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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2
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2answers
62 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
608 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
18 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
25 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
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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
33 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 ...
0
votes
2answers
34 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
53 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
113 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
94 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
35 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
156 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
45 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
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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
16 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
108 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
13 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
76 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
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0answers
52 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
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0answers
38 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 ...
3
votes
1answer
51 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
14 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
41 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
16 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 ...
0
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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
15 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
89 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 ...
5
votes
2answers
106 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
27 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
19 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
48 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
50 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 ...
3
votes
4answers
43 views

Finding a matrix representation of the transpose transformation

Define $T : M_{n×n}(\mathbb{R}) → M_{n×n}(\mathbb{R})$ by $T(A) := A^t$. I know this transformation is linear and just takes a matrix and spits out it's transpose. I also know that the transpose is ...
0
votes
1answer
40 views

Questions about Eigenspace

I'm learning about Eigenspaces and have a few questions. Do eigenspaces, eigenvalues, and eigenvectors correspond to a tranformation or can a single vector space $V$ have an eigen-stuff? Is an ...
2
votes
1answer
42 views

Necc. and suff. conditions for a canonical transformation.

Let $\mathbf{P} = C^{−1}\mathbf{p} + B\mathbf{q}, \mathbf{Q} = C\mathbf{q}$, where $C$ is a symmetric nonsingular matrix. Determine necessary and sufficient conditions on $C$ for the transformation ...
0
votes
1answer
34 views

is it possible to decompose nonperiodic sinusoidal signal?

Using Fourier series we can decompose any any signal into it's elementary signals but condition is that signal should be periodic and sinusoidal one. Now, is it possible to decompose nonperiodic ...
-1
votes
1answer
35 views

Do all n x n matrices over the reals represent linear transformations?

Do all $v \in M_n (\mathbb{R})$ represent linear transformations? To add to that a bit to further clarify for myself: Looking up the def. of a transformation it is any function $f$ mapping a set $X$ ...
1
vote
1answer
18 views

Equivalent operations on Bezier curve points as control points?

In this question Explicit Bezier Curves: Lerping between curves same as lerping control points?, it shows that linearly interpolating between the result of evaluating two explicit bezier curves is the ...
1
vote
1answer
42 views

When does $ \langle gI, t \rangle = \langle I, g^{-1} t\rangle $ hold true?

Consider $I, t \in \mathbb{R}^d$ and $g$ is some element in a group of transformations (for example like the affine group in $\mathbb{R}^2$). I was wondering when the inner product $ \langle gI, t ...
2
votes
1answer
34 views

Find $Z$ transform of given signal

Given the discrete signal $h(n)=r^n\frac{\sin{[(n+1)\theta]}}{\sin{\theta}}$ if $n \geq 0$ and $h(n)=0$ otherwise, find the $Z$ transform of $h(n)$. What I did: We know that ...
-2
votes
1answer
21 views

About the proof of Zeta Transform

I have to prove $$Z[k^n]=(-1)^nD^n\left(\frac{z}{z-1}\right)$$ where $$D=z\frac{d}{dz}$$ and $n$ varies over the set $\mathbb{Z}$ My book doesn't give me any advice; how can I go further?
0
votes
0answers
7 views

About the notation in zeta transforms

My book writes: $$Z[n^k]=(-1)^kD^k(\frac{z}{z-1})$$where $D=z\frac{d}{dz}$ and $n$ varies over the set $\mathbb{Z}$. What does $D^k$ mean?
1
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0answers
20 views

Change of Basis Matrix: Cartesian to Spherical Laplacian

I was looking at how a change of basis matrix, $[P_{\beta\leftarrow\alpha}]$, is made. While this is a bit more advanced that than what was taught at the course, I wonder what would be the change of ...
1
vote
0answers
26 views

Singularities in the Gauss Hypergeometric Function

I am evaluating the following term in a series: $$I_k = \int\!x^{-3(2k+1)}(1+\lambda x^4)^{-1/2}\,\mathrm dx$$ When I plug this into WolframAlpha, I get the following result: $$I_k = ...
0
votes
0answers
40 views

How to compare ZOH and tustin

I'm discretizing some continuous time systems. Now there (MATLAB) are of course different types of discrtization methods, among them tustin (bilinear), euler backwards, euler forward etc. Often one ...