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

7
votes
0answers
275 views

Way to Tietze's Transformation Theorem

during our knot-theory lecture we have talking about the following theorem: Given two finite presentations of the same group, one can be obtained from the other by a finite sequence of Tietze ...
5
votes
0answers
60 views

Applying PCA on covariance matrix in order to generate a new random variable.

Let $\mathbf{x}$ be a random $n\times1$ real vector, $\mathbf{x}\in\Bbb{R}^n$, which is distributed normally with mean $\bar{\mathbf{x}}$ and covariance matrix $\Sigma_x\in\Bbb{R}^{n\times n}$, i.e. ...
5
votes
0answers
169 views

How to use polynomial or conformal transformation

In my research, I came to a transformation problem. The simple version is an initial circle (or sphere) region is advected by some deformational flow. After some time the circle will be deformed into ...
4
votes
0answers
58 views

Transform recurrence relation

Is it possible to transform following recurrence relation $a_n=4a_{n-2}-a_{n-4}$, $a_0=1$, $a_1=0$, $a_2=3$, $a_3=0$ so that it will have nonnegative coefficients? Number of terms, of course, can be ...
4
votes
0answers
292 views

Geometric intuition for Jordan normal forms (invariant subspaces, shearing, scaling, etc.)

I'm trying to visualize what a linear operator does to a vector space if that operator can be put into Jordan normal form. For concrete motivation, let's take $V = \mathbb{R}^3$, with some linear ...
4
votes
0answers
223 views

how do you map a sphere to a cube

I want to map a sphere to a cube in order to create a panoramic tour like the one given here But I don't know how can you obtain images like This image is one of the cube's faces. What I tried was ...
4
votes
0answers
179 views

What is known about the transformation of a power series in which $z^n$ is replaced with $z^{n^2}$?

Say we have the function $$G(z) = \sum_{n \geq 0} g_n z^n.$$ Is there a name for the transform T defined so that $$(T(G))(z) = \sum_{n \geq 0} g_n z^{n^2}?$$ Is there anything known about this ...
3
votes
0answers
71 views

Fast Hankel Transform

Can someone please explain what would be the expression for weights(Ho) in a Fast Hankel Transform.I found this in a paper and could not find any satisfactory answers .
3
votes
0answers
14 views

Find a matrix to represent the mapping of a factor module

I have a problem from my past paper I can't figure the logic to, even after seeing the answers. The question goes 【Q】Let $V=\mathbb{R}[X]_{<4}$ be the vector space of real polynomials of degree ...
3
votes
0answers
40 views

Proof that Determinant is Scale Factor

I've seen a lot of supposed properties of linear transformations that're never proven -- just often repeated. These include: The determinant is the scale factor between the volume of region in your ...
3
votes
0answers
80 views

Changing coordinate system with non standard definitions

The standard coordinate transformation to polar coordinates is $$ \begin{cases} x=r\cos(\varphi)\\ y=r\sin(\varphi) \end{cases} $$ with $r\in[0,\infty), \ \varphi\in[0,2\pi)$ The question is whether I ...
3
votes
0answers
134 views

Proof for a summation-procedure using the matrix of Eulerian numbers?

I've discussed a procedure for divergent summation using the matrix of Eulerian numbers occasionally in the last years (initially here, and here in MSE and MO but not in that generality and thus(?) ...
2
votes
0answers
42 views

The Adrian Transformation of a function in $\mathbb{R}^{2}$

Recently I came upon a problem (if you would call it that, more of a thought experiment), which was phrased something like this: Rotate the area formed by $\int_{-1}^12dx$ around the curve ...
2
votes
0answers
31 views

The Composition of Two rotations

So far I rewrote the halfturns of d,c,b,a to halfturn (p,n)(m,l) where n=m because lines c and d are parallel so I can make ambiguous lines n and p parallel too. I also know that lines c,d can be ...
2
votes
0answers
66 views

Transforming the cubic Pell-type equation for the tribonacci numbers

The Lucas and Fibonacci numbers solve the Pell equation, $$L_n^2-5F_n^2=4(-1)^n\tag1$$ The tribonacci numbers $z = T_n$ are positive integer solutions to the cubic Pell-type equation, $$27 x^3 - 36 ...
2
votes
0answers
85 views

Given the minimal polynomial, find the largest invariant subspace

Let the linear transformation T on the vector space $V$ over $\mathbb{Q}$ have minimal polynomial $(x^{7} - x^{3})$. a) What is the largest invariant subspace W of V for which $T (W) = W$? b) Find a ...
2
votes
0answers
18 views

Finding the matrix ${\left[ T \right]_E}$

Let the matrix ${\left[ T \right]_{B \to E}}$, the matrix where: $${\left[ T \right]_{B \to E}}{\left[ v \right]_E} = {\left[ {T(v)} \right]_B}$$ It's given that: $${\left[ T \right]_{B \to E}} = ...
2
votes
0answers
75 views

Finding transformation from $T : \Bbb R^5 \rightarrow \Bbb R^4 $ …

Is there a Linear Transformation from $T : \Bbb R^5 \rightarrow \Bbb R^4 $ so $$\operatorname{Ker}T = \{( x,y,z,t,w) \in \Bbb R^5 \; | \; x = 2y, \text{ and, } z = 2t = 3w\}$$ if so find an example of ...
2
votes
0answers
138 views

From inverse Weierstrass function to Jacobi elliptic/inverse elliptic functions?

As a conclusion to a previous question on integrals, I get an answer in terms of inverse Weierstrass elliptic function : $$ f\left(x\right)=\wp^{-1}\left( \beta + \frac{9\beta^2-1}{3(x-\beta)} \right) ...
2
votes
0answers
90 views

Invariant functions under integral transforms

We all know Fourier transform has invariants such as $e^{-x^2}$, and another MSE post has shown the non-existence of invariant function under Hilbert transform using Fourier transform. I am wondering ...
2
votes
0answers
110 views

Get 2D coordinate transformation matrix based on points in a system and their angles in the other?

I'd like to get the parameters (rotation angle,$\Theta$, and translation coefficients, $x_0$ and $y_0$)) of a transformation for translating and rotating points in a coordinate system to another. As ...
2
votes
0answers
64 views

Heat equation $\frac{\partial \theta}{\partial t}=\kappa \frac{\partial ^2\theta}{\partial x}$ using two transformations to solve

Consider the heat equation $$\frac{\partial \theta}{\partial t}=\kappa \frac{\partial ^2\theta}{\partial x}$$ for an infinite rod. We use the transformation $q_1=\frac{x^2}{kt}$ and $q_2=\frac{\theta ...
2
votes
0answers
324 views

Derive Student T distribution using transformation theorem

I am trying working on an exercise that asks me to show that If $ X_1 \in N(0,1) $ and $ X_2 \in \chi^2(n) $ are independent random variables, then $ X_1 / \sqrt{X_2/n} \in t(n) \, $ where $ ...
2
votes
0answers
68 views

Stabilize Variance for Statistics (Transformation)

Problem: When $Y (> 0)$ has mean and variance equal to $\mu$ and $\mu/n$ respectively, it is shown in the textbook that the appropriate transformation of Y to stabilize variance is the square root ...
2
votes
0answers
41 views

Inverting a discrete linear transformation

Consider the transformation from the set $\{a_n\}_{n=0}^N$ to the set $\{p_j\}_{j=0}^N$: $$ p_j = \sum_{n = 0}^Na_n\phi_n(x_j)$$ where $\{\phi_n(x)\}_{n=0}^N$ is a set of basis functions (linearly ...
2
votes
0answers
66 views

Perhaps an easy algebra problem, but it still evades me

I need help spotting a corresponding transformation Let $x,y$ be some variables and $$z=z(x,y)$$. We have a transformation $X(\lambda):(x,y,z)\to (x',y',z')$, such that $$x'= x\exp(a\lambda)\\ ...
2
votes
0answers
38 views

Identity for reducing the power in the parameters of Hypergeometric functions

Is there any identity/formula for reducing/increasing the power in the parameters of the Gauss Hypergeometric function $ _2F_1(a,b;c;z^d)$ (d is a real) let's say to z? Is there also any identity for ...
2
votes
0answers
224 views

Hodograph transformation and implicit solution of a non-linear PDE

I am trying to understand how can one apply the Hodograph transformation to a non-linear PDE. I read that this transformation implies the representation of the solution in the implicit form . So, if I ...
2
votes
0answers
80 views

Is there an intuitive understanding of what a walsh coefficient is?

I am working with Walsh coefficients. I know the intuitive understanding is almost that that they are the degree of connectivity, but it is there a better way of thinking about it? What is the ...
2
votes
0answers
97 views

bound on Hilbert transform

Consider $\widehat{Tf(\xi)}=m(\xi)\hat{f}(\xi)$, where $m(\xi)=(1-\vert\xi\vert)1_{[-1,1]}$, i.e. $T$ is the operation of taking Fourier transform and multiplying with the function $m(\xi)$. I am ...
2
votes
0answers
345 views

Understanding Discrete Cosine Transformation

I'm currently working on some software and a key component is 2D DCT. But my question is more general, as I'm trying to understand the DCT in general, let's say from engineers point of view. For ...
2
votes
0answers
212 views

transformation of coordinate systems by rotation

I am trying to convert a set of coordinates from ECEF (Earth Center Earth Fixed) to ENU (East North Up). The operation is performed by applying a rotation matrix as shown in: ...
2
votes
0answers
135 views

Transform 3D vectors between planes using a matrix

I've got 6 points in 3D space: $A,B,C,D,E,F$, that represent 4 vectors. $AB$ is perpendicular to $AC$ and $DE$ is perpendicular to $DF$. I need to find a transformation matrix M, that transforms $AB$ ...
2
votes
0answers
134 views

Hyperbolic Universal Covering Space

I have been working with Ricci flow in the euclidean and hyperbolic space but have been having considerable trouble determining how to generate a universal covering space for complex hyperbolic ...
2
votes
0answers
401 views

How to find the center of an (scaled) ellipse?

This question is an extension of How to find the center of an ellipse?. The solution there works well, but in Javascript the floating point calculations are not that accurate. The workaround is to ...
1
vote
0answers
23 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 ...
1
vote
0answers
13 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
22 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 = ...
1
vote
0answers
18 views

Remove Multiplicative Constant from Hypergeometric Function

I have a function of the form $$f(x;\lambda) = {}_2F_1\left(a,b;c;-\frac{e^{2x}}{\lambda}\right)$$ I need to invert this function to solve for the constant $\lambda = f\left(x\right)$. I could do ...
1
vote
0answers
25 views

Fourier transform from Laplace transform

So what I did was Laplace transform $f(t)$ to $F(s)$ and then plug in $s=jw$. However, when I tried this for $cos(3t)$, $$F(jw)={jw\over9-w^2}$$ was the answer. I don't know if this is correct, and ...
1
vote
0answers
56 views

Strange results on training ANN by BP to “do” a kind of simple Discrete-Fourjer-Synthese

Here's the result of an experiment I'm trying to do atm. I've checked it, It really does a FourjerSythese! Can someone please explain me how these strange matrixs' can do that ? For those of you ...
1
vote
0answers
39 views

Is this chain of inequalities correct?

Is this chain of inequalities correct? If not how to make it works? $$\frac{\ln \left( 1+x^3+y^3 \right)}{\sqrt{x^2+y^2}} \le \frac{\left( x^3+y^3 \right)}{\sqrt{x^2+y^2}} \le \frac{ \left( ...
1
vote
0answers
29 views

Solids of Revolution around other functions.

Recently I've been thinking about solids of revlution, and thought about an interesting experiment. Can you rotate functions around, for example, the line $f(x)=x$? And consequently, could you rotate ...
1
vote
0answers
18 views

How can I transform coordinate systems based on quaternion data?

I have a single rigid body object, and its orientations in quaternion with respect to two coordinate systems, each is called original and prime, respectively; therefore, I have two quaternions ...
1
vote
0answers
30 views

mobius transformation form $M= B(0,1)\setminus\overline{B(1,\sqrt2)}$ to a sector.

I know that the intersections of the two circles need to be sent to $0$ and $\infty$ in order to get a sector $S = \{x+iy: x>0,0<y<x\}$ The intersections of the two balls were i and -i. So i ...
1
vote
0answers
40 views

Transformation that f(A+B)=f(A)+f(B) and f(AB) = f(A)f(B)

if $f:M_{n*n}(F) \rightarrow M_{m*m}(F) $ and f transformation identity to identity matrix and $f(AB) = f(A)f(B) , f(A+B)=f(A)+f(B)$. now we want to prove there is an integer like $k$ that $m = k*n$
1
vote
0answers
32 views

inverse laplace tranform

I have a simple question, There are some functions f(t), g(t) and lets say F(s) and G(s) for the form of Laplace transform of f(t) and g(t), respectively. While I am solving differential equation ...
1
vote
0answers
27 views

How do I calculate 3D movement based on yaw, pitch and roll?

I'm creating a 3D game demo and I need to calculate the position of the player in the space (i.e. the player's x, y and z coordinates). I understand that this would be affected based on the camera ...
1
vote
0answers
28 views

Reducing heat equation into nondimensional form

I want to get nondimensional form of heat equation $u_t=a(x,t)u_{xx}$. For the case of $a(x,t)=a(t)$, by setting $A(t)=\int_0^ta(\eta)d\eta$ and $t=\phi(\tau)$, where $\phi$ is the inverse mapping ...
1
vote
0answers
39 views

Mapping A point from one 3D Coordinate System to Another 3D coordinate System with Euler Angles between the two systems given

Suppose I have a point in the green coordinate system, and I wish to describe it in reference to the orange coordinate system. I know the roll, pitch, and yaw of the green system with respect to the ...