In Euclidean geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions. The result of uniform scaling is similar (in the geometric sense) to the original. A scale factor of 1 is normally allowed, so that congruent shapes are also classed as similar. Uniform scaling happens, for example, when enlarging or reducing a photograph, or when creating a scale model of a building, car, airplane, etc.
More general is scaling with a separate scale factor for each axis direction. Non-uniform scaling (anisotropic scaling) is obtained when at least one of the scaling factors is different from the others; a special case is directional scaling or stretching (in one direction). Non-uniform scaling changes the shape of the object; e.g. a square may change into a rectangle, or into a parallelogram if the sides of the square are not parallel to the scaling axes (the angles between lines parallel to the axes are preserved, but not all angles). It occurs, for example, when a faraway billboard is viewed from an oblique angle, or when the shadow of a flat object falls on a surface that is not parallel to it.
When the scale factor is larger than 1, (uniform or non-uniform) scaling is sometimes also called dilation or enlargement. When the scale factor is a positive number smaller than 1, scaling is sometimes also called contraction.
In the most general sense, a scaling includes the case in which the directions of scaling are not perpendicular. It also includes the case in which one or more scale factors are equal to zero (projection), and the case of one or more negative scale factors (a directional scaling by -1 is equivalent to a reflection).
Scaling is a linear transformation, and a special case of homothetic transformation. In most cases, the homothetic transformations are non-linear transformations.
Hello,
In studying linear regression more deeply, I learned that scaling play an important role in multiple ways:
a) the range of the independent variables ##X## affects the values of the regression coefficients. For example, a predictor variable ##X## with a large range typically get assigned...
Hello guys, I have conducted an experiment and got some results.
I have 3 variables to vary, for example, five x1, five x2, and two x3
and 2 observation results, like y1, y2
I already make y1 y2 and x1 x2 x3 dimensionless
since plot is 2D, what I am doing now is just plot when x3=1, x2=1, plot...
It is given that a theory is invariant under the length scaling:\begin{align*}
x &\rightarrow \lambda x \\
\phi(x) &\rightarrow \lambda^{-D} \phi(\lambda^{-1} x)
\end{align*}for some ##D## to be determined. The action of a real scalar field is here:\begin{align*}
S = \int d^4 x...
Hi,
I have some measurements for pressure drop of Helium at room temperature and I would like to scale it to other temperatures. Taking into account that, i) the flow is turbulent, ii) the pressure drop, ##\Delta p##, happens always in the same piping and iii) there is only variation on the...
as you see from %[scale=0.8]
I tried to scale the picture but it didn't after removing the %
\begin{tikzpicture}%[scale=0.8]
[declare function = {
tilde_y(\q) = sign(\q)*sqrt(abs(\q) / (1 - (abs(\q)-3)^3));},
pics/coordinates/.style args={(#1,#2),(#3,#4)}{ code={
%\draw[help lines] (#1,#2) grid...
Attempting this question without any guidance from my professors unfortunately as they did not teach this bit. Searched online and also there aren't many questions like this.
From what I know,
(I) Having n-1 means you should shift right by 1, which means x[0] is now equals to 0? So x[n-1] = [0 5...
Hello everyone,
When working with variables in a data set to find the appropriate statistical model (linear, nonlinear regression, etc.), the variables can have different range, standard deviation, mean, etc.
Should all the input variables be always standardized and scaled before the analysis...
I'm not familiar with the concept of dimensional scaling at all. Anyone please help me with this problem.
I need to understand how they obtained eq(16) from eq(15) in the paper here so I can do the same when I change Q0 from a constant to a variable function in time. Summary as below:
Scaling...
I have been reading Manton & Sutcliffe for some time now and can't quite wrap my head around something.
If you take the Hopf invariant N of a topological soliton ϕ then its Skyrme-Faddeev energy (which I hope I've gotten right up to some constants)
E=∫∂iϕ⋅∂iϕ+(∂iϕ×∂jϕ)⋅(∂iϕ×∂jϕ) d3x
satisfies...
The energy spectrum of a particle in 1D box is known to be
##E_n = \frac{h^2 n^2}{8mL^2}##,
with ##L## the width of the potential well. In 3D, the ground state energy of both cubic and spherical boxes is also proportional to the reciprocal square of the side length or diameter.
Does this...
We're working on a project that plots flux density of a light curve with respect to time. To do this, we had to scale data from different wavelengths so we had just the one variable for the flux. Essentially we took each value for flux density and multiplied it by three over the frequency raised...
Hi all,
For my studies I chose a course on scaling up and down of industrial processes (mostly focussed on the chemical industry), but for our project we (a group of students who knew almost nothing about nuclear reactors) chose to look if the approach (dimensional analysis) can be applied to...
When considering the forward FFT of a mathematical function sampled at times ##t = 0, \Delta, \ldots, (N-1) \Delta##, following the usual convention, we have something like
$$
H(f) = \int_{-\infty}^{+\infty} h(t) e^{-2 \pi i f t} dt \quad \Rightarrow \quad H_k = \sum_{n=0}^{N-1} h_n e^{-2 \pi i...
In the picture, there is a problem where the t is in units of square root(l/g), and V in square root(gl)
I am wondering
1. What it means when time is in units other than time? Does it mean that when solving I have to take time/squareroot(l/g)
2. How did they get square root(l/g).
Thank you...
Hello! I just discovered (maybe a bit late) that most fitting programs (Python lmfit or scipy, for example) have a parameter (by default turned on) that allows a scaling of the covariance matrix for calculating the errors (usually called scale_covar or something similar). After some reading I...
Hi everybody,
After watching the first lecture by Walter Lewin from MIT, I'm finding hard to follow the part in what he talks about scaling arguments. I've been watching around the Internet for resources to get my head around it but I couldn't find much or maybe I'm not using the right...
This is a pretty simple question, I am just trying to clear up confusion. Let ##D## be the rectangle in the plane with vertices ##(-1,0),(-1,1),(1,1),(1,0)##. Let ##\lambda >0##. Then what exactly does the set ##\lambda D## look like? Is it correct to say that, for example, ##2D## is the...
A=[ 8147 6324 9575 9572
9058 0975 9649 4854
1270 2785 1576 8003
9134 5469 9706 1419];
D=diag([T^(-3)*L^2 T^(-3)*L^2 T^(-1)*L T^0*L^0]);
I have matrix A whose first two columns are of the units T^3/L^2, third columns unit is T/L, and the last...
Hey! :o
Let $S_i$, $i=1,2$ the scaling with scaling factor $r_i$ and center $Z_i$.
Let $r_1r_2\neq 1$. I want to show that $S_2\circ S_1$ is a scaling and to calculate the center and the scaling factor.
When $Z_1\neq Z_2$ I want to show that $Z\in Z_1Z_2$. I don't really have an idea. Could...
The essential question is can this tech be scaled up in the vacuum of space to add a substantial tangential acceleration to the orbital velocity vector of a piece of debris such as a speck of paint 1cm^2 in cross section. I am looking to deflect that speck of paint into a container or out of...
Lets assume we are mapping one face of earth. we place a plane touching the Earth at 0 lattitude and 0 longitude. Now we take the plane of projection. suppose that we expand the projection unevenly. The small projectional area of a certain lattitude and longitude is expanded by a factor which is...
Hi guys.
We're on a new topic in math now which has to do with ratios and proportions but with scaling.
A few questions I have are
How can I express the following scales in ratio?
1/2" = 1"
1/8" = 1'0"
Also we're doing a Job and Drawing scale
A dimension on a job is 24 in. Using scale...
Homework Statement
If we multiply all the design dimensions of an object by a scaling factor (f), its volume and mass will be multiplied by f^3. a) By what factor will its moment of inertia be multiplied? b) IF a 1/48 scale model has rotational kinetic energy of 2.5 Joules, what will be the...
I am given the solution to the first part of the problem, however not the second part - would appreciate for someone to double check my work! Cheers.
1. Homework Statement
If a scale model of the solar system is made using materials of the same respective average density as the sun and...
I am aware of only two fields where the renormalization (sub)group ideas can be systematically and
unambiguously applied: particle physics and equilibrium critical behaviour.
1.- Are there any others?
2.- What are these ideas used for in fluid mechanics?
3.- When cosmologists speak about...
In my fictional universe, I have several reaches and transformations and whatnot. Each power level is determined by the average power of a human.
Humans are usually in a power scale of 1-5.
This means that someone can be 5 times the strength of someone at maximum.
Now. I need to figure out...
Hello guys,
I have to code Jacobian Free version of GMRES with scaling and reordering algorithms separately. But I have serious problems about the convergence of inner gmres iterations and I have doubts on my formulation about jacobian-vector product for scaled equations since its bookkeeping...
See the title. I'm not sure that this is the right place to post this question, but I'm not sure it fits any better on any of the other boards.
Let's say you have a phase transition. The correlation length will scale as:
ξ = |TC-T|ν
This diverges on both sizes of the phase transition. Now...
Hey everyone, I understand how to normalize a second order system, but I wanted to know if the same steps are taken when the parameters of the system are not scalar but matrices. For example
where the parameter phi, and gamma are both 3x3 matrices and X is a 3x1 vector.
From what I've see...
Hello Physics Forums!
I'm not an engineer/scientist but I work with antennas and I'm having trouble deciphering the above antenna radiation pattern diagram.
I am used to seeing radiation patterns expressed in polar plots. Usually the scale is set to dBi (where the isotropic antenna = 0dBi)...
This question is slightly related to my other plane, but I didn't want to post in the same thread as I didn't want the other one to get derailed.
If I have blueprints for an RC plane, and I scale down everything by a ratio (let's say 1/2), would it still work just as well? Or would it follow...
Homework Statement
Consider two solid dielectric spheres of radius ##a## separated by a dis-
tance ##R## (##R\gg a##).
One of the spheres has a charge ##q## and the other is
neutral. We scale up the linear dimensions of the
system by a factor of two. How much charge should reside on the first...
Given a Positive Definite Matrix ## A \in {\mathbb{R}}^{2 \times 2} ## given by:
$$ A = \begin{bmatrix}
{A}_{11} & {A}_{12} \\
{A}_{12} & {A}_{22}
\end{bmatrix} $$
And a Matrix ## B ## Given by:
$$ B = \begin{bmatrix}
\frac{1}{\sqrt{{A}_{11}}} & 0 \\
0 & \frac{1}{\sqrt{{A}_{22}}}...
I am multiplying a lognormal distribution by an function to scale it larger. While I know that scaling a lognormal distribution by a constant multiplier yields a lognormal distribution, in this case the multiplier is not a constant. Instead, smaller values from the lognormal distribution are...
Neil Turok, Director of the Perimeter Institute of Theoretical Physics in Ontario, Canada suggests scaling invariance is a fundamental property of nature, including spacetime. that nature does not recognize any kind of scale, including Planck scale.
if true how would this affect the leading...
Good evening people of PF! I have recently encountered a problem from Himmelblau's Basic Principles and Calculations in Chem. E. which asks to set up an energy balance for a tank, and then non-dimensionalize the differential equation before solving it. It's not the most complex task, but it's...
Homework Statement
If the size of the nucleus ( in the range of 10-15m to 10-14m) is scaled up to the tip of a sharp pin, what roughly is the size of an atom ? Assume the tip of the pin to be in the range 10-5m to 10-4m
Homework EquationsThe Attempt at a Solution
It's scaled up by a...
I guess this is just a maths problem about algebra. I'm learning to solve Schrodinger equation numerically, and right now I'm just dealing with the simplest examples like harmonic potential, square well, etc. The problem is that sometimes my program gives some strange results and I suspect it is...
Hi all,
I have an FEM model that I am doing a modal analysis of. I wanted to check that how I am computing the physical displacement is the correct way, as I've read a lot of about normalising modes, participation factors, effective masses, etc. and I'm not 100% sure on it.
I've got the...
For the distance function ##(\Delta s)^2 = (\Delta r)^2 + (r \Delta \theta)^2##, the rotation matrix is ##R(\theta) = \begin{pmatrix} cos\ \theta & - sin\ \theta \\ sin\ \theta & cos\ \theta \end{pmatrix}##.
That means that for the distance function ##(\Delta s)^2 = (\Delta r)^2 +...
Hi PF!
I'm doing some scaling over a PDE and I understand the math side of things but I do not understand the physical side of what we are finding.
For example, suppose we have some PDE, say 2-D continuity for it's simplicity ##u_x + v_y = 0##. Let ##L## be the length of a side of a flowing...
So according to the ideal voltage loss equation V = IR if I double I (per se) I will double the loss in voltage. This is a bit odd and I just want to make sure my intuitive explanation is correct.
I am assuming voltage loss is mainly due to some electrons bouncing off the resisting material and...
Hi!
I was wondering how I would go about solving a problem like this: (I have experience with up to multi-variable calculus and moderate level physics)
So I'm trying to see if it's plausible to create electromagnets, which fit into gloves and/or boots, that would allow a person to climb around...
http://advances.sciencemag.org/content/1/1/e1400066
Settlement scaling and increasing returns in an ancient society
Scott G. Ortman, Andrew H. F. Cabaniss, Jennie O. Sturm, Luís M. A. Bettencourt
Science Advances 01 Feb 2015:
Vol. 1 no. 1 e1400066
DOI: 10.1126/sciadv.1400066
The basic...
Homework Statement
[/B]
b) For a Form factor of form ##\theta_{(1-r)}## and ##\frac{1}{1 + e^{\frac{r-R}{a}}}##, how will these change when ##r \rightarrow 2r##?
c) How would one accelerate and observe scattered protons?
Homework EquationsThe Attempt at a Solution
Part(b)
[/B]
Rate of...
Consider the Hamiltonian ##H = - \frac{d^2}{dx^2}+gx^{2N}##.
Scaling out the coupling constant ##g##, the eigenvalues scale as ##\lambda \propto g^{\frac{2}{N+2}}##.
So, we can drop the g dependence and just consider the numerical value of the eigenvalues and the associated spectral functions...
Scaling - Inverse relationship between uncertainty and mass
I’m trying to express Heisenberg's Uncertainty Principle in a simplified formula that is not boundary unlimited and still capture what I believe is an inverse relationship between uncertainty and mass - the "scaling hypothesis".
I...