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.
Hi,
\phi(x) is an interpolating scaling function (also called fundamental function or Dubuc-Deslauriers function) as given on pages 155 to 158 in these lecture notes: http://pages.unibas.ch/comphys/comphys/TEACH/WS07/course.pdf
Why does the follwoing yield:
\int_{-\infty}^{\infty}\phi(x) dx...
Homework Statement
I need to show that if X ~ r(a1,B) Y ~ r(a2,b) where r means gamma distribution then if X and Y are independent
i) X+Y ~ r(a1+a2,B)
ii) cX ~ r(a1,cB)
Homework Equations
The Attempt at a Solution
i) i use the mgfs of x and y and ended up with mgf(x+y) =...
Hi all
According to the textbook Signal and Systems by Oppenheim (2nd edition) pages 685 and 686, if the Laplace transform of x(t) is X(s) with ROC (region of convergence) R, then the Laplace transform of x(at) is (1/|a|)X(s/a) with ROC R/a.
Consequently, for a>1, there is a compression...
Homework Statement
Prove that
\displaystyle \int_{-\infty}^{\infty} \delta (at - t_0) \ dt = \frac{1}{ | a |} \int_{-\infty}^{\infty} \delta (t - \frac{t_0}{a}) \ dt
For some constant a.
The Attempt at a Solution
Edit: Looking at this again, I really don't understand where this is coming...
Hello and thank you for taking the time to read this.
I am making a number generator that generates a number based on a pareto distribution.
The problem is, the distribution essentially goes from 0 to infinity. How would I go about scaling the values so I get a range between 0 and 1...
Hi, I developed Matlab code to solve the diffusion equation using the central difference equation, with an added term at the end. The equation is the following:
dS/dt=Ds*d^2S/dx^2-(Vmax*S/Km+S)
In my code, the length of the space domain is very small, 1E-6. I would like to scale...
Im writing a program in python to simulate the propagation of a gaussian beam through a thick lens and to the focussing point using Fourier optics.
Due to the strength of the focussing I need a lot of data points so that I have a decent resolution at the focus. To speed things up and to...
The standard definition of coordinates on Penrose diagrams seems to be something like \tan(u\pm v)=x\pm t. This is what Wikipedia gives, and Hawking and Ellis also give a transformation involving a tangent function, although I haven't checked whether the factors of 2, etc. agree. Neither source...
* corresponds to matrix product
I'm working on a method of visualising graphs, and that method uses eigenvector computations. For a certain square matrix K (the entries of which are result of C_transpose*C, therefore K is symmetric) I have to compute the eigenvectors.
Since C is mXn, where...
This paper
http://www.ece.mtu.edu/faculty/wfp/articles/j_comp_appl_math.pdf
says (end of item 2.) that it is possible to multiply the arguments to a Kummer M function by constants and later rescale them to get the correct result. But how?
Homework Statement
Restate the vertical projectile problem in a properly scaled form. (suppose x<<R).
d2x/dt^2=-g(R^2)/(x+R)^2
Initial conditions: x(0)=0, dx(0)/dt=VoFind the approximate solution accurate up to order O(1) and O(e), where r is a small dimensionless parameter. (i.e. the...
Homework Statement
Prove that \delta(at)=\frac{1}{abs(a)}\delta(t)
Hint: Show that \int\phi(t)\delta(at)dt=\frac{1}{abs(a)}\phi(0)
(the limits of integration are from -inf to +inf btw, I couldn't find how to put them in..)
Homework Equations
The Attempt at a Solution
Ok. I...
I'm trying to make a bar graph from data which were taken at irregular intervals over 100m. I want the x-axis to be scaled instead of showing each data point one after another. (i.e. I'm coming out with 1-4-5 when I want 1-----4-5). I hope this makes sense - I'm not sure of the proper...
Using the defining property of the dirac delta function,
\int{dx f(x) \delta(x-c)}
show that
\delta(ax)=\frac{1}{|a|}\delta(x)
I think all I need to do is make the right u substitutions and it will come out right, but I can't think of how to make the substitutions...after a long time working...
I am reading that diffusive scaling in one dimension means that "increasing the size of a cell by a factor of 20 increases average diffusion time by a factor of 400". I can't find anything on diffusive scaling. Can anyone give me an explanation of this?
Thanks
I am doing a project to build a scaled wave tank.
"This project involves the design, construction and testing of a wave tank.
When studying the hydrodynamics of a scale model or prototype, it is important to choose an appropriate scale.
To determine tank and wave dimensions it is necessary to...
Differential geometry (which includes general relativity) often introduces the length differential, expressed as ds2=gabdxadxb, to introduce the covariant form of the metric tensor gab. However, this formulation scales ds2 incorrectly. The appearance of an index as a superscript, as in dxa...
Homework Statement
A statue is to be 'scaled down.' It will have its size changed without changing its shape. It starts with an initial volume of 4.25 m^3 and ends up with a final volume of 0.250 m^3.
If the height of the original statue was 215 cm, calculate the height of the smaller...
Let's say I have a vector x in \mathcal{R}^3. Let's also suppose that any vector x undergoes the transformation x' = kx (where k is a positive real).
Obviously, normalizing the vector will give us a quantity which is invariant to uniform scaling. In fact, \frac{\mathbf{x'}}{|\mathbf{x'}|} =...
Hi there,
I am plotting some data onto a hammer plot using ipy- matplotlib. The axes on the plot range from alpha= -pi:pi and beta=-pi/2:pi/2. I have a value (lets call it delta) for each point on the plot where (alpha=integer,beta=integer). I have arranged the delta values in ascending order...
One of the outstanding questions I have in physics relates to scaling relations.
Say you're presented with a problem like: Find X--which has units of y--given that the relevant (dimensional) quantities of this problem are A, B, C, and D. Then you construct a solution using these quantities...
Hi everybody,
I'm trying to reproduce daubechies basic building graph and daubechies wavelet function graph (φ(r)=0 if r≦0 or 3≦r). And i found this algorithm. I would appreciate if there is anybody could help me to understand the function defined below as function [s,w] = cascade(n,cs,cw)...
I have a question about Witten's original 1998 paper on AdS/CFT
http://arxiv.org/abs/hep-th/9802150
Since the AdS metric diverges at the boundary, the boundary metric is only defined up to a conformal class Eq. (2.2),
ds^2 \to d\widetilde{s}^2 = f^2 ds^2
Similarly, the solution for...
Hi, suppose i have various graphs which each have many nodes. In one graph the nodes x and y values may be within the range 0-1000, in another the x values may range from 100-500 and the y values from 300-800. Basically, the ranges always vary and there is no consistency.
I need a way to...
Homework Statement
Many undamped mechanical vibrations are described by the Differential Equation
x′′ + Kx = 0
that describes a position x(t) with t being time and K a positive constant.
a) What are the MKS units for K?
b) Introduce s=t/T for some characteristic...
Homework Statement
I am analyzing power spectrum of the series obtained using different approaches. I have 100 points series. First method is obtaining power spectrum by taking squared FT of series and divided by the period. Second method is taking IFT from ACVF of the series. In both cases...
Hi all!
I am just wondering why scaling in DIS is approximately realized for the middle (Bjorken-) x-values and not e.g. for the high or low x-values? Is there any depicitive reason?
Thanks for the ideas!
Blue2script
Hi
I understand most of the steps in the determination of the time scale. But i don't really understand the step in equation 6.96.
The first attachment is the full details of the time scale, and the second attachment is the part which I am stuck on.
I just want to know, how they get...
How does Wien's scaling law
\frac{u(\lambda)}{T^5} = \frac{f(\lambda T)}{\lambda^5T^5}
imply that if \frac{u(\lambda)}{T^5} is plotted as a function of \lambda T, all experimental data will lie on a single curve?
Geometrically, matrix multiplication of an nxn matrix is the scaling, and rotation of a vector in n dimensions true? So when you find the inverse of a matrix, what you're actually doing is finding a transformation such that in the 'transformed space' the vector is a unit vector.
If the inverse...
I don't understand where to even start with this problem. This book has ZERO examples. I would appreciate some help.
Show that by a suitable scaling of the space coordinates, the heat equation
u_{t}=\kappa\left(u_{xx}+u_{yy}+u_{zz}\right)
can be reduced to the standard form
v_{t} = \Delta...
Homework Statement
Derive the scaling law for static buoyancy force of a solid sphere in a liquid with a density of γ.
Assume the sphere is made of a material with a density of γs (γs< γ).
Homework Equations
The Attempt at a Solution
I calculated the the buoyancy force of the...
Homework Statement
a system has the following property:
Its response to a sum of inputs is the sum of its responses to the inputs.
(a) Prove that the scaling property holds for any integer scaling factor
(b) Prove that the scaling property holds for any rational scaling factor...
Does anyone know of any particularly good texts on scaling laws?
I'm doing some research into nanotechnology, and I'd like some valuable references.
If you're unfamiliar, scaling laws are just like they sound: mathematical relationships describing, as one application, mechanical concepts as...
Hi all.
To me, scaling means adopting properly scales in nondimensionalization.
However, as I see in the book "A modern introduction to the mathematical theory of water waves" by R.S.Johnson, the author distinguish the two processes in a way that confuses me much. (Sec. 1.3.1 and 1.3.2)
Can...
Hello All,
I'm doing some fun at home experiments with my 1:6 scale RC car and I
wanted do some camparisons with its full scale counter part.
Mainly how jumping the two off a ramp (Duke's of Hazzard style).
I wanted to know if you can help me with confirming some of my scaling...
i am taking a +/-5 voltage signal is supposed coming from a torque sensor. this is supposed to be equal to 100 N*m. However, -5.04 volts is actually equal to 100 N*m and 4.85 volts is actually equal to 100 N*m. I need to scale this signal to a maximum input of 200 mV for display on a panel...
Hi guys,
I'm a high school student at a private high school in the LA county called Bishop Amat Memorial. I'm currently in Honors Physics, so far I've been have a smooth and fine time until my teacher hit us with a project concerning scaling...
His philsophy of teaching for this project is...
Hi all!
I always come across "scaling" in various subjects.
Like in Fluid mechanics, I am confused about why there is a need to choose some characteristic scales and non-dimensionalize the differential equation?
I am confused about this...can anyone expain me the rationale behind this...
http://arxiv.org/abs/astro-ph/0606048
About universes with scale-related total masses and their abolition of presently outstanding cosmological problems
Authors: H.J. Fahr, M. Heyl
Comments: Submitted to AN. 7 pages
Cosmological consequences of a strictly valid total energy conservation...
http://arxiv.org/abs/astro-ph/0605488
Challenges for scaling cosmologies
Authors: Luca Amendola, Miguel Quartin, Shinji Tsujikawa, Ioav Waga
Comments: 14 pages, 3 figures
A cosmological model that aims at solving the coincidence problem should show that dark energy and dark matter follow...
Hi all!
I am having problems with understanding the scaling process of the N-S equations in fluid dynamics.
From textbooks, I see that each quantity say velocity, time, length...etc are all divided some some reference values in order to obtain some dimensionaless quantity V*, t*, p*, g* etc...
A particle of mass M moves along a straight line with initial speed v_i. A force of magnitude Fpushes the particle a distance D along the direction of its motion.
assume that the particle's mass is increased to 3M.
B.)By what multiplicative factor R_k does the initial kinetic energy...
Hi all,
I am not sure if I'm posting on the right place. I am currently working on a computational project. It's about simulating a system with two non-test masses and a bunch of test masses. The instruction sheet says that we should "scale the problem carefully by setting the units such that...
physics 40s scaling and proportions ...please help
the physics of a wire supporting a sphere...
A wire of .2cm in diameter is just strong enough to support a steel sphere. What must the diameter of the wire be if the following changes are made to the sphere?
A. new sphere has 4x the...
I am having dificulty with this homework problem. It doesn't seem to give me enough information. I am asked to find the volume and surface area of an object of unknown volume and shape. Here is the problem:
A sculptor builds a model for a statue of a terrapin to replace Testudo (go UMD!)...
I am presently taking my first course in signals and systems and I have been charged with proving the scaling property of the impulse function; that:
delta(a(t-to)) = 1/abs(a)*delta(t-to)
I am seriously miffed and need some help.
Hi to all,
my problem is the following:
I have different different quality metrics which are able to assess image quality by investigating several parameters in an image.
The problem with those metrics is their scale. Some of them range from 1 to 10 (10-highest quality) some of them from 0...
In critical phenomena, we can enlarge the block size(momenta fluctuation) by Kadanoff transformation, say
k \rightarrow bk (b<=1) , and scale the new Hamiltonian by k' = k/b, x'=bx to recover to the original block size.
In QFT, similarly integrating out the high momenta produces the effective...