2^2=1$, how do I transform it into $y_1y_2=1$? When I factor the first equation I get $(x_1+x_2)(x_1-x_2)=1$, so I thought $y_1=(x_1+x_2)$ and $y_2=(x_1-x_2)$. ...
1
vote
1answer
28 views

Fourier transform of a $H(x)$ product distribution

So I am given this simple example, where $T \in \mathcal{S}(\mathbb{R})$: \begin{equation} T=(\mu +\lambda x+\beta x^2)H(x) \end{equation} where $H(x)\in \mathcal{S}(\mathbb{R})$ (also notated as the ...
1
vote
1answer
58 views

Fourier Transform of a Polynomial

Lets say you are given \begin{equation} f(x)=1+x^3 \end{equation} and the definition of Fourier transform: \begin{equation} \hat{f}(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-ikx}f(x)dx, ...
3
votes
0answers
41 views

On the Fourier transform of $f(x)=\ln(x^2+a^2)$

I would like to derive the Fourier transform of $f(x)=\ln(x^2+a^2)$, where $a\in \mathbb{R}^+$ by making use of the properties: \begin{equation} \mathcal{F}[f'(x)]=(ik)\hat{f}(k)\\ ...
2
votes
1answer
14 views

How to see transformations on polytopes?

I have a polytope in six dimension with extreme points $(1,0,0,0,0,0)$ $(0,1,0,0,0,0)$ $(0,0,1,0,0,0)$ $(1,1,0,1,0,0)$ $(1,0,1,0,1,0)$ $(0,1,1,0,0,1)$ $(1,1,1,1,1,1)$ $(0,0,0,0,0,0)$ Each of the ...
1
vote
0answers
26 views

On the convolution of $f(x)=\sin x/x$ and $g(x)=1-|x|$

I am having trouble with computing the convolution of $f(x)=\sin x/x$ and: \begin{equation} g(x)=\begin{cases} 1-|x|,& -1 \leq x \leq 1 \\ 0, & x \notin [-1,1] \end{cases} \end{equation} I ...
2
votes
0answers
15 views

What does the s-Transform (exponential transform) mean conceptually? What does it show us?

I don't understand the conceptual idea. If I have PDF, and I calculate its s-transform for some s, what do I know that I did not know before?
1
vote
2answers
58 views

On the Fourier transform of $f(x)=e^{-x^2+2x}$

So, I have the $f(x)=e^{-x^2+2x}$ and to take the FT of it, I complete the square: \begin{equation} f(x)=e^{-x^2+2x \pm1}=e^{-(x-1)^2}e \end{equation} Then, by knowing that the FT of $g(x)=e^{-x^2}$ ...
9
votes
4answers
606 views

Integral becomes improper after a substitution

I'm suprised about the following phenomenon which I would like to discuss with you. Consider the proper integral $$\int_{\pi/4}^{\pi/2}\frac{1}{\sin(x)}dx.$$ Since $\sin(x)$ is a diffeomorphism on ...
1
vote
1answer
17 views

Transformation of two i.i.d. uniform random variables

G'day folks, I'm trying to work through a problem in preparation for an exam and it's got me stumped. The question is: Let $X_{1}$ and $X_{2}$ be i.i.d. $U(0,1)$ random variables. Let ...
2
votes
1answer
16 views

Interpolate between 3D plane and 3D hemisphere

I have a simple 3D plane whose points (different $x, y$ values, but all $z = 0$) need to be mapped to 3D Cartesian coordinates in order to form a hemisphere. However, I also would like to be able to ...
0
votes
0answers
8 views

Theory that studies varous transformations

It is obvious that Fourier transform, Laplace transform and integration itself are similar things. So, which kind of mathematics generalizes such transformations?
1
vote
0answers
20 views

Fourier cosine transformation

Good day! I'm studying right now some transformation and I encountered the following equation: $$(2\pi)^{-n/2} \int_{-\infty}^\infty\cdots\int_{-\infty}^\infty \exp\left(-\frac{1}{2} ...
1
vote
0answers
29 views

Is my proof true or not ? $rk(A) + rk(B) \ge rk(A+B)$

I know that my question has already an answer here, but I have proved it another way and I want to see whether my proof is true or not? If we assume $A$ , $B$ and $A+B$ are respectively the matrices ...
-1
votes
0answers
36 views

How windowing is done in Short time Fourier transform?

From Fourier transform we can get features localised in Frequency domain but we neglect all time domain features. So we use Short time Fourier transform (STFT) in which we do some windowing to ...
-1
votes
0answers
30 views

How condition for existence of Fourier transform is valid?

The condition for Discrete time Fourier transform to exist for function $f(n)$ is given as $$\sum_{-\infty}^\infty |f(n)| < \infty.$$ In case of continuous Fourier transform the difference is ...
0
votes
2answers
33 views

Convergence of random variable times function: $nX_n$

If $X_n\xrightarrow[]{p}X$, can I prove that $n(X_n-X)\xrightarrow[]{p}0$ if $X$ is a natural number. I know that if $Y_n$ is bounded in probability $Y_nX_n\xrightarrow[]{p}0$, or that if $n$ is a ...
3
votes
1answer
19 views

Get the known Laplace's equation

Let $u(x,y), x^2+y^2 \leq 1$, a solution of $$u_{xx}(x,y)+2u_{yy}(x,y)+e^{u(x,y)}=0, x^2+y^2\leq 1$$ Show that $\min_{x^2+y^2 \leq 1} u(x,y)= \min_{x^2+y^2=1} u(x,y) $. We suppose that ...
1
vote
1answer
48 views

Condition for existence of Fourier transform?

We can convert signal into frequency domain using Fourier transform. But I think we can't compute Fourier transform of any signal . Fourier transform also should have some limits. So I want to ask ...
6
votes
1answer
102 views

Why is the momentum a covector?

Can someone tell me why the momentum is an element of the cotangent space? More detailed: if we have some smooth manifold M and the cotangent space $T_{x}M^{*}$ I know that the momentum p is an ...
2
votes
2answers
89 views

How do the components of a cross product transform?

Let $x^{j}$ and $y^{k}$ be the components of two vectors $x,y\in \mathbb{R}^{3}$. According to the way the compontents of $x$ and $y$ transform when we change the basis, we know they are ...
0
votes
0answers
30 views

What do real and imaginary parts of phase spectrum represent?

In frequency domain, we can compute phase spectrum of a signal. Usually phase spectrum is complex valued. So my question is what do real and imaginary parts of phases of phase spectrum represent ? ...
2
votes
1answer
151 views

How to decompose matrix transformations

Let us assume $A$,$B$ and $C$ are known affine transformation matrices in homogeneous 2D space. If it should happen that $C=A^m B^n$ for some unknown $m,n$, is there a way to detect this short of ...
0
votes
1answer
28 views

How to transform angles to a transformation matrix?

I'm working on an open source project. I need to transform three angles (X, Y, Z) to a matrix. The matrix is a standard 4x3 homogeneous transformation matrix, where the right column describes the ...
0
votes
0answers
42 views

The definitions of “transformation” and “isometry”

Let $T$ be a mapping from the plane to itself. In the context of Euclidean geometry, can $T$ be called a "transformation", or is this word reserved for cases where $T$ is bijective? Is there ...
0
votes
0answers
20 views

Understandning Radial Fourier Analysis

I'm currently studying living cells. In order to characterize their form, we use "Radial Fourier Analysis" as described here. I can't, however, seem to find more information about this topic (Radial ...
3
votes
1answer
15 views

Why can the transformation derived from a list of points and a list of their transformed counterparts not be affine or linear?

Some context (original question below): I wanted to know if there's a nice concise formula to calculate the transformation based on a list of points and another list of the transformed points. This is ...
0
votes
1answer
14 views

Transformation of a function: is this integral improper?

I've strumble solving this integral that I obtained after a transformation. Consider $f_{X,Y}(x,y)=e^{-(x+y)}, x,y>0$. Let $V=X^2$ and $R=\frac{X}{X+Y}$. I want to get $f_{V,R}$. So $x=\sqrt{v}$ ...
0
votes
2answers
98 views

Why Fourier series has summation and Fourier transform has integration symbol in their respective formulae?

Fourier transform for aperiodic signal is given by $$ X(\omega) = \int\limits_{t=-\infty}^{+\infty} x(t) e^{-j \omega t} dt. \quad (1) $$ Fourier series for periodic signal is given by $$ y(t) = ...
0
votes
0answers
14 views

Double transformation

I am stuck in determening the following distribution, could anyone tell me the mistake I am making? Consider the uniform distributed stochast $U$ on $[0, 1]$, independent from $X$ and $Y$, $X$ and ...
0
votes
0answers
10 views

Transform Coordinate system

I would like some help to understand a specific transformation for a coordinate system change as I am not sure about it. I got some sample code so I can see how it is calculated but dont understand ...
8
votes
0answers
71 views

How to explain the topic of Fourier transform interactively? [closed]

This is a soft question . In the walk-in for the lectureship, I have decided to give demo lecture on the topic of Fourier transform. The principal of the institution ask me to take lecture ...
1
vote
0answers
47 views

Compare between Short Time Fourier Transform and Wavelets

Fourier transform is localised in only frequency domain but Short time Fourier transform(STFT) is localised both in time and frequency domain same as in wavelets. I want to know How are STFT and ...
3
votes
0answers
33 views

Find transformation matrix with respect to another basis

I understand how we can find the transformation matrix $D$ with respect to another basis $B$, by using a transformation matrix that we already know, say $A$: $$D = C^{-1}\cdot A\cdot C$$ Where $C$ is ...
0
votes
1answer
32 views

Laplace transform,Fourier transform and Z transform mathematical equations

Fourier transform $x(w)$ of signal x(t) is given by $$ x(w) = \int\limits_{t=-\infty}^{+\infty} x(t) e^{-j w t} dt -(1)$$ Laplace transform $x(s)$ of signal x(t) is given by $$ x(s) = ...
2
votes
0answers
12 views

Finding the transformation matrix of a projective transformation in RP^2

So I want to understand how to find the matrix that represents the projective transformation that sends 4 given points to 4 given images, I know that 4 points are necessary to determine it but I can't ...
0
votes
0answers
39 views

How do I express each natural number as sum of serie?

I have many attempts to express each natural number as a sum of series which I meant not to take all convergents series that are giving us 1 as a result I want only how to let e.g : 1 defined ...
0
votes
0answers
14 views

Fourier transform of a polynomial function with both real and complex roots

I am given the following function: \begin{equation} f(x)=\frac{x}{x^3-7x^2+16x-10} \end{equation} which has the following roots: \begin{equation} x_1=1 \in \mathbb{R}, \quad x_{2,3}=(3 \pm i) \in ...
1
vote
0answers
41 views

Distribution Function and Transformation [on hold]

Mike is randomly shoot an arrow at 1m bar. Jackie will shoot an arrow at randomly the right side of Mike shoot. X is the point, arrow of Mike, Y is the point, arrow of Jackie. X,Y are from the ...
0
votes
0answers
34 views

Fourier transform of $f(x)=\frac{x}{1+x^4}$ and $g(x)=\frac{x^2}{1+x^4}=xf(x)$

Let $f(x)=x/(1+x^4)$, the improper integral of which exists. I computed the Fourier transform of $f$, to be: \begin{equation} ...
0
votes
0answers
13 views

Representing cartesian unit vectors in terms of (u, v) during Jacobian transformation

If r(u, v)=f (u, v)i+g (u, v)j= xi+yj defines a plane, I need to know how dxi=(∂x/∂u)duu+(∂x/∂v)dvv I do not understand the summation of u and v components. Given that x is a scalar function of (u, ...
4
votes
1answer
88 views

Is this an inversion through the origin?

I have a polar vector $e$ with $|e|=1$, and I perform a transformation $T$ that maps all other polar vectors such that $e \cdot T (s) = - e \cdot s$. One such $T$ is inversion through the origin. What ...
1
vote
2answers
78 views

Is Fourier series used always for periodic signals and Fourier transform for aperiodic signals only?

I want to ask basic question. In our mathematics classes ,while teaching the Fourier series and transform topic,the professor says that when the signal is periodic ,we should use Fourier series and ...
0
votes
2answers
25 views

Finding an ordered basis to diagonalize Transpose matrix.

We define $T : M_{n \times n}R \to M_{n\times n}R$ by $T(A) = A^t$. We can write the matrix representation of this transformation as: $[T]_\beta^\beta = \begin{pmatrix} ...
0
votes
0answers
18 views

Graphical transformation : reflect and shift

I know that x[-n] will be reflection of x[n] along y-axis and x[n+k] will shift x[n] to left by k points. Now if I take x[n] 1. x'[n]=x[-n] should reflect along y axis 2. x'[n+k]=x[k-n] should shilf ...
1
vote
1answer
47 views

Mean Value Theorem

Good Day! I,m aware of the basic concept of mean value theorem but the application of it in proving makes me confuse, this is how it goes: By mean Value theorem: $$2 - t^{n-1} (1+t) = (1 - t)[θ^{n – ...
-1
votes
1answer
42 views

Fourier synthesis of periodic signals

I was reading the Fourier synthesis of periodic signals But I didn't understand the sentence i.e. "Although the calculation of $a_0, a_1, b_1, a_2, b_2$, is a mathematically straightforward ...