Homogeneity Definition and 59 Threads

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.

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  1. murshiddreamengineer

    I Proof of Lorentz transformation

    What are the supporting arguments for the assumption that space and time are homogeneous?
  2. Agent Smith

    B Chi-Square Tests for Homogeneity & Association

    ============================================================================================= From the above example questions, we have ##2## different kinds of Chi-square tests. 1. One for homogeneity 2. One for association (independence/dependence). The answer guide says that if we take...
  3. cianfa72

    I Spatial homogeneity condition for a free particle Lagrangian

    Hi, reading "Mechanics" book by Landau-Lifshitz, they derive from spatial homogeneity that the Lagrangian ##L## of a free particle cannot explicitly depend on spatial coordinates ##q## in an inertial frame. However my point is as follows: suppose to consider the Lagrangian ##L= \frac 1 2...
  4. G

    I Active transformation use-case in homogeneity

    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...
  5. G

    I Writing the Lagrangians for different frames depending on how "the ball is dropped"

    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 =...
  6. G

    I Some doubts about determining experiments

    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...
  7. G

    I Reference frames in terms of homogenity/isotropy

    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...
  8. G

    I Galilean relativity in terms of homogeneity

    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...
  9. P

    I Violation of Special Relativity Principle?

    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...
  10. Leo Liu

    I Two-sample t test vs. chi-squared test for homogeneity

    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...
  11. P

    I What proof do we have that the universe is homogenous?

    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...
  12. A

    I Small Scale Homogeneity and Expansion Rate Variance

    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...
  13. V

    I How can an expanding Universe look homogeneous?

    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?
  14. weezy

    I Explore Homogeneity of Space: Effects on Relativity Theory

    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...
  15. Andrea Vironda

    Exploring the Homogeneity of Space & Time in Lagrangian Mechanics

    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...
  16. T

    I Online sourses to learn chi square test of homogeneity

    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...
  17. S

    Can inviscid fluids rightly be called Newtonian?

    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...
  18. F

    I Spatial homogeneity and the functional form of two-point functions

    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...
  19. H

    Deriving the Lorentz Transformation from the Homogeneity of Spacetime

    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...
  20. S

    Solving a differential to show the homogeneity of space.

    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)...
  21. R

    Are spatial and temporal dimensions interchangeable?

    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...
  22. Goddar

    Homogeneity and isotropy in Big Bang model

    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...
  23. RyanH42

    Newtonian Friedmann Equation, Referance frame, Homogeneity

    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...
  24. andrewkirk

    Thermodynamics - does homogeneity follow from additivity?

    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...
  25. ChrisVer

    Does Isotropy Necessarily Imply Homogeneity in the Universe?

    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.
  26. C

    Litterature on Statistical Homogeneity and Isotropy

    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...
  27. I

    Homogeneity of space and the form of the Lagrangian

    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...
  28. kostoglotov

    Partial derivatives Q involving homogeneity of degree n

    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) =...
  29. D

    Homogeneity criteria (Thermodynamics)

    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...
  30. A

    Exploring the Homogeneity of the Hot Observable Universe

    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...
  31. S

    Why Does the Constant 't' Appear in the Derivative of a Homogeneous Function?

    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...
  32. C

    Dark Energy an effect of inhomogeneity?

    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...
  33. C

    Is hyperbolic space consistent with homogeneity?

    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...
  34. T

    Big Dipper Cosmic Ray Hotspot Homogeneity question

    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
  35. andrewkirk

    A Cosmological Principle: Trying to define large-scale homogeneity

    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...
  36. S

    The 'Huge-LQG' quasar 'structure' does not violate homogeneity

    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...
  37. K

    Linear equations and homogeneity of space and time

    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...
  38. S

    How Can We Quantify Homogeneity in Physical Systems?

    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...
  39. D

    Understanding Space Homogeneity & Lorentz Transformations

    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?
  40. P

    Finding Dimensional Homogeneity

    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...
  41. L

    What does homogeneity mean (in the cosmological context)?

    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...
  42. M

    Superposition and homogeneity - graph analysis

    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...
  43. P

    Is it common to have dimensionally wrong equations in physics?

    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 ...
  44. P

    Why do we need inflation to explain the homogeneity of CMB?

    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...
  45. S

    What is the Scale of Homogeneity in the Universe and How is it Measured?

    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...
  46. J

    Why can't we always add quantities with the same dimensions?

    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...
  47. E

    Conjugate Homogeneity for Self-Adjoint Operators: Proof and Explanation

    (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...
  48. LarryS

    Homogeneity of Space and Time vs Gravity

    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.
  49. L

    Can a 3-Space Be Isotropic About Two Distinct Points Without Being Homogeneous?

    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...
  50. S

    Homogeneity holds but additivity does not. I'm stuck

    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...
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