What is generalized: Definition and 215 Discussions

A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model). As such, they are the essential basis of all valid deductive inferences (particularly in logic, mathematics and science), where the process of verification is necessary to determine whether a generalization holds true for any given situation.
Generalization can also be used to refer to the process of identifying the parts of a whole, as belonging to the whole. The parts, which might be unrelated when left on their own, may be brought together as a group, hence belonging to the whole by establishing a common relation between them.
However, the parts cannot be generalized into a whole—until a common relation is established among all parts. This does not mean that the parts are unrelated, only that no common relation has been established yet for the generalization.
The concept of generalization has broad application in many connected disciplines, and might sometimes have a more specific meaning in a specialized context (e.g. generalization in psychology, generalization in learning).In general, given two related concepts A and B, A is a "generalization" of B (equiv., B is a special case of A) if and only if both of the following hold:

Every instance of concept B is also an instance of concept A.
There are instances of concept A which are not instances of concept B.For example, the concept animal is a generalization of the concept bird, since every bird is an animal, but not all animals are birds (dogs, for instance). For more, see Specialisation (biology).

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  1. Z

    What is the oscillator model in a generalized Snyder scheme?

    What is the oscillator model in a generalized Snyder scheme?How to derive the formula?
  2. D

    I Equation of motion: choice of generalized coordinates

    I am looking at a textbook solution to the following problem of finding the equation of motion of a half disk. In the solution, the author considers the half disk has a COM at the black dot, and to find the instantaneous translational velocity of the center of mass (he considers rotational...
  3. VVS2000

    A Independence of generalized coordinates and generalized velocities

    How can I make sense of this and further how to think of this in the context of phase space diagrams?
  4. J

    A The Generalized Capabilities of the Standard Model Lagrangian?

    If the standard model Lagrangian were generalized into what might be called "core capabilities" what would those capabilities be? For example, there are a lot of varying matrices involved in the standard model Lagrangian and we can generalize all of them as the "core capability" of matrix...
  5. wrobel

    A Why don't we multiply generalized functions?

    Because it drives to contradictions. Here is a nice example from E. Rosinger Generalized solutions of nonlinear PDE. We can multiply generalized functions from ##\mathcal D'(\mathbb{R})## by functions from ##C^\infty(\mathbb{R})##. This operation is well defined. For example $$x\delta(x)=0\in...
  6. H

    A Is there a generalized second law of thermodynamics?

    Hi Pfs, There are different kinds of entropies. I discoved the free entropy. https://arxiv.org/pdf/math/0304341.pdf the second law says that the total entropy cannot decrease when time goes by. Is it always the same "time" for the different entropies? the author, Voiculescu, wrote articles...
  7. Mayhem

    [Quantum Chemistry] Generalized wave function in covalent bonding

    Is there a general expression for the wave function $\psi$, which describes the electronic properties of an arbitrary covalent bond? For example is it equal to some sort of trigonometric expression?
  8. J

    A Generalized Forces and QED/QCD

    In Lagrangian mechanics we learn about generalized forces. However, I haven't seen these explicitly mentioned in books on QFT. Can the Lagrangians of QED or QCD be expressed in terms of generalized forces or is there some connection there, in particular to the Nielsen form.
  9. S

    MHB Perfecting My Proof of Generalized Vandermonde's Identity

    My tests are submitted and marked anonymously. I got a 2/5 on the following, but the grader wrote no feedback besides that more detail was required. What details could I have added? How could I perfect my proof? Beneath is my proof graded 2/5.
  10. 1

    I Why did I lose 60% on my proof of Generalized Vandermonde's Identity?

    My tests are submitted and marked anonymously. I got a 2/5 on the following, but the grader wrote no feedback besides that more detail was required. What details could I have added? How could I perfect my proof? Beneath is my proof graded 2/5.
  11. H

    I Lagrangian with generalized positions

    Hi Pfs When instead of the variables x,x',t the lagrangiean depends on the trandformed variables q,q',t , time may be explicit in this lagrangian and q' (the velocity of q) may appear outside. I am looking for a toy model in which tine is not explicit in L but where the velocities appear somhere...
  12. Ced

    Can You Help Solve This Generalized Work Problem with an Illustrative Image?

    Here is an image for better illustration, I only managed to solve for (a) but I'm not sure if I did it right. As for (b) and (c), I have no idea how to do it. My answer for (a): => Ki + Ui + Wext = Kf + Uf => 0+mgh1-LμmgCosΘ = 1/2mv^2 + mgh2 =>1/2v^2 = gh1- gh2 - LμgCosΘ => V = √2g(h1 - h2 -...
  13. K

    A Two equations of generalized forces

    Wikipedia article under generalized forces says Also we know that the generalized forces are defined as How can I derive the first equation from the second for a monogenic system ?
  14. e2m2a

    A Generalized Diophantine equation and the method of infinite descent

    There is an entry in Wikipedia at this link: https://en.wikipedia.org/wiki/Pythagorean_triple Under elementary properties of primitive Pythagorean triples, general properties,sixth bullet from the bottom of this section, there is this generalized Diophantine equation: x^2p + y^2p = z^2 Where: p...
  15. Alwar

    I How to Find the Generalized Eigenvector in a Matrix ODE?

    Hi, I have a set of ODE's represented in matrix format as shown in the attached file. The matrix A has algebraic multiplicity equal to 3 and geometric multiplicity 2. I am trying to find the generalized eigenvector by algorithm (A-λI)w=v, where w is the generalized eigenvector and v is the...
  16. Ssnow

    Quantum References for generalized canonical commutation relations

    Hi to all, I ask if somebody of the Physics community know good references for article where the author works with generalized canonical commutation relations ( I mean that the author works with ##[x,p]=ic\hbar## with ##c## a real constant instead of ##[x,p]=i\hbar##). Thank you for the answers...
  17. patric44

    How does Generalized Work have dimensions of Work?

    hi guys my analytical mechanics professor asked a question the other day about, how come the generalized forces##Q_{\alpha}## doesn't need to have a dimension of force, and the generalized coordinated ##q_{\alpha}##as well doesn't need to have a dimension of length, but the generalized work...
  18. Z

    I Why does the summation come from?

    I want to take the derivative of a composite function that looks like $$f( g(x), h(x) ).$$ I know from Wolfram that the answer is $$\frac{ df( g(x), h(x) ) }{ dx } = \frac{ dg(x) }{ dx }\frac{ df( g(x), h(x) ) }{ dg(x) } + \frac{ dh(x) }{ dx }\frac{ df( g(x), h(x) ) }{ dh(x) }.$$ We can...
  19. patric44

    Proof of the generalized Uncertainty Principle?

    hi guys i am trying to follow a proof of the generalized uncertainty principle and i am stuck at the last step : i am not sure why he put these relations in (4.20) : $$(\Delta\;C)^{2} = \bra{\psi}A^{2}\ket{\psi}$$ $$(\Delta\;D)^{2} = \bra{\psi}B^{2}\ket{\psi}$$ i tried to prove these using the...
  20. Mayan Fung

    Microcanonical ensemble generalized pressure

    In the discussion of the pressure in macrocanonical ensemble, I found in textbook that: ##dW = \bar p dV## (##dW## is in fact d_bar W, yet I can't type the bar) The derivation goes like: ##\bar p = \frac{1}{Z} \sum_{r} e^{-\beta E_r} (-\frac{\partial E_r}{\partial V}) = ... = \frac{1}{\beta}...
  21. T

    I Simple Generalized Eigenvalue problem

    Good Morning Could someone give me some numbers for a Generalized EigenValue problem? I have lots of examples for a 2 x 2, but would like to teach the solution for a 3x3. I would prefer NOT to turn to a computer to solve for the characteristic equation, but would like an equation where the...
  22. benorin

    Munkres Topology ch1 ex #9 - Generalized DeMorgan's Laws

    So formulating them was easy, just set ##C:=D\cup E## in (1) and set ##C:=D\cap E## in (2) to see the pattern, if ##\mathfrak{B}## is a non-empty collection of sets, the generalized laws are $$A-\bigcup_{B\in\mathfrak{B}} B = \bigcap_{B\in\mathfrak{B}}(A-B)\quad (3)$$...
  23. U

    Generalized Lorentz Transformation

    Summary: The problem is to generalize the Lorentz transformation to two dimensions. Relevant Equations Lorentz Transformation along the positive x-axis: $$ \begin{pmatrix} \bar{x^0} \\ \bar{x^1} \\ \bar{x^2} \\ \bar{x^3} \\ \end{pmatrix} = \begin{pmatrix} \gamma & -\gamma \beta & 0 & 0 \\...
  24. R

    Choosing Free Variable for Generalized Eigenvector

    As you can see from my eigenvalues, here I've got a repeated roots problem. I'm wondering if it matters which variable I can choose to be the free variable when I'm solving for the generalized eigenvector. I think both are equally valid but they look different from one another and I'd like to...
  25. A

    How is the Coriolis generalized potential obtained

    The Coriolis potential last term of (42) is obtained by integration through r and R from last term of (40). I do not understand why we do not need to integrate through v as well, since the Coriolis force depends on v? Homework Equations Equation (41) is wrong I think, L must be replaced by...
  26. chmodfree

    I Generalized Momentum is a linear functional of Velocity?

    Generalized momentum is covariant while velocity is contravariant in coordinate transformation on configuration space, thus they are defined in the tangent bundle and cotangent bundle respectively. Question: Is that means the momentum is a linear functional of velocity? If so, the way to...
  27. E

    I Linear Algebra and Identity Operator Generalized to 3D

    I'm just getting into 3D quantum mechanics in my class, as in the hydrogen atom, particle in a box etc. But we have already been thoroughly acquainted with 1D systems, spin-1/2, dirac notation, etc. I am trying to understand some of the subtleties of moving to 3D. In particular, for any...
  28. Mr Davis 97

    I How can we use induction to prove the generalized associative law?

    I am trying to prove the generalized associative law with induction, but am being tripped up by one aspect. I am reading a solution and it says for the induction step argue that any bracketing of the product ##a_1 \cdot a_2 \cdot \cdots a_n## must break into two subproducts ##(a_1 \cdot \cdots...
  29. W

    A How Do Generalized Linear Models Extend Beyond Standard Linear Regression?

    Hi, Just wanted to see if I understood the meaning of Generalized Linear Models: In the case of Standard ( "Non-generalized") Linear models, a dependent variable y is a linear function of a dependent variable x. In a Generalized Linear Model (GLM), a dependent variable y is linear in some...
  30. Peter Morgan

    A Generalized free fields as dark matter?

    @vanhees71 reminds us that which suggests something I've wondered about for a while, whether dark matter might be adequately modeled by generalized free fields, which do not have asymptotic free states. Ray Streater, in Rep. Prog. Phys. 1975 38 771-846, "Outline of axiomatic relativistic...
  31. J

    Generalized coordinates- scalar product

    Homework Statement a: In plane polar coordinates, find the scalar product of the vector (0,1) with itself. b: What would be the r, θ components of the unit vector in the θ direction? Homework Equations Scalar product of 2 vectors = AαgαβBβ The Attempt at a Solution For part a, I used the...
  32. nomadreid

    I Generalized coordinates basic question

    From "A Student's Guide to Langrangins and Hamiltonians", Patrick Hamill, Cambridge, 2017 edition. Apologies: since I do not know how to put dots above a variable in this box, I will put the dots as superscripts. Similarly for the limits in a sum. On page 6, "we denote the coordinates by qi...
  33. mertcan

    Generalized Ohm's Law: Current Density & Capacitance

    initially my attachment/picture has been cut off that link http://sun.stanford.edu/~sasha/PHYS780/PLASMA_PHYSICS/phys780_2014_l13.pdf page 6 Also I would like to put into words that divergence of current density is accepted as 0 in continuous loop( no capacitors exist...). But if you look at...
  34. rocdoc

    Dirac's Generalized Hamiltonian Dynamics Theory?

    I wondered if anyone might know of any open access materials, possibly lecture notes, on the content of the following papers or books. P.A.M Dirac, 1950, Can. J. Math. 2,147 "Generalized Hamiltonian Dynamics" P.A.M Dirac, 1933, Proc. Camb. Phil. Soc., 29, 389 "Homogenous variables in classical...
  35. W

    Generalized Velocity: Lagrangian

    Homework Statement [/B] In this example, I know that I can define the horizontal contribution of kinetic energy to the ball as ##\frac{1}{2}m(\dot{x} + \dot{X})^2##. In the following example, Mass ##M_{x1}##'s horizontal contribution to KE is defined as ##\frac{1}{2}m(\dot{X} -...
  36. P

    LaTeX Generalized Product Rule D^(n-m) (x^2 -1)^n (LaTeX inside)

    Homework Statement Show: ##D^{(n-m)} (x^2-1)^n = \frac{(n-m)!}{(n+m)!} (x^2-1)^m D^{(n+m)} (x^2-1)^n## Hint: ##D^{(n-m)} (x^2-1)^n = D^{(n-m)} [(x-1)^n (x+1)^n]## Homework Equations [/B] Leibniz Rule for Differentiation: $$D^k (uv) = \sum_{j=0}^k \binom{k}{j} D^j (u) D^{(k-j)} (v)$$ The...
  37. J

    A Newton's Generalized Binomial Theorem

    I'm trying to expand the following using Newton's Generalized Binomial Theorem. $$[f_1(x)+f_2(x)]^\delta = (f_1(x))^\delta + \delta (f_1(x))^{\delta-1}f_2(x) + \frac{\delta(\delta-1)}{2!}(f_1(x))^{\delta-2}(f_2(x))^2 + ...$$ where $$0<\delta<<1$$ But the condition for this formula is that...
  38. L

    B Generalized Linear equation of a cube

    As per my understanding, a linear equation with two variables form a line segment (ax=by+c or ax+by=c) and linear equation with three variables form a plane (ax=by+cz+d or ax+by+cz=d). Am I right? And if I am right, does an equation with four variables form a cube?
  39. R

    I Amplitudes of Fourier expansion of a vector as the generalized coordinates

    When discussing about generalized coordinates, Goldstein says the following: "All sorts of quantities may be impressed to serve as generalized coordinates. Thus, the amplitudes in a Fourier expansion of vector(rj) may be used as generalized coordinates, or we may find it convenient to employ...
  40. Pushoam

    Generalized Galilean transformation

    Homework Statement Write the Galilean coordinate transformation equations for the case of an arbitrary direction for the relative velocity v of one frame with respect to the other. Assume that the corresponding axes of the two frames remain parallel. (Hint: let v have componentsvx, vy, vz.)...
  41. S

    How to plot generalized hypergeometric function in ROOT?

    Hello everyone I am trying to write code in ROOT.I want to plot generalized hypergeometric function pFq with p=0 and q=3 i.e I want to plot 0F3(;4/3,5/3,2;x) as a function of x using TF1 class.I am not getting how to plot this function in ROOT.Kindly help me out. Thanks in Advance
  42. S

    Rolling ball and generalized co-ordinates

    Consider a sphere constrained to roll on a rough FLAT HORIZONTAL surface. A book on classical mechanics says it requires 5 generalized co-ordinates to specify sphere's configuration: 2 for its centre of mass and 3 for its orientation. I did not understand why 3 for orientation. I guess only 2...
  43. H

    Generalized force and the Lagrangian

    Homework Statement A particle of mass m slides without rolling down on a inclined plane, Find the generalized force and the Lagrangian equation of motion of mass m. Homework Equations T = (mx'^2)/2 Generalized force Q=-d/dx(V) The Attempt at a Solution To find the generalized force first I...
  44. S

    A Generalized Forces: Understanding i & j - Physics Help

    hello guys i like physics specially the classical dynamics but am finding it hard to understand those letters (i , j ) now am studying about the generalized forces corresponded with generalized coordinates and there is an equation in the attached pic with this thread can anybody help me and...
  45. donaldparida

    Generalized version of work-energy theorem

    I know that for rigid bodies only the work-energy theorem states that the net work done on the body equals the change in kinetic energy of the body since a rigid body has no internal degrees of freedom and hence no other forms of energy such as potential energy. Is there a most generalized form...
  46. KT KIM

    I Lagrangian Equation with Generalized Force term

    In basic level classical mechanics I've known so far The Lagrangian Equation is Like this But in the little deeper references, they covers Lagrangian Equation is Like this Qi is Generalized force, and Qi also contains frictions that's what reference says But I still can't grasp. What is the...
  47. K

    I Generalized version of the Fourier Transform

    Hello everyone, I was trying to develop a sort of generalized version of the Fourier Transform. My question in particular is: Given a function f(x,u), is there a function g(x,u) with \int_{-\infty}^\infty f(x,u)g(x,u')\mathrm{d}x=\delta(u-u') For f(x,u)=e^{2\pi ixu} the solution would be...
  48. Ibrahim Mustafa

    I What are the generalized Sundman transformations?

    I would like to give me a simple definition about the generalized Sundman transformations and how we use it to solve the second order differential equation.
  49. O

    A Generalized Coordinates and Porn

    Yes, that is a serious title for the thread. Could someone please define GENERALIZED COORDINATES? In other words (and with a thread title like that, I damn well better be sure there are other words ) I understand variational methods, Lagrange, Hamilton, (and all that). I understand the...
  50. D

    I Defining a generalized coordinate system

    (Note that the title of this thread might be incorrect - I'm just drawing on the vocabulary people use when discussing Lagrangian Mechanics...) Hi, I'm trying to set up a coordinate system to represent points in space where one of the coordinates is the distance along a parametric curve, one is...