What is Homogeneous: Definition and 404 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.

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  1. K

    Nonhomogeneous System: Similar Coefficients & Solutions?

    This might belong in the HW section, but since it's specific to Linear Algebra I posted it here. Alright, so we have a homogeneous system of 8 equations in 10 variables (an 8 x 10 matrix, let's call it A). We have found two solutions that are not multiples of each other (lets call them a and...
  2. N

    Must ensembles be homogeneous?

    An ensemble is a collection of systems, all prepared in the same way. Does this mean that all the systems are in the same state? I have seen some authors create ensembles where 30% of the systems are in a state, s, and 70% of the systems are in a state, t . As far as measurements go, this...
  3. P

    What Values of k Give Infinite Solutions in This Homogeneous System?

    Homework Statement 1. Determine the values of k such that the following homogeneous linear system has infinitely number of solutions. My problem is: I cannot to find the value of k Homework Equations 2x – ky + z = 0 -x + y – 3kz = 0 kx – 2y + 2z = 0 The Attempt at a Solution After I...
  4. G

    Example of a homogeneous, but not isotropic system

    Hi, I have some trouble understanding if linear momentum and angular momentum (and their conservation laws) are completely independent or not. For example, one can calculate the angular momentum of a uniformly moving body with respect to a fixed point in space and show that it is indeed...
  5. M

    2nd order, linear, homogeneous proof

    Hi, I was wondering if someone could provide either a bit of intuition or a mathematical proof (or both) as to why if the Wronskian (W(f,g)) does not equal 0 for all t in an interval, then the linear combinations of the two functions f and g encompass ALL solutions. Is there any particular...
  6. DavitosanX

    Homogeneous 2nd Order DE from spring pendulum

    I'm currently taking a Classical Mechanics course, and we're studying the lagrangian equation. After a few exercises, I thought I'd try to come up with the motion equations for a pendulum where the mass hangs from a spring. The resulting differential equations take a form that I don't really...
  7. _N3WTON_

    The Principle of Superposition for Homogeneous Equations (DiffEq)

    Homework Statement Verify that e^x and e^-x and any linear combination c_1e^x + c_2e^{-x} are all solutions of the differential equation: y'' - y = 0 Show that the hyperbolic sine and cosine functions, sinhx and coshx are also solutions Homework Equations Principle of Superposition for...
  8. E

    Linear homogeneous D.E. with constant coefficients - known solutions

    My task is to find Linear homogeneous D.E. with constant coefficients which has solutions: $$\\\varphi 1(x)=x^2,\varphi 2(x)=e^{-3x},\varphi 3(x)=cos(5x)$$ Any idea?
  9. C

    Variational Principle for Spatially Homogeneous Cosmologies/KK-theory

    These questions applies to both spatially homogenous cosmological models, and multidimensional Kaluza-Klein theories: Suppose we have a manifold M, of dimension m, for which there is a transitive group of isometries acting on some n-dimensional homogeneous subspace N of M. Thus there exists a...
  10. V

    Hydrogen Atom in homogeneous magnetic vector potential

    Hey! I did an quantum mechanical analysis of a Hydrogen Atom in a homogeneous magnetic vector potential (I know that it might be impossible to create this kind of field) out of curiousity. I showed it to some professors of mine, but they all said that they don't have time. So I decided to post...
  11. PeteyCoco

    Green's Function of a homogeneous cylinder

    I've been reading this article for a prof this summer: http://arxiv.org/pdf/1302.0245v1.pdf I'm having some trouble following the math in Appendix B: Green's Function Of A Homogeneous Cylinder (page 9). Can someone explain to me why there is a factor of \frac{1}{\rho\rho'} in front of the Green...
  12. J

    Non homogeneous density in a gas

    Hi, have searched around but can't find what I am looking for (probably because I am not entirely certain what its called) An elemental gas in a contained volume, say microcanonical for simplicity, will have atoms that bounce against the walls. Is there an equation for the average density of...
  13. B

    What are the implications of spatial homogeneity in cosmological models?

    In a spatially homogeneous model, spacetime is filled with a one-parameter set of invariant hypersurfaces H(t). Spatial homogeneity means that the metric on each H(t) is described in terms of constants. Meaning that the metric becomes a function of time only. I guess that this means that...
  14. V

    Homogeneous gravity field time evolution position and momentum operato

    Homework Statement I am trying to solve Problem 21 from this sheet: Homework Equations The equation describing the time evolution of operators is given in the problem. The Attempt at a Solution I have found the commutators of the position and momentum operator with the...
  15. A

    Homogeneous differential equations

    Is this a homogeneous DE? 3y'''' + 21y'' + y' + 6y = 0 So... since a(n-1)y''' is missing, would this still by definition be a homogeneous differential equation?
  16. B

    Homogeneous spacetime - Lie groups

    All Bianchi type spacetimes have metrics that admits a 3-dimensional killing algebra. They are in general not isotropic. Bianchi type IX have a killing algebra that is isomorphic to SO(3), i.e. the rotation group. But what does it mean? If the fourdimensional spacetime is invariant under the...
  17. Y

    MHB Solve Homogeneous Function f(x,y) w/ Euler's Rule

    Hello, I need some help with this question here, I'll explain why in a second. The question is: f(x,y) is a homogeneous function of order 3. It is known that: \[f_{x}(2,1)=7\] and \[f_{y}(8,4)=5\] find f(12,6). Now, I know of Euler's rule, which includes the partial derivatives, but in...
  18. 1

    Is Homogeneous Line Universe Possible in Higher Dimensions?

    I have been trying to work out the solutions to a homogenous line universe using special relativity, and have found that, as per special relativity, one of the solutions is $$v = tanh(d)$$, where $v$ is the velocity and $d$ is the distance of recession of galaxies in this one dimensional case...
  19. N

    Need help in understanding W paremeter for homogeneous coordinates

    First I would like to apologize first if this is the wrong place for posting this problem. I don't really understand what is the importance of w in the homogeneous coordinate (x,y,z, w). One of the example i have read is about a parrallel line extended to infinity, and both line would...
  20. M

    Homogeneous initial value problem

    Homework Statement 4y" + 4y' + 5y = 0 y(0) = 3 y'(0) = 1 Homework Equations yh = e^ax(c1cosbx + c2sinbx) The Attempt at a Solution For the roots I got -1/2 + i and -1/2 - i so my a = -1/2 and b = 1 then I have to differentiate yh = e^(-1/2x)[c1cosx + c2sinx] this is where I get this...
  21. MarkFL

    MHB Yahoo Answers: Linear Homogeneous Recurrence - JunkYardDawg

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  22. MarkFL

    MHB Solving 2nd Order Homogeneous ODE - Joe's Question on Yahoo Answers

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  23. J

    Solve Homogeneous D.E. integrating

    Homework Statement Dy/Dx = (Y-x)/(Y+x) Homework Equations Y=ux dy=udx+xdu The Attempt at a Solution Dy/Dx = (Y-x)/(Y+x) Plug in my substitutions udx+xdu(1/dx)=(ux/ux+x) - X/(ux+x) Simplify u+x(du/dx)=(ux)/x(u+1) - (x)/((x)(u+1)) u+x(du/dx)=u/(u+1) -(1)/(u+1)) u+x(du/dx)=u-1/(u+1)...
  24. C

    Homogeneous function of degree n

    I can't see the proof clearly. Gradient [f(tx,ty)] dot <d/dt (tx) , d/dt (ty)> = d/dt [f(tx,ty)] = n * t^(n-1) * [f(x,y)] I don't see how the t^(n-1) term disappears. HELP
  25. K

    Solution to homogeneous wave equation

    Homework Statement Prove by direct substitution that any twice differentiable function of (t-R\sqrt{με}) or of (t+R\sqrt{με}) is a solution of the homogeneous wave equation. Homework Equations Homogeneous wave equation = ∂2U/ ∂R2 - με ∂2U/∂t2 = 0 The Attempt at a Solution Could you...
  26. Z

    MHB Solving Homogeneous Function Confusion: ln(Y/X) in Numerator

    Hey people! I'm confused as to why the ln(Y/X) part of the numerator is not considered in the calculation of the degree of numerator. Any help or websites to browse through for the answer would be appreciated!
  27. M

    2nd order linear homogeneous DE

    when you solve a 2nd order linear non-homogeneous DE, where it is equal to a constant as in Kirchoff's 2nd Law and the roots of the auxiliary equation are imaginary then you have superposition of 2 solutions. so the particular solution is equal to a constant k and you can solve for this by...
  28. J

    General tendency of a homogeneous floating balloon in a wind current

    Homework Statement Any floating homogeneous balloon in a planar uniform wind current will always "tend" to present to the wind flow a section of maximal drag. 2. The attempt at a solution I have three possible solutions to this "problem": 1- Speed gradient justification: When an object...
  29. A

    Is the Higgs Field Homogeneous or Does It Vary with Particle Interactions?

    is the higgs field always homogeneous, or do particles such as electrons cause it to be concentrated about certain points? Are there waves in the higgs field?
  30. K

    MHB Homogeneous, linear, first-order, ordinary differential equation mistake

    I am trying to solve a homogeneous, first-order, linear, ordinary differential equation but am running into what I am sure is the wrong answer. However I can't identify what is wrong with my working?! $$\frac{dy}{dx}=\frac{-x+y}{x+y}=\frac{1-\frac{x}{y}}{1+\frac{x}{y}}.$$ Let $z=x/y$, so that...
  31. J

    General homogeneous shrinking core problem

    Hi Guys, First post here. I'm just wondering if anyone could lend a helping hand in the following derivation. It is taken from Ishida AIChE J 14 (1968) 311 (also very similar to that derived by Ausman Chem Eng Sci 17 (1962) 323) and concerns the derivation of the general non-catalytic shrinking...
  32. J

    Non-Homogeneous ODEs with Coupled Equations: Solving with Fourier Series?

    how do we solve an ODE which has forcing function in terms of Fourier series? i have attached a pdf file of the problem.
  33. Gh778

    Pendulum with not homogeneous attraction

    I would like to study a pendulum in 2D, not on Earth but only with one fixed mass attract a disk. Considered this fixed mass like a point. No friction on this theoretical study. The disk can move only around a part of circle. There are only 2 forces, F the attraction and N the force from wall...
  34. C

    Homogeneous function of degree n

    A function f is called homogeneous of degree n if it satisfies the equation [f(tx,ty,tz)]=(t^n) *[f(x,y,z)] for all t, where n is a positive integer and f has continuous second order partial derivatives". I don't have equation editor so let curly d=D I need help to show that...
  35. Y

    Solving non homogeneous heat equation

    \frac{\partial{u}}{\partial{t}}=\frac{\partial^2{u}}{\partial{r}^2}+ \frac {2}{r} \frac {\partial{u}}{\partial{r}}+\frac{1}{r^2}\left[\frac{\partial^2{u}}{\partial{\theta^2}}+\cot\theta \frac{\partial{u}}{\partial {\theta}} +\csc\theta\frac{\partial^2{u}}{\partial{\phi}^2}\right]+q(r,\theta,t)...
  36. T

    Forces in a homogeneous electric field

    Hello everyone, I was wondering if anyone could help me out with this one: Problem statement: A charged metal ball is hanging from a nylon wire in the space between both plates of a loaded capacitor. The ball has a mass of 9.2 grams. The distance between the hanging point and the ball's...
  37. M

    The homogeneous strength of the Higgs field

    Dear Physics Forum, As I understand it, the Higgs field is a quantum field that stretches throughout our universe. Particles that carry mass (for example protons and electrons) acquire this property by interacting with the (local) Higgs field. I assume this interaction can be written in the...
  38. MarkFL

    MHB Thingsto Do's question at Yahoo Answers regarding a first order homogeneous IVP

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  39. M

    Show that this equation is homogeneous. PLEASE HELP, relatively simple

    I have attempted two questions, see below. 1. The relationship between energy, E, and momentum, p, is E = P^2C^2 + M^2C^4 for a relativistic particle, Show that the equation is homogeneous 2. The current, I, in a wire is given by I = nAev where n is the number of electrons per unit volume, A...
  40. N

    MHB How to Solve This Homogeneous Differential Equation?

    Hi, I actually made a similar thread here: Solve this homogeneous type equation and a user pointed out a mistake in my workings, however i could still not manage to get the solution. So I was wondering if someone could help with the last few parts Question: 2xyy' = y^2 - x^2y' = (y^2-x^2)/(2xy)...
  41. MarkFL

    MHB CuRio$ty's question at Yahoo Answers regarding a linear homogeneous recursion

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  42. MarkFL

    MHB Paul's question at Yahoo Answers regarding a 3rd order linear homogeneous ODE

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  43. H

    System with homogeneous equation in denominator help

    Homework Statement The system is declared as follows: 8/(2*x - y) - 7/(x + 2*y) = 1 4/((2*x - y)^2) - 7/((x + 2*y)^2) = 3/28 Homework Equations The Attempt at a Solution I define 'x' to equal k*y and I replace it inside the equation: 8/(2*k*y^2) - 7(k*y + 2*y) = 1...
  44. P

    Find a linear homogeneous equation with given general solution

    I need help finding a linear homogenous constant-coefficient differential equation with the given general solution. y(x)=C1e^x+(C2+C3x+C4x^2)e-x 2. I tried to come with differential equation but this is it I can 't seem how to begin
  45. M

    He most general form of the metric for a homogeneous, isotropic and st

    What is the most general form of the metric for a homogeneous, isotropic and static space-time? For the first 2 criteria, the Robertson-Walker metric springs to mind. (I shall adopt the (-+++) signature) ds^2=dt^2+a^2(t)g_{ij}(\vec x)dx^idx^j Now the static condition. If I'm not mistaken...
  46. S

    Solving Homogeneous System of 3 Equations & 4 Variables

    Hi, I'm solving out a homogeneous 3 equations and 4 variables system so I considered one variable as known term but the determinant of the matrix is 0, how do I use Cramer in this case ? these are the 3 equations 2x + 3y - z - 2v = 0 4x - 3y - 5z + 5v = 0 8x + 3y - 7z + v = 0...
  47. D

    Integrating factor for a 2nd order homogeneous linear ODE

    Homework Statement Consider the general linear homogeneous second order equation: P(x)y'' + Q(x)y' + R(x)y = 0 (1) We seek an integrating factor μ(x) such that, upon multiplying Eq. (1) by μ(x), we can write the resulting equation in the form [μ(x)P(x)y']' + μ(x)R(x)y = 0...
  48. M

    Form the differential equation & homogeneous

    Hi
  49. S

    Construct a 2 by 3 system that has these particular and homogeneous eq

    Homework Statement Construct a 2 by 3 system Ax = b with particular solution xp = (2,4,0) and homogeneous solution xn = any multiple of (1,1,1). The answer is {{1,0,-1},{0,1,-1}} x = {{2},{4}} which has xp and xnull = (c,c,c). Homework Equations Ax = b The Attempt at a Solution I...
  50. pellman

    What are the hamilton equations of motion for homogeneous lagrangians?

    For a Lagrangian L(x^k,\dot{x}^k) which is homogeneous in the \dot{x}^k in the first degree, the usual Hamiltonian vanishes identically. Instead an alternative conjugate momenta is defined as y_j=L\frac{\partial L}{\partial \dot{x}^j} which can then be inverted to give the velocities as a...
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