What is Homogeneity: Definition and 56 Discussions
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, size, weight, height, distribution, texture, language, income, disease, temperature, radioactivity, architectural design, etc.); one that is heterogeneous is distinctly nonuniform in one of these qualities.
I'm sorry that I have asked so many questions about this subject and the endless discussion that I caused. This was not my intention at all. There is nothing I'd love more than to close this chapter of confusion.
I'd appreciate if you could read my exact questions and only try to answer them...
I wanna be checking homogeneity of space(only interested in vertical) for simplicity and example we can do is "ball is dropped". To check homogeneity, we use either passive or active transformation and I'm interested in lagrangians.
I heard that we can write lagrangians such as: ##L =...
Imagine experiment is such as I drop a ball from some height vertically only.
What’s the right way to do 2nd experiment in order to check homogeneity of space.
Way 1: I move a little bit and drop the ball (same height, it’s just I moved - ball as well, but not in terms of height)
Way 2: We...
Question 1: in the non-inertial frame, space is non-isotropic. If we're in an accelerated train frame, and we face forward(the same direction where train is accelerating) and drop a ball, ball moves backward. If we face backward and repeat the experiment, dropped ball moves forward to us. So we...
I have a question related to Landau's book. In that, he says:
As an example, I'd like to bring a car and the ball hung inside the car and we can look at it from 2 different frames of reference.
Frame of reference is me(I'm inside the car): If car moves with constant speed, nothing happens to...
From the michelson-morley experiment, if a clock were to measure the time period of light hitting the mirror and returning back, it would be 2L/c, where L is the distance between the laser nd the mirror. For a moving observer, the time period would have a factor of *gamma*, the boost factor...
The purpose of t test is to find how close the two means of the two samples given are; whereas the result of ##\chi^2## homogeneity test indicates the likeness of the two distributions of two populations (or maybe samples--I am not sure). Can anyone please tell me the differences between them...
In the book, it states that a universe is isotropic if it looks the same regardless of which direction you look at large enough scales. This seems fairly easy to prove these days with observations from galaxy surveys and the CMB. However, how can we possibly prove that the university is...
From the continuity equation ##\frac{\partial \rho}{\partial t}+\rho (\nabla • u)=0## where ##\rho## is the mass density and is homogeneous and ##u## is the velocity of expansion or contraction.
For an expanding volume this becomes ##\nabla•u=\frac{\dot v(t)}{v(t)}=\Theta## which gives the rate...
Observation shows that the Universe is homogeneous (and isotropic) at the large scale, while one expects to see inhomogeneity (increasing density at greater distances) on the past light cone due to expansion. This seems inconsistent. Am I misunderstanding something here?
This is a basic assumption that's made during the derivation of results of special theory of relativity is that space is homogeneous i.e. space intervals shouldn't be given preference based on our choice of origin. However I want to understand more about this assumption and its importance...
Hi,
i know that The homogeneity of space and time implies that the Lagrangian cannot contain
explicitly either the radius vector r of the particle or the time t, i.e. L must be a function of v only
but the lagrangian definition is ##L=\int L(\dot q,q,t)##, so velocity appears in the definition...
I want to study chi square test of homogeneity from any authentic source- book / website especially problems where samples are compared for more than one attribute.
What are some relevant sources?
Relevant background:
I was studying examples from random online sources before I saw this book...
I am studying an inviscid fluid. I am trying to characterise the fluid. Does it make sense to call it Newtonian or should I avoid this designation? What I mean is - if there are no viscous stresses then does it make sense to characterise it's response to viscous stresses? (That box that doesn't...
Consider a two-point function $$f(\mathbf{r}_{1},\mathbf{r}_{2})$$ If one requires homogeneity, then this implies that for a constant vector ##\mathbf{a}## we must have $$f(\mathbf{r}_{1},\mathbf{r}_{2})=f(\mathbf{r}_{1}+\mathbf{a},\mathbf{r}_{2}+\mathbf{a})$$ How does one show that if this is...
Homework Statement
Show that the isotropy and homogeneity of space-time and equivalence of different inertial frames (first postulate of relativity) require that the most general transformation between the space-time coordinates (x, y, z, t) and (x', y', z', t') is the linear transformation...
Homework Statement
The final part of the problem I am trying to solve requires the proof of the following equation:
\frac{d}{dr}(\frac{rf'(r)-f(r)+f^2(r)}{r^2 f^2(r)})=0[/B]Homework Equations
I've been given the ansatz:
f(r)=(1-kr^2)^{-1}
leading to
f'(r)=2krf^2(r)...
According to general relativity, time is a dimension, one of four dimensions that form 4D spacetime - a structure which is mathematically symmetrical and homogeneous.
Should not all four dimensions, therefore, be mathematically interchangeable? Assuming that we are 3-dimensional bodies...
Hi there.
I'm having a hard time understanding the precise meaning of the so called "cosmological principle":
My understanding of the general Big-Bang model is that far enough back in time the observable universe came down to something very small (compared to now), very dense, very hot... Ok, i...
Hi all
I want to ask a question about NFE(Newtonian Friedmann Equation).I know that NFE is not usefull to describe universe.But we can have a general idea about universe to use that formula.
I know that the only spatial coordinate system is CMB referance frame and NFE is derived from...
In Herbert Callen's text 'Thermodynamics and an introduction to thermostatistics' 2nd edition, he introduces four postulates of thermodynamics in the first chapter. The third postulate incorporates an 'additivity property' which is stated as 'The entropy of a composite system is additive over...
quating from Cosmological_principle
How can one see that if the universe appears isotropic from any two locations it must also be homogeneous?
And why would we need three points for a sphere?
Thanks.
When dealing with cosmological perturbations, there are a lot of different notions that are thrown around in the literature like statistical homogeneity and isotropy. However, these terms are often not motivated and clearly defined.
Could anyone recommend any good references where these notions...
I was reading that the homogeneity of space can lead to the conclusion that the lagrangian of a free particle is not explicitly dependent on its position. At the moment, this does not come very intuitively to me. By homogeneity, I understand that if you displace the initial position of a...
Homework Statement
Show that if f is homogeneous of degree n, then
x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} = nf(x,y)
Hint: use the Chain Rule to diff. f(tx,ty) wrt t.
2. The attempt at a solution
I know that if f is homogeneous of degree n then t^nf(x,y) =...
Homework Statement
The problem is this one:
Consider a monocomponent fluid, isolated and in equilibrium,
a) Find the homogeneity criteria that must fulfill the number of microstates Ω(U,V,N).
b) If Ω(U,V,N)=exp(a*Vα*Uβ) when a>0 use the result in a to find the condition that have to fulfill...
From what I understand, the observable universe began as homogeneous and very hot. if the universe was very hot, doesn't that mean that particles are vibrating at very fast speeds? after all, isn't heat simply kinetic energy of particles? if this is the case, then how could the universe be...
I've been reading a book on economics and they defined a homogeneous function as: ƒ(x1,x2,…,xn) such that
ƒ(tx1,tx2,…,txn)=tkƒ(x1,x2,…,xn) ..totally understandable.. they further explained that a direct result from this is that the partial derivative of such a function will be homogeneous to the...
This new paper Local Large-Scale Structure and the Assumption of Homogeneity claims that a combined analysis of several surveys indicates that there is a substantial local under-density in the universe on the order of 800 MPC in size.
Previous work done by other authors suggests that an...
The FRW metric is usually expressed as
$$ds^2 = -dt^2 + a(t)^2 ( \frac{dr^2}{1-kr} + r^2 d\Omega^2))$$
where ##k=-1,0,+1## respectively for a hyperbolic, flat or spherical space. The spatial part of this metric can be derived by considering a 3-sphere embedded in a four-dimensional flat space...
I thought this was an interesting article. I wondered does it create issues for the isotropic homogeneous view of the Universe when 25% of the highest energy cosmic rays come from one spot?
http://news.yahoo.com/big-dipper-hotspot-may-help-solve-100-old-135703814.html
I have been trying to pin down a precise definition of large-scale homogeneity, in the context of saying, per the Cosmological Principle, that all constant-time hypersurfaces (CTHs) of a foliation are large-scale homogeneous.
Here is my attempt:
Let M represent any coordinate-independent...
There was some excitement a few months ago about the discovery of the Huge-LQG quasar structure, claimed to be the "largest structure in the Universe", which was said to violate the cosmological principle and the assumption of homogeneity of the Universe. Some previous threads on this topic on...
Einstein, in his paper "On the Electrodynamics of Moving Bodies", part 1, sec. 3, writes: "Primarily it is clear that on account of the property of homogeneity which we ascribe to time and space, the equations must be linear." What has the homogeneity of space and time to do with the degree of...
We all know what it means to be homogeneous in a "hand waving" sort of way. And, of course, there are abstract mathematical definitions for a homogeneous space. I have been unable to find a physical measure of homogeneity which could be applied to a ensemble of particles, box of rocks, or the...
I often read sentences like, "if space is homogeneous, then the Lorentz transformation must be a linear transformation." What exactly does it mean to say that space is homogeneous, and how does it imply that the Lorentz transformations are linear?
Homework Statement
Which one of the following equations is dimensionally homogeneous?
Where:
F= force (N)
m= mass (kg)
a= acceleration (m/s2)
V= velocity (m/s)
R= radius (m)
t= time (s)Homework Equations
1. F=ma
2. F=m(V2/R)
3. F(t2-t1)=m(V2-V1)
4. F=mV
5. F=m(V2-V1)/(t2-t1)The Attempt at a...
First, I must stress that I am not asking this question in relation to the proof, or disproof, of any form of rotating Universe. I am only asking in order to understand the meaning of “homogeneity” in the cosmological context.
Secondly, I know that there are many threads that reference...
Homework Statement
hi guys,
Can someone please help me understand superposition and homogeneity in regards to the following graph.
(current voltage characteristic of a diode)
To be honest i don't understand the terms. The exact question I am asked is Use two test points on Graph to...
Hi,
I have two very specific questions.
I was trying to read this paper :
http://downloads.bbc.co.uk/looknorthyorkslincs/sun_climate_connection.pdf
and i noticed that equation 3 and equation A6 were different :
Eq3 : L(t)=Q(t)/£(t) (not dimensionally correct)
and
Eq A6 ...
Can't we simply assume that the initial condition for the universe is perfectly spherically symmetric, and the problem is solved? In other words, can't we make the CMB homogeneous just by imposing homogeneous initial conditions? The fluctuations can be explained by quantum effects. Of course...
Above what scale is the universe considered to be homogeneous? What sort of measure is used? I've looked at a few cosmology texts and they don't really discuss it much. Weinberg states 300 million LY. With 250 million LY diameter voids and 1370 million LY long filaments, 300 million LY seems...
According to principle of homogeneity, quantities having same dimensions can be added and subracted...but isn't it false ?
because according to the principle ,
we can add quantities having dimensions M0L0T0
I.E we can add plane angle and solid angle,
we can add angles in different...
(aT)∗ = \bar{a}T∗ for all a ∈ C and T ∈ L(V,W);
This doesn't make much sense to me. Isn't a supposed to be=x+iy and
\bar{a}=x-iy? Not a fan of complex numbers. And
this proof also confuses me.7.1 Proposition: Every eigenvalue of a self-adjoint operator is real.
Proof: Suppose T is a...
Is the homogeneity of space (conservation of momentum) and the homogeneity of time (conservation of energy) violated in the curved space-time of a gravitational field? Thanks in advance.
http://camoo.freeshell.org/27.16wrong.pdf"
Mistake by the author?
Laura
Latex source below for quoting purposes but the .pdf may've been edited since then.
Exercise 27.16 asks you to show why a connected 3-space can't be
isotropic about 2 distinct points without being homogeneous...
Homework Statement
Give an example of a function f:R^2 -> R such that f(av) = a(f(v)) for all a in R and all v in R^2 but f is not linear.
Homework Equations
f(v + w) = f(v) + f(w) (Additivity)
The Attempt at a Solution
I really can't think of a function that will satisfy these...
If the assumptions of homogeneity and isotropy lead to the conservation laws of linear momentum, angular momentum, and energy, would a cosmology that drops either of those assumptions lead to violations of those conservation laws which in turn could be observed? My first guess would be yes, but...
...on the gauge-invariance of classical electrodynamics ?
I'm thinking of flat Minkowski space-time of SR in which a charged particle moves and generates an electromagnetic field described by the well-known Lie/nard - Wiechert potential.
In this situation can we say all spacetime is...