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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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How to find the angle required to rotate a parent frame such that one of the axes of its child frame points to a particular direction? [closed]

My problem is exactly similar to the one discussed here. I want to calculate the the angle required to rotate a parent frame such that a particular axis of the child frame points in the direction of a ...
user22930's user avatar
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What symbol should I use for non linear proportional to?

Until now, if I had a function $f$ and a monotonic transformation of it $g$, I would write $f \propto g$, however, a colleague of mine pointed out that it's usually considered "proportional to&...
Alberto Sinigaglia's user avatar
3 votes
2 answers
117 views

Precisely sketch the mapped region and write the equations of its boundaries.

Find the formula for the fractional linear transformation $f$ that maps the points $0$, $-i$, and $2i$ respectively to the points $-1$, $0$, and $3$. Determine and write where the mapping $f$ maps the ...
math123's user avatar
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4 votes
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Write the conformal mapping that maps the region and sketch both regions as well.

Write the conformal mapping that maps the region $$ D = \left\{ z \in \mathbb{C} \mid |z| > 1 \land 0 \leq \arg(z) < \frac{\pi}{3} \right\} $$ to $$ D' = \left\{ z \in \mathbb{C} \mid |z - 2i| &...
math123's user avatar
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1 answer
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Calculating $\int_{-\infty}^{\infty}e^{ixt}\cos(at)e^{-\frac{t^2}{2}}\,dt$ and $\int_{-\infty}^{\infty}\cos(at)e^{-\frac{t^2}{2}}\,dt$ [closed]

Problem Statement: Derive the formulas: $$\begin{align*} \mathcal{F}\{\cos(at)f(t)\} &= \frac{1}{2} \left( F(x + a) + F(x - a) \right)\\ \ \mathcal{F}\{\sin(at)f(t)\} &= \frac{1}{2i} \left( F(...
user1718's user avatar
3 votes
1 answer
124 views

For any functions $f(t)$, $g(t)$, and their Fourier transforms $F(x)$, $G(x)$, the following relation holds

For any functions $f(t)$, $g(t)$, and their Fourier transforms $F(x)$, $G(x)$, the following relation holds: $$ \int_{-\infty}^{\infty} f(t) G(t) \, dt = \int_{-\infty}^{\infty} F(x) g(x) \, dx. $$ ...
lolip123's user avatar
2 votes
2 answers
82 views

Let $F(z)$ be the Laplace transform of the function $f(t)$. Derive two formulas

Let $F(z)$ be the Laplace transform of the function $f(t)$. Derive the formulas: $$ \mathcal{L} \left\{ \int_0^t f(u) \, du \right\} = \frac{F(z)}{z} $$ and $$ \mathcal{L} \left\{ \frac{f(t)}{t} \...
Markus's user avatar
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1 answer
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Coordinate transformations of a scalar field

I'm very confused about coordinate transformations and the statement that scalar fields remain unchanged under coordinate transformations. Consider a coordinate transformation $$ x \rightarrow x' $$ ...
bennnn's user avatar
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Hough transformation: get line function from Hough parameters (rho, theta)

I have two an input image (256px x 256px) with a line and Hough space image (256px x 256px) with a mark where the corresponding ...
Schelmuffsky's user avatar
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How to combine the $4$-dimensions of spacetime into 1 dimension?

I have been thinking about the possibility of representing all points in a $4$-dimensional spacetime coordinate system $\mathbb{R}^{1,4}$, as points on one line $P$ (or axis of a $1$-dimensional ...
A.M.M Elsayed 马克's user avatar
2 votes
0 answers
73 views

Rotation angle between two parallel transport (Bishop) frames

I am conducting academic research on steerable needles with multiple sections and I have run into a roadblock. Each needle is composed of two or more sections connected in series and each section can ...
user1346036's user avatar
4 votes
3 answers
260 views

How to define a line in 3d space in terms of $\rho, \theta$, and $\phi$?

I have been struggling recently to come up with a solution for parameterizing a line to be swept by a vector in spherical coordinates. More specifically, imagine a line segment defined in the form $\...
anonymous user's user avatar
2 votes
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Volume preserving transformation in the Circular Restricted Three-Body problem

the Lagrangian of the planar circular restricted three-body problem in the rotating coordinate frame is: $\mathcal{L}(x,y,\dot{x},\dot{y})=\frac{1}{2}(\dot{x}-\Omega y)^2 + \frac{1}{2}(\dot{y}+\Omega ...
Hajarl's user avatar
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Decoupling Linearly Coupled Wave Equations

I'm currently working numerically with wave equations and I was wondering if one can always decouple two wave equations, with potentials, which are linearly coupled. The system I'm talking about is ...
Afraxad's user avatar
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Concave transformation and effect on function arguments

Consider $g(x)$ as an increasing concave function such that the following equation holds: $p.g(a) + (1 - p).g(b) = p.g(A - k1) + (1 - p).g(B - k1)$ where $a$, $b$, $A$, $B$ are non-negative real ...
eddie-v's user avatar
-1 votes
1 answer
24 views

Transformations of Function

Im having trouble conceptualizing why when we transform a function, we need to describe $x$ and $y$ as functions in the new coordinate system. For example with polar coordinates $x$ and $y$ are now ...
TreyarchPi's user avatar
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1 answer
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Transformation of multi-variate probability densities using reciprocal of inverse function jacobian

If $x$ and $y$ are $d$-dimensional vectors. We transform probability densities $P_x(x)$ and $P_y(y)$ according to $x = g(y)$. Transform is given by $P_y(y) = P_x(x)|\det \mathbf{J}| = P_x(g(y))|\det \...
xTom's user avatar
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1 answer
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Can anyone see a way to linearize this function for linear regression?

I have the following function: $$f(x) = \dfrac{a_1}{(x+b_1)^2+c_1} + \dfrac{a_2}{(x+b_2)^2+c_2}.$$ From multiple measurements of $f$ at known $x$ values I would like find the values of $a_1,a_2,b_1,...
HazCam's user avatar
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1 vote
0 answers
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Given a standard normal r.v. $ X $, how can I compute the transformation $ Y = \sqrt{|X|} $?

I am working through a problem which seems closely related to the Chi-square distribution, however keep getting stuck on the computation and intuition behind the task. So far, I have expressed the ...
vAlkanol's user avatar
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Intuition of Transformed Random Variable

Find the PDF of $Y=X^2$ given that the PDF of $X$ is $f(x)=2x,0<x<1$ Calculate the inverse and the derivative: $$y=x^2\implies x=v(y)=\sqrt{y}\implies v'(y)=\frac{1}{2\sqrt{y}}$$ Applying the ...
Starlight's user avatar
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1 vote
0 answers
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Trouble on transforming a PDE into an ODE and solving it

I have encountered an issue in a PDE (A Green's function actually). I am solving it in (d+1)-dimensions and I use Poincare coordinates, meaning I have a dimension "z" and I also have d-...
Βασίλης Γερμανίδης's user avatar
0 votes
1 answer
67 views

If $X\sim N(\mu, \sigma^2)$ and $\Phi$ is the CDF of a standard Normal random variable, what is the distribution of $\Phi(X)$?

Let $\Phi$ be the cumulative distribution function of a standard Normal random variable, $Z\sim N(0,1)$. Let $X\sim N(\mu, \sigma^2)$ follow any Normal distribution. We know that $\Phi(Z)\sim \...
cgmil's user avatar
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2 votes
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Different Formulations of Differentials as Generating Transformations: $e^{t A}f(x) = h(x,t); A=\frac{\partial_x}{g'(x)} \& h(x,t)=f(g^{-1}(g(x)+t))$

For context and introduction please see: Perspective on Differentials Generating Transformations of Functions - with Fourier Transformations $e^{a\partial_x} f(x) = f(x+a)$ Here I used Fourier ...
theta_phi's user avatar
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2 votes
1 answer
42 views

Perspective on Differentials Generating Transformations of Functions - with Fourier Transformations $e^{a\partial_x} f(x) = f(x+a)$

An analytic function $f(x)$ can be transformed by the exponential of a differential operator. The most known and easiest example is $ e^{a \partial_x} f(x) = f(x+a) $ Generally this is shown by Taylor ...
theta_phi's user avatar
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1 answer
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transforming a polynomial function

I was exploring transforming polynomials (sorry if this is the wrong term). Essentially, I found a way to rewrite polynomials in different equivalent forms analogous to changing a quadratic from ...
James S.'s user avatar
2 votes
1 answer
21 views

Confused on the result of Sequence of Geometric Transformations

Here is the question. Consider the parent function $f(x)=\frac{1}{x}$. Now do the following sequence of transformations. $1.$ Shift up by $4$ units. $2.$ Shift left by $2$ units. $3.$ Vertically ...
LifeIsMath's user avatar
1 vote
2 answers
113 views

Increase and decrease length of 2 sides of a Triangle while keeping third side's length and opposite angle constant constant

I need to find a math formula that will help me with this kind of problem. I need to use it on my game's player controller which will act as a flying/hover line on surface. Basically, what i want is ...
Vasile Nebunu's user avatar
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1 answer
33 views

Change vector's basis from 3d spherical to cartesian coordinates [duplicate]

I am writing a simulation program. I have a vector field in spherical coordinates which I need to transform into Cartesian coordinates. I understand how this works in $2D$ case - simple enough, I just ...
Maciej's user avatar
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0 votes
1 answer
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Orthonormal system build upon a Fourier Transform

I'm currently taking a Fourier Transform course and I have to solve the next problem: "Suppose you have a function $f$, continuous in $[0,1]$ and with compact support also $[0,1]$ (that is, $f(x)=...
Pedro Mateo piqueras's user avatar
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0 answers
22 views

Transform of a differential operator to get a constant coefficient equation and how to set up the characteristic equation?

Say I have a Sturm-Liouville ODE: $$\Big(x y(x)'\Big)' + \lambda y(x) = 0$$ And I want to solve this problem by subbing in: $x\frac{d}{dx} = \frac{d}{du}$ which makes the problem: $$\tag{1}\frac{d^2 ...
Researcher R's user avatar
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0 answers
31 views

Find the probability distribution density of a random process at the output of a nonlinear circuit

I know: Probability distribution density at the input of the circuit $$ W(x) = \frac{\mu}{2\left(1+\mu^2\left(x-m\right)^2\right)^{3/2}} $$ Non-linear function $$ f(x) = a_0 + a_1 \cdot x - x $$ ...
Антон's user avatar
0 votes
1 answer
72 views

Help on transformation of boundary conditions

I was working the transformation in this paper A new algorithm for solving classical Blasius equation by Lei Wang The boundary value problem is He used the transformations $$y=f''(\eta),x=f'(\eta)$$ ...
Mohamed Mostafa's user avatar
5 votes
0 answers
130 views

Existence of a postive measurable set such that $T^{-k}(E)\cap E=\emptyset$ for a particular $k\ge 1.$

Let $(X,\mathcal B,\mu)$ be a atomless probability measure space and $T:X\to X$ be a non-singular transformation such that $\mu\left(\{x\in X: T^n(x)=x\}\right)=0$ for every $n\ge 1.$ Let $A\in \...
abcdmath's user avatar
  • 2,007
2 votes
1 answer
105 views

Analytical transformation or an effective numerical method for calculating $\sum_{n=1}^{\infty}{K_0(\frac{n}{a})\sin(n)}$ series

I have a series for that I need to get a fast calculation: $$\sum_{n=1}^{\infty}K_0\!\left(\frac{n}{a}\right)\sin(n)$$ where $K_0$ is the $0^{\text{th}}$ modified Bessel functions of the second kind, ...
gearquicker's user avatar
0 votes
0 answers
12 views

Transformation and image of a plane under a given transformation when one of the variables is constant.

Consider the transformation given by: \begin{cases} x = \frac{\sin(\sigma)\cos(\phi)}{\cosh(\tau) - \cos(\sigma)} \\ y = \frac{\sin(\sigma)\sin(\phi)}{\cosh(\tau) - \cos(\sigma)} \\ z = \frac{\sinh(\...
Angelo's user avatar
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0 votes
1 answer
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Graphing a function which is a product or sum of 2 functions?

Background: I was solving the last exercise of a chapter from S.L. Loney's Trigonometry book, where the exercise was to graph functions like sin(x) + cos(x). Now, here we are supposed to manipulate ...
Ayush Singh's user avatar
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0 answers
21 views

Decomposition of many particle coordinate system into translation, rotation, and vibration

Consider the space $\left(\mathbb{R}^3\right)^N$ of configurations of $N$ three-dimensional points. Given for each point a "mass" $m_n$ and a reference configuration $x^o_n$, is it always ...
creillyucla's user avatar
1 vote
1 answer
65 views

Is it possible to take the RGB-space and reverse the direction of the grayscale axis while "preserving" the rest of the cube? If so, how?

In Python, I have been attempting to generate a slightly different-from-normal RGB cube with the only difference being that the grayscale is reversed, just for fun. My code generates the RGB cube and ...
DarthEwok07's user avatar
5 votes
1 answer
146 views

Existence of canonical form for cubic and quartic form?

I am a post graduate student who is currently studying some optimization for quadratic form. From the class lecture, I know that we can always turn any quadratic functions into theirs corresponding ...
Tuong Nguyen Minh's user avatar
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0 answers
44 views

Optimal orthogonal transformation - the naive way

I am solving the following problem: For given position vectors $\vec{x}_1,...,\vec{x}_n$ and $\vec{y}_1,...,\vec{y}_n$ we have following equations $Q\vec{x}_i+\vec{b}=\vec{y}_i$, $i=1,...,n$ $Q^TQ=I$ ...
Matjaž Pogačnik's user avatar
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0 answers
16 views

How to approximate a sigmoid function, given the two points on x which correspond to where the curve stops at its y min and y max [duplicate]

Given a sigmoid equation L / ( 1 + e ^ (-kx - m)) How can I determine "k" and "m" given inputs "x1" and "x2" (see picture)? ...
Sheldonfrith's user avatar
2 votes
1 answer
98 views

If seven vertices of a hexahedron lie on a sphere, then so does the eighth vertex.

I'm trying to prove https://imomath.com/index.cgi?page=inversion (Problem 11) by projective geometry: If seven vertices of a (quadrilaterally-faced) hexahedron lie on a sphere, then so does the ...
auntyellow's user avatar
1 vote
1 answer
69 views

A domain-covariant notation for functions?

Note: I'm using the terms "covariant" and "contravariant" a bit loosely in this question. The standard function notation seems to be naturally codomain-covariant and domain-...
KKZiomek's user avatar
  • 3,875
0 votes
2 answers
46 views

From $z = \dot{\Theta}^2 \operatorname{sgn}(\dot{\Theta})$ to $\dot{\Theta} = \operatorname{sgn}(z) \sqrt{\left| z \right|}$

I read on a scientific paper (*) the following equations: $$ z = \dot{\Theta}^2 \operatorname{sgn}(\dot{\Theta}) $$ and then: $$ \dot{\Theta} = \operatorname{sgn}(z) \sqrt{| z |} $$ Could you tell me ...
Federica Guidotti's user avatar
0 votes
0 answers
46 views

Applying a Vector Field Defined in Cylindrical Coordinates $(r, \phi, z)$ to Points in Cartesian Coordinates $(x, y, z)$

I have written a Python script that defines a meshgrid. The meshgrid represents a 3D grid in Cartesian (x, y, z) coordinates, and for each point, I want to apply a function. After applying the ...
David's user avatar
  • 161
0 votes
0 answers
22 views

Pseudo-Inverse of mapping with identical null-spaces

I have two functions $f_a(x) = y_a$ and $f_t(x) = y_t$, where the dimension of the domain is larger than range. Finally, I want to compute the linearised mapping from $y_t$ to $y_a$ via ${}^t J_a = \...
scleronomic's user avatar
1 vote
1 answer
29 views

Vertical Stretch transformation Question for rational function with variable in numerator and denominator. [closed]

I'm trying to help a student with following rational equation question: Describe the transformations of $$g(x) = \frac{-4x - 2}{7x +1}$$ from the graph of $$f(x) = \frac{1}{x}.$$ The given answers ...
DavidGslade86's user avatar
0 votes
0 answers
14 views

Ratio Scale transformation

I am in a pickle, and I would genuinely appreciate it if you could guide me. I spent 2 years to find a way to develop a ratio scale, and I did it; however, it cannot be used the way it is. The scale ...
Parham's user avatar
  • 1
0 votes
0 answers
36 views

Questions on change of variables

I came across the following change of variables to turn equation into an elliptic curve. (Source: https://www.youtube.com/watch?v=6eZQu120A80&t=3022s&ab_channel=ImperialCollegeLondon, the part ...
Mardia's user avatar
  • 325
0 votes
2 answers
70 views

Transform a vector of positive and negative values to sum up to 0 [closed]

Is there a transformation that produces a vector with sum $0$? There are positive and negative values and the transformation does not need to be preserve the weights. E.g.: $f(x_1, x_2, x_3) = (x_1', ...
Philohippo's user avatar

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