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. T

    Generalized aerodynamic force matrix

    Hello guys, Do you know a good source for the definition of the generalized aerodynamic force matrix? I know that the term ij is the work done by the pressure on the mode j that works with the displacements of mode i integrated over the wing domain. This formulation is derived using the...
  2. 0

    Generalized change of variables?

    Does change of variables generalize to situations other than integration?
  3. K

    Measurement Issues: POVM, Neumann and generalized

    [SOLVED] Measurement Issues: POVM, Neumann and generalized What is a POVM measurement? How is it different from Nuemann measurement theory and what would be the most general measurement formulation? Please keep things discrete and not continuous. -KM
  4. C

    Want to learn more about Delta Function and Generalized Function

    I am reading a electrical engineering book about digital signal processing, in the process, those Fourier transform and discrete time Fourier transform of constant and the exponential function lead to the delta function. I understand how to manipulate them formally, but I have serious trouble...
  5. V

    Adding X as an Instrument Variable in GMM Estimation

    Hello, I have a question concerning the GMM equation specification. Say we partition each day in 7 intraday intervals. We want to estimate the 7 intraday interval moments for a variable Y observed in those 7 intervals over a period of T days meaning we have t = T*7 total observations for...
  6. radou

    Understanding Generalized Coordinates in Goldstein's Classical Mechanics

    I have just started to read Goldstein's classical mechanics, and he got me a bit confused: is it correct to think of polar and spherical coordinates as of generalized coordinates? the way I got it, every coordinate system different from the standard cartesian-one is a set of generalized...
  7. B

    Generalized likelihood ratio test

    This is the problem (t for theta): X ~ Expo(t) = t * e ^ (-t * x), x>0, t >0 0 otherwise Test H0: t <= 1 vs. H1: t >1 using the generalized likelihood ratio test where you have a random sample from X {X1, X2, ... , X50} and the sum of all Xi = 35. Use alpha =...
  8. T

    Are Generalized Pell Numbers Already Known and What are Their Properties?

    Hi, In the following document: Generalized Pell Numbers, I've defined what I call "Generalized Pell Numbers". They provide a way for computing: 1+\sqrt[m]{m}. I'd like to know if these numbers are already known or not, and if someone knows about other properties they have or if someone is...
  9. S

    The generalized Lambert W-function

    Can anyone here help me derive the generalized Lambert function? I'm working on a solution for an ODE from the homework group which involves this function. This is what I have so far: The W-function is defined as the inverse of the following: f(x)=xe^x=y then...
  10. P

    Generalized Lorentz Transformation for an Accelerated Frame of Reference

    This paper by Robert A. Nelson derives an exact, explicit coordinate transformation between an inertial frame of reference and a frame of reference having an arbitrary time-dependent, nongravitational acceleration and an arbitrary time-dependent angular velocity...
  11. F

    Lagrangian mechanics (problem with generalized coordinates)

    Dear friends, Well, I’ve got a problem to solve but I’m not going to ask you to do it for me. Instead, what I need is an explanation of what I am doing wrong. The problem is as follows: we have a rod of mass m and length l hanging of a rail (don’t know how to call it). It moves as the...
  12. R

    Generalized Pell Equation and Primes

    I have a conjecture that the equation X^2 - 2Y^2 = P has solutions in odd integers if P is a prime of the form 8*N+1. I know of a paper that requires one to find Q such that Q^2 = 2 mod P inorder to solve these equations using continued fractions. To get to first base in proving my conjecture...
  13. M

    Generalized Triangle Inequality

    Can someone please give me a definition of the generalized triangle inequality?
  14. J

    Need help with a generalized formula for factoring

    Hello, I’ve come upon a problem in my transitional mathematics course, wherein I need to prove a generalized formula for the factorization of X^n-Y^n where the first term is (X-Y). I have deduced a formula making use of the summation of X*Y over the range of positive integers ending at n...
  15. humanino

    Generalized parton distributions

    In some of my previous posts, I referred to GPDs, a recently developped formalism describing hadronic states. I am afraid that this formalism remains largely ignored partly because few introduction review are only recently available. See for instance : Markus Diehl, Phys.Rept. 388 (2003)...
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