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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263 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 ...
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57 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
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166 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
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56 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
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0answers
280 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
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219 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
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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
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77 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
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133 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
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41 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
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26 views

How to solve an inverse relationship (cooking temp/time)

How to figure out exactly the "add a little more time" to the question: cook at 425 deg for 18 minutes ... if I have several things in the same oven and need to set the oven at 375. I can't use a ...
2
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25 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
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61 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
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78 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
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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
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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
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126 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
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79 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
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104 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
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63 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
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312 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 $ ...
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67 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 ...
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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
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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
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219 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
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75 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
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96 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 ...
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331 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 ...
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211 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
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134 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
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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 ...
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396 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 ...
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10 views

Identifying a subclass of the class of monotonic transformations

Let $u$ be a continuous function from $R$ to $R$. Then $v$ is called a positive monotonic transformation of $u$ if $u(x) < u(y)$ if and only if $v(x)<v(y)$ and similarly for greater than and ...
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34 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( ...
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19 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 ...
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16 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 ...
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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 ...
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35 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 $m = k*n$
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28 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 ...
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0answers
15 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 ...
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21 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 ...
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25 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 ...
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23 views

DPE problem invlolving Fourier transforms / partial eq.

Don't even know where to start with this question! would really appreciate some guidance.
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33 views

Fourier Tansform of derivative on Wolfram Alpha

If I'm not mistaken, the Fourier Transform of the nth order partial derivative of a function with respect to x, using the transform variable k is: (i*k)^n * [F(k)] so for the 1st order derivative ...
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38 views

How can I show that the AR process is nonstationay if x(n) has nonzero mean?

This is a first-order-real-valued autoregressive (AR) process $y(n)$ that satisfies the real-valued difference equation $y(n)+a_1y(n-1)=x(n)$ where $a_1$ is a constant and x(n) is a white noise ...
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0answers
28 views

How to apply the chain rule for partial derivatives to transformations?

I'm currently working to solve the Black-Scholes model partial differential equation (it's a model for a.o. stock option prices). The Black-Scholes equation for a calloption C(S,t) is given by $ ...
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25 views

probability integral transformation and distribution of P= P[ |T| <= |t|] .

The task is to find the distribution of P. where , P=P[ |T| <= |t|]. (T is a continuous random variable with PDF f(t)). now , I tried to make the following two arguments : 1.P= P[ |T| <= |t|] ...
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46 views

Is there an interesting interpretation of the ROWS of an affine transform matrix?

Context: I have a question about affine transform matrices in 3-space. Matrices are 4x4, with the right-most column being the translation, and the bottom row being [0,0,0,1]. In discussions I read ...
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46 views

Prove that the continuous $f: \mathbb C \to \mathbb R$ has a global max and min

I am having this continuous transformation $f: \mathbb C \to \mathbb R$ and $\ f\ (\mathbb C)$ is bounded Now I have to prove that there are a global maximum and a global minimum. My thoughts: I ...
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36 views

Transforming 1D Burger's Equation into infinitely many coupled ODE's

I've been working on the following problem but I can't justify my steps, would a savvy mathematician kindly tell me what, if any, violations I've made. Problem: Show Burger's equation can be written ...