Questions tagged [transformation]

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), (rigid-transformations).

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Is there any method to deal with nonconvex objective function $\frac{f(x)}{a^Tx+b}$ where $f(x)$ is convex

Consider the following problem \begin{equation} \begin{aligned} \min_{x} & \quad \frac{x^T D x + d^Tx}{a^Tx+b}\\ s.t. & \quad Cx=0\\ & \quad 0 \leq x_i \leq 1, \forall i=1, 2, 3, \cdots, n ...
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Time transformation derivative of another transformation

I am not sure how to solve the derivate of a transformation with respect to another transformation. Se have: $$\Psi_k: q_k(t) \rightarrow \hat{q_k}(t)=q_k(t)+\epsilon \psi_k(q(t), \dot{q}(t),t)+O(\...
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Can we simplify objective function based on the property of optimal solution?

Consider the non-convex optimization problem \begin{equation} \begin{aligned} \max_{x} & \quad f(x)\\ s.t. & \quad 0 \leq x\leq 1 \end{aligned} \tag{1} \end{equation} where $f(x)$ is non-...
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114 views

Let $F:M\rightarrow N$ and $G:N\rightarrow P$ be surface transformation. Show that $(G\circ F)^{*}=F^{*}\circ G^{*}$

$Let F:M\rightarrow N$ and $G:N\rightarrow P$ be surface transformation. Show that $$(G\circ F)^{*}=F^{*}\circ G^{*}$$ Here is definition : Let $F:M\rightarrow N$ transformation of surfaces. $i$) If $...
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11 views

When is a mapping called transformation?

Fourier transform and Gelfand transform are isomorphisms, while Laplace transform is not. Does a transformation need to be injective or something?
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Homegraphy with Known Camera Pose

I want to know if there is a process where given camera at certain arbitrary pose, we could compute the how an image would appear there via Homegraphy? In the following picture if we place the camera ...
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1answer
56 views

Find an estimate for $\int_{\pi/2}^\pi \sin(x) dx$ using the Monte Carlo Simulation

I want to find an estimate for $$\int_{\pi/2}^\pi \sin(x) dx$$ I want to use the monte carlo simulation method. I've plotted the graph of $\sin(x)$ in the given interval. The total area $$\begin{align*...
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How to understand transformations on operators

This question is related, or maybe is better to say inspired, to/by this other one about quantum mechanics. From what I currently understand the action of a transformation $T$ on another matrix, or to ...
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How two definitions of Hankel transform compatible? [closed]

For suitable function $f:[0,\infty)\to \mathbb{C}$ some author define its Hankel transform of order $\alpha>-\frac{1}{2},$ by $$\mathcal{H}_\alpha f(R)=C_\alpha \int_0^\infty f(r)j_\alpha(Rr) r^{2\...
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Is there any initiuition to transforming a equation of pair of straight lines to the equation of circle?

We know that for the following lines to intersect the coordinate axes in concyclic points,the following condition must be met; $$ a_1a_2 = b_1b_2 $$ Where, $$ L_1 : a_1x+b_1y+c_1=0$$ $$ L_2 : a_2x+...
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52 views

Find PDF of Y=1/X²

$f(x)=1/(2θ) , -θ≤x≤θ$ then Pdf of $Y=1/X^2$. I have tried this question in cdf method as $P(Y≤y)= P(1/X^2≤y) = P(-1/√y≤X≤1/√y) = F(1/√y)-F(-1/√y)$. Therefore PDF of $Y= (-1/2y^{3/2})f(1/√...
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New equation of function rotated 180 degrees? [duplicate]

Consider the graph of If the equation was rotated 180 degrees around the point what would the new equation be? I tried to flip it using but I can't get the exact slope right at the point of ...
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Calculate the translation required to zoom in on a pivot point

I cannot think of a better way to explain this. I hope it makes sense. Mathematical question: Let the origin be $O=\left[0,0\right]$. Let the pivot point be $P=\left[P_x,P_y\right]$. Let the scale ...
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68 views

Hypergeometric transformations

Denoting $_2F_1(a,b;c;z)$ as the hypergeometric function we have $$ _2F_1(a,b;c;z)=\sum_{k=0}^{\infty} \frac{(a)_{k}(b)_{k}}{(c)_{k}} \frac{z^{k}}{k !},\quad |z|<1. $$ Euler's transformation is $$ {...
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Find a linear transformation $T\begin{pmatrix}-1\\-2\end{pmatrix}$

$T:\mathbb{R}^2 \Rightarrow\mathbb{R}^3, T\begin{pmatrix}1\\0\end{pmatrix}=\begin{pmatrix}-1\\4\\3\end{pmatrix}$ and $T\begin{pmatrix}0\\1\end{pmatrix}=\begin{pmatrix}4\\-1\\1\end{pmatrix}$. Find $T\...
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Transformation of a parabola to a circle in optimization problem

I was working with two optimization problems and have a feeling that they are dual of each other. First problem: we know that some convex orange set lies inside red unit circle and I want to find its ...
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Transform heat equation to add drift/transport term

Let $f \in C^{1,2}((0,\infty)\times \mathbb R)$ be a solution to the heat equation: $$ \partial_t f(t,x) =\partial_x^2f(t,x). $$ Given a constant $c\in \mathbb R$, is there a reasonable transformation ...
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Word for a transformation without an origin

Is there a precise term for a transformation which does not have an origin? For example, a translation of $\mathbb{R}^2$, or a deformation which is equally applied to each tile in a tiled plane, so ...
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Find the equations of circles passing through $(1, -1)$ touching the lines $4x+3y+5=0$ and $3x-4y-10=0$

Find the equations of circles passing through $(1,-1)$ touching the lines $4x+3y+5=0$ and $3x-4y-10=0$ The point of intersection of the lines is $(\frac25,-\frac{11}5)$ If we want this point of ...
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Definition of inversion in a circle

Let $C$ be a circle with the middle point $O$ and the radius $r$, we say that the points $P$ and $P'$ are inverse points with respect to $C$ if: $$|OP|·|OP'|=r^2$$ Can anyone tell what is the ...
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Transforming a complex function into $F(z) = u(x,y)+iv(x,y)$

I have the following function: \begin{equation*} F(z) = \frac{1}{2}\left(z+\frac{1}{z}\right)+ik\ln(z), \quad k \geq 0 \end{equation*} where $\ln$ is the main branch of the complex logarithm. And I ...
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coordinate transformation geometry and rotation sense

I have the situation depicted in the figure below. Horizontal and vertical axes are $(x,y)$ respectively. The axes $(x',y')$ are denoted by dotted lines and they are rotated by $-31^\circ$ and their ...
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How to estimate the transform (basis) that will maximally sparsify the original signal with a series of available observations of said signal?

*I am new to this, so please do let me know if my formatting, tags, etc. are not optimal. Say, I have a series of, or $j$ observations of a vector signal with the length of $n$, denoted by $X_{n\times ...
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How $y = mx+c $ satisfies Linear Transformation from $X $ to $Y$

I was watching 3b1b video on linear transformations. In that he said that linear transformations of matrices preserves origin from the original space to transformed space. Now I got to prove this fact ...
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Can I infer that if I'm told $f(x)=\ln x$ then $f(y)=\ln y$

I have a lengthy economics question, I am told that $f(x)=\ln x$ (the natural log of $x$) but I am also given the functions $f(y)$ to work with. Since $f(.)$ is a function which applies a ...
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28 views

Changing PDE with scaled and shifted Fourier transform

I have defined a transform ($p,q,c$ are constants) $$V_{p,q}(\omega,\tau,z) = \frac{1}{2\pi}\int e^{-ih(\tau - (p+q)z/c)}U_{p,q}(\omega,h,z)dh$$ so that $$U_{p,q}(\omega,h,z) = \int V_{p,q}(\omega,h,z)...
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175 views

Transformation (?) of continuous random variables

Background Suppose that we are using a simplified spherical model of the Earth's surface with latitude $u \in (-\frac {\pi} 2, \frac {\pi} 2)$ and longitude $v \in (-\pi, \pi)$; then (if the radius is ...
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62 views

Why is this exponentially distributed [duplicate]

At some point in my notes they essentially imply that when $u\sim \mathcal{U}(0, 1)$ then $-\log(u)\sim \text{ExponentialDistribution}(\lambda=1)$. Clearly this isn't true since by the integral ...
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Prove condition implies isometry on $S^2$

Let $(p,q)$ and $(p',q')$ be any two pairs of points of $S^2$ such that the distances satisfy $d(p, q) = d(p',q')$. Show that there is a spherical isometry $T: S^2 \longrightarrow S^2 $ such that $T(q)...
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Continous function transform equals zero, then function equals zero in $\mathbb{R}$

Im proving the following statement: Let $\alpha>0$, the gaussian weight $\omega(z)= \frac{\alpha}{\pi}e^{-\alpha \left | z \right |^2}$ is the unique continuous, radial weight verifying $\int_\...
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63 views

Linear Transformations from $\mathbb R$ to $\mathbb R$

Is my iff statement correct? $f:\mathbb R\to\mathbb R$ is a Linear Transformation iff there exists a unique $a\in\mathbb R$ such that for all $x\in\mathbb R$, $f(x)=xf(a)$ So if I am given any linear ...
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Help understanding notation in matrix transformation

In my notes it's written that, for two basis $B={e_i}$ and $B'=e'_i$, the transformation rule is: $e'_j = c^i{}_j e_i$ This I can understand, every vector of the $B'$ basis is a linear combination of ...
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25 views

Line segment multiplication

I was just reviewing the inverse curve and there is a condition, the two lines $OP$ and $OQ$ should multiply to the radius $k$ squared: $OP\cdot OQ = k^{2}$. By calculus I found the magnitude of each ...
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How do I deduce the range of variable which endpoints of it are infinite values of change of variable?

$x_{1}[k]$ is a function with an argument $k$ $$f(z)=\sum_{k=-\infty }^{+\infty} \left( x_{1}[k] \left( \sum_{n=-\infty }^{+\infty} \frac{x_{2}[n-k]}{z ^{n} } \right) \right)$$ $$p:=n-k$$ $$n:-\infty\...
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1answer
27 views

Finding the transformation matrix that belongs to the linear map

I am given a linear map with: $$ \begin{align} f(1,1,0) &= (3,7,1) \\ f(1,0,1) &= (3,4,2) \\ f(0,2,1) &= (-1,2,1) \end{align} $$ and I have to find a $3 \times 3$ matrix that belongs to ...
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1answer
40 views

Find projective transformation

Given four projective lines $L_1,L_2,L_3,L_4$ in projective plane, such that no three lines intersect in the same point and another four lines $M_1,M_2,M_3,M_4$ such that any three also do not ...
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22 views

Is there some systematic way to find a transformation satisfying this equation?

I am looking for a way to systematically find a solution $T(\vec{x})$ that satisfies the relationship $$ |J_T(\vec{x})|\left[2T(\vec{x})-\vec{1}\right]=\begin{bmatrix}x_1+2x_2x_3-x_2-x_3\\x_2+2x_1x_3-...
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1answer
25 views

Transformation-matrix for square to circle?

I'm striving for a general explanation of integral transformation. So far I've been told some variable substitutions (like polar coordinates) without really getting the gist of it. However I've just ...
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40 views

'simplify' integration by transformation: Parallelogram

I have to calculate the volume (?) of the function $f(x,y) = 3\,x-2\,y$ through the area that is spanned by the points $(1,0)\quad (0,1) \quad (-1,0) \quad (0,-1)$ A simple parallelogram. I'd solve it ...
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33 views

Transforming function while maintaining constant definite integral

Suppose we have the function $f(x) = x$. The definite integral of this function from 1 to a is: $$ \int_1^a x dx = \frac{a^2-1}{2} $$ Now suppose we want to make $f(x)$ nonlinear, so it may take the ...
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25 views

Derive the distribution & Construct a Confidence Interval

For the pdf $f(x,\theta)=\frac{1}{2\theta}e^\frac{-|x|}{\theta}, -\infty<x<\infty, \theta>0$ derive $Y_i=|X_i|$. This is what I've done: $Y_i=|X_i| => X_i = \pm Y_i$ Then |J| = 1 where J ...
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1answer
33 views

Convert function to convex on infinite range, knowing gradient and hessian

Let's take a function $f=sech(x)$ as an example. It is strongly convex in a limited range of $x$. It is further assumed that: $f$ is a "black box", $x$ as input ; output is the value of the ...
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Can any 3D transformation be represented by a composition of one translation, one rotation and one scale matrix?

Can any 3D transformation be represented by a 4x4 matrix $M=TSR$ where $T$ is a translation Matrix, $S$ is a scale matrix and $R$ is a rotation matrix ?
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Contour plot: Maintaining direction, inverting magnitude?

This question is motivated by a specific application, and might require some introduction. I want to create the illusion of flow by animating a contour plot, similar to the example here. This illusion ...
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1answer
50 views

How to find a unitary (transformation) matrix between two given vectors?

Suppose that $\vec{b}=U\vec{a}$ for two normalized vectors $\vec{a}$ and $\vec{b}$. When the two vectors are known, can we find a unitary matrix $U$ that satisfies the transformation $\vec{b}=U\vec{a}$...
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34 views

Find the values to make the transformation one to one.

Find the preimage of set $\Omega$ given by a transformation from cartesian $(XYZ)$ to canonical cylindrical coordinates $(R \Theta Z)$ and also find the values of $r$, $\theta$ and $z$ such that this ...
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23 views

Literature on co- and contravariant formulation

I am currently working on the topic of co- and contravariant formulation in physics. Unfortunately, my literature uses the topic very superficially. Is there any mathematical book with definition, ...
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proving an equation is invariant under a given transformation

I'm trying to prepare for an upcoming test and I was going through my textbook and came across the the question below. I've tried for a while but was unable to solve it. Any tips/hints would be ...
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33 views

distance of marginal distribution vs distance of distribution

I have two random raviable $\xi ,\eta \in \mathbb R^d$ and their pdf are $p_{\xi}(x),p_\eta(x): \mathbb R^d \to \mathbb R^+$. $\xi$ and $\eta$ satisty that $\forall \alpha \in \mathbb R^d,\|\alpha \|...
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28 views

transformation formula in English

I've tried several notations like "integral transform" and "coordinate transformation", but I don't happen to find the following theorem, easily known as "Transformationssatz&...

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