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. L

    A question about mechanics and generalized coordinates.

    Hello, I wasn't quite sure where to make this topic, so I hope I didn't do wrong by putting it here. The question I'm having is somewhat difficult to describe and I guess it's more of a mathematical question really, but since I'm learning mechanics now and came up with it, I thought it...
  2. G

    Did I make a mistake in finding the third generalized eigenvector for A?

    I remember reading a theorem that said that for an n x n matrix A, there exists a basis of Cn consisting of generalized eigenvectors of A. For A = [1 1 1; 0 1 0; 0 0 1] (the semicolons indicate a new row so that A should be 3 x 3 with a first row consisting of all 1's and a diagonal of 1's)...
  3. kreil

    Proving Identity for Generalized Sum S(x)

    Homework Statement In order to solve the problem I am working on, I have to prove the following generalized problem, S(x)=\sum_{n=0}^{\infty} n x^n =\frac{x}{(x-1)^2} for |x|< 1 I evaluated this sum using Wolfram Alpha. Clearly it looks related to the geometric series solution, but I am...
  4. C

    Help with generalized linear model

    I have a data set {X(t) = (x(t), y(t))}_t=1,...,N and I'm interested in modelling the changes from t to t+1, using some metric d(X(t),X(t+1)) The issue is that x(t) has some dependence on y(t), and I'd like to account for this: if there is a large change in y(t) we expect there to be a...
  5. J

    Can't prove generalized De Morgan's Law

    Homework Statement Let B be a non-empty set, and supose that {Sa : a\inB} is an B- indexed family of subsets of a set S. Then we have, (\cup a\in B Sa)c = \bigcapa\in B Sac. Homework Equations The Attempt at a Solution I tried to show that the two were both subsets of each...
  6. K

    Generalized version of cannon ball problem

    For All p in Natural Number, Is \exists n , n > 1, \sum^{n}_{k=1} k^p = C^2 where C is arbitary natural number (not constant) ??
  7. L

    Deviance of Binomial generalized linear model

    The formula for the deviance of a binomial generalized linear model is: D = 2\sum[y_i \log(\frac{y_i}{\hat{y}_i})+(n_i-y_i)\log(\frac{n_i-y_i}{n_i-\hat{y}_i})]. where the responses y are Binomial(n_i, p_i), and \hat{y}_i = n_i\hat{p}_i. The second log in that equation is undefined when...
  8. JK423

    Generalized momentum - Physical meaning

    We know the the generalized momentum is P=mu + qA Can someone explain to me, what's the physical meaning of the quantity 'qA'? The particle's momentum that we measure is just 'mu', right?
  9. H

    Generalized Eigenspace and JOrdan Form

    Homework Statement For each linear operator T, find a basis for each generalized eigenspace of T consisting of a union of disjoint cycles of generalized eigenvectors. The find a Jordan canonical form J of T. a) T is the linear operator on P2(R) defined by T(f(x)) = 2f(x) - f '(x)Homework...
  10. W

    WCFSGS'S Version: Generalized Second Law of Thermal Dynamics

    WCFSGS' Version: Generalized Second Law of Thermodynamics We have known that there has been some generalization to the second law of thermodynamics. We like to present here the Version of WCFSGS about this generalization. At this moment, we are not quite sure if our version is different from...
  11. U

    Generalized uncertainty principle

    So I'm working on the proof of the generalized uncertainty principle and there is a step that I'm not fully understanding. There is a line were it says that for any complex number we can write the inequality as [Re(z)]^2 + [Im(z)]^2 >/ [Im(z)]^2. why are we able to get rid of the real part on...
  12. S

    Generalized Complex Circle: Finding the Radius and Center

    Homework Statement Let have the problem to find the complex generalized cirlce of radius r Homework Equations |z-c|^2 = r^2 The Attempt at a Solution hvor r is the radius and c the center.. by expanding the above z\overline{z} - z\overline{c} - \overline{z}c +...
  13. K

    Generalized Cantor Set

    "Given (rn), rn E (0,1), define a generalized Cantor set E by removing the middle r1 fraction of an interval, then remove the middle r2 fraction of the remaining 2 intervals, etc. Start with [0,1]. Take rn=1/5n. Then the material removed at the n-th stage has length < 1/5n, so the total...
  14. A

    Rigid bodies: generalized forces according to the Euler-Lagrange formalism

    I am trying to derive the dynamic equations of an aerial vehicle with 6 degrees of freedom (a quadrotor to be precise). I am using - two coordinate systems: the Earth frame and the body frame; - the Euler-Lagrange formalism: generalized coordinates {x,y,z,phi,theta,psi}, respectively, the...
  15. P

    Generalized Laplace transform

    "Generalized" Laplace transform Hello, I'm having trouble proving injectivity of what might be called a "generalized" Laplace transform (not the one by Varma). Let f be a rational function and C be a fixed closed contour in the complex plane, (such that C contains not pole of f): The operator...
  16. Z

    Generalized ZETA function

    given the function Z(s)= \prod _{k=0}^{\infty}\zeta (s+k) with \zeta (s) being the Riemann Zeta function the idea is if ALL the roots have real part (i mean Riemann Hypothesis) is correct, then what would happen with the roots of Z(s) ?? what would be the Functional equation relating...
  17. J

    Coupled Generalized Momentum & Hamiltonian Mechanics

    I have a brilliantly engineered system of a bead-on-a-circular-loop (mass=m) rigidly attached to a massive block (mass=M) on one side and a spring on the other. The spring motion is constrained to be in x-direction only, while the bead is free to move on the wire anyway it wants to (no \phi...
  18. Born2bwire

    Solving Gen. Eigen Probs w/ Real Sym Indefinite A & Definite B

    I have a system that ideally creates a real symmetric negative definite matrix. However, due to the implementation of the algorithm and/or finite-precision of floating point, the matrix comes out indefinite. For example, in a 2700 square matrix, four eigenvalues are positive, the rest are...
  19. G

    Generalized Gaussian integers

    Homework Statement If \omega is and nth root of unity, define Z[\omega], the set of generalized Gaussian integers to be the set of all complex numbers of the form m_{0}+m_{1}\omega+m_{2}\omega^{2}+...+m_{n-1}\omega^{n-1} where n and m_{i} are integers. Prove that the products of generalized...
  20. marcus

    Lewandowski et al's generalized spinfoams

    http://arxiv.org/pdf/0909.0939 MTd2 spotted this paper back when it came out on 6 September and posted a reminder on another thread suggesting that we should discuss it. We should. It looks like a paper that is both important in the development of LQG and also exceptionally clear and...
  21. S

    Vector proof of generalized parallelogram law for quadrilaterals

    Homework Statement The following theorem in geometry suggests a vector identity involving three vectors A, B, and C. Guess the identity and prove that it holds for vectors in Vn. This provides a proof of the theorem by vector methods. "The sum of the squares of the sides of any...
  22. F

    GR from generalized inner product?

    The kinetic energy of a free particle is sometimes viewed geometrically as the inner product of velocity with momentum, where velocity is seen as a vector in the tangent space to the configuration space of a particle, and momentum is viewed as a vector in the tangent space of the phase space of...
  23. T

    How Do You Choose Generalized Coordinates for a Timber Beams and Springs System?

    Homework Statement We two beams of timber, of identical length joined together at the middle, perpendicular forming a "X" in a sense. Underneath the end of each beam we have a spring attached, thus 4 in total. 3 have identical spring constants and the forth is greater than the other 3. We...
  24. Z

    Could GR generalized to non-integer dimension?

    could GR generalized to non-integer dimension?? let us suppose that the dimension of space time is NOT an integer then , could we generalize GR to obtain an expressions of Tensor, Covariant derivatives... in arbitrary dimensions ?? let us say 4.567898.. or similar, i mean GR in non integer...
  25. P

    Generalized coordinates of a couple harmonic oscillator

    Homework Statement Suppose there is a square plate, of side a and mass M, whose corners are supported by massless springs, with spring constants K, K, K, and k <= K (the faulty one). The springs are confined so that they stretch and compress vertically, with unperturbed length L. The...
  26. quasar987

    The generalized rank-nullity theorem

    If one has a short exact sequence 0-->A-->B-->C-->0 of finitely generated abelian groups, how does one show that rank(B)=rank(A)+rank(C) ? We have that A embeds in B and C is isomorphic to B/A. The natural thing to try to use I think is the uniqueness of the decomposition of a finitely...
  27. T

    Are Generalized Coordinates Necessary for Simplifying Complex Systems?

    Are generalized coordinates, as used in Legrangian mechanics, just a different name for coordinates on a chart in a manifold? The idea of generalized coordinates never quite "clicked" with me, but after reading a paper today, it seems that they are just an implicit way of working with manifolds...
  28. N

    A Generalized trigonometric identity for Cos(x_1++x_n)?

    Hi Everyone, Do there exist any explicit formula for Cos(x_1+x_2+...+x_n) as a sum of products of Sin(x_i) & Cos(x_i)? Or we need to expand using Cos(A+B), Sin(A+B) again & again? If it exists then what is about Sin(x_1+x_2+...+x_n)? [It is understood that there will be 2^(n-1) number of...
  29. D

    How Can I Derive a Set of 3x3 Matrices Analogous to the Pauli Matrices?

    If I am to try and derive a set of 3x3 matrices analogous to the Pauli matrices, how would I go about doing this? I want to find the basis for all complex 3x3 matrices (analagous to the 3 Pauli matrices and the identity matrix for all complex 2x2 matrices) to expand a complicated matrix into so...
  30. T

    Understanding the Generalized Binomial Formula: Exploring the Role of n+1/2

    I stuck at the second "="...i know it goes like this because the formula...i just someone explain to me why it works like that. thank you soooo much!
  31. S

    Determinant formula in monomials - can it be generalized?

    Determinant formula in monomials -- can it be generalized? I ran across this question in one of the Usenet groups (fr.sci.maths), and after doing a double take and realizing what was actually being asked I realized I don't know the answer, and after searching a bit I haven't turned it up, so I...
  32. D

    Why do we need generalized eigenvectors for matrices with repeated eigenvalues?

    So I understand that if an nxn matrix has n distinct eigenvalues that you can diagonalize the matrix into S\LambdaS^{-1}. This is important because then this form has lots of good properties (easy to raise to powers, etc) So when there are not n distinct eigenvalues, you then solve...
  33. M

    Generalized eigenspace invariant?

    Hey, Is the generalized eigenspace invariant under the operator T? Let T be finite dimensional Linear operator on C(complex numbers). My understanding of the Generalized Eigenspace for the eigenvalue y is: "All v in V such that there exists a j>=1, (T-yIdenitity)^j (v) = 0." plus 0. thanks
  34. F

    Development of generalized metric

    We all know how the metric of GR is a generalization of the flat Minkowski spacetime metric. But I wonder if the SR metric is generalized from the kinetic energy term of Newtonian physics. There the kinetic energy is (1/2)m*v^2=(1/2)m*dx*dx. If the mass/2 plays the role of the metric, then this...
  35. G

    Derivate of generalized function

    Hi, I have to show that if the derivate f'(x) of a generalized function f(x) is defined by the sequence f'_n(x) where f(x) is defined f_n(x)[\tex] then \int_{-\infty}^{\infty}f'(x)F(x) dx = - \int_{-\infty}^{\infty}f(x)F'(x) dx I use the limits for generalized functions and...
  36. F

    MATLAB Solving Generalized Crank-Nicolson Problem with MATLAB

    can anybody help me about the attached problem about generalized Crank-Nicolson scheme. I need MATLAB code. How can i write a tri-diagonal solver for this problem?
  37. gel

    Can Quantum Mechanics Be Generalized Beyond Bell's Theorem?

    I have a couple of questions on what is possible within quantum mechanics, and the physical justifications (if any). My question is a bit subtle and tricky to explain, but I'll try to explain as well as I can. Hopefully someone here can spread a bit of light on this. This problem first puzzled...
  38. T

    What are generalized probability functions?

    Okay I posted a question a few days ago about Luders Rule but didn't get any responses. I've studied this stuff in Hughes (The Structure and Interpretation of Quantum Mechanics) a bit more so I can ask a slightly different question. Hughes says you can create a "generalized probability...
  39. R

    Geometric interpretation of Generalized MVT

    Homework Statement I am trying to see the geometric interpretation of the generalized MVT. It is not a homework problem, but would like to know how to interpret the equation Homework Equations [f(b)- f(a)]* g'(x) = [g(b)- g(a)]* f'(x) The Attempt at a Solution On...
  40. E

    Generalized coordinates: Understanding Kinetic Energy

    Homework Statement When I use generalized coordinates how do I know that I can add the kinetic contributions from each to get the total kinetic energy? How do I know that you are not "counting the same KE twice"? e.g. if you have a double pendulum how do you know that you can just add the...
  41. P

    Lagrangian depend upon upon my choice of generalized coordinates?

    does the lagrangian depend upon upon my choice of generalized coordinates
  42. strangerep

    Gauge Transformations and (Generalized) Bogoliubov Transformations.

    I've been discussing some things with Samalkhaiat over in the conformal field theory tutorial. A part of that conversation (indicated by the new title) was drifting away from CFT matters, so we both thought it was better to move it into the Quantum Physics forum, to minimize pollution of the...
  43. L

    Generalized PDEs with an apparent contradiction

    I have a question which has perplexed me for a time and thought maybe someone here would have some insight that might prove useful. My research involves a generalization of first order partial differential equations. The simplest case can be defined in the following manner: Let V be an arbitrary...
  44. D

    How Does Generalized Wick's Theorem Evaluate Multi-Operator Contractions?

    I have the following contour integral form of Wick's theorem (C indicating contraction): C[A(z):BC:(w)]=\frac{1}{2 \pi i} \int _w \frac{dx}{x-w} C[A(z)B(x)]C(w) + B(x)C[A(z)C(w)] Does anybody know how to evaluate contractions like C[:AB:(z)C(w)]?
  45. S

    Generalized jordan curve theorem

    Is anyone here familiar with the proof (using homology) of the generalized Jordan curve theorem, that a subspace of S^n homeomorphic to S^(n-1) divides it into two components? It can be found on page 169 of Hatcher's algebraic topology book, which can be downloaded from...
  46. Q

    Bases of Generalized Eigenvectors

    Hi all. So I'm a bit confused about finding a basis of generalized eigenvectors for an operator that is not diagonalizable. I have some "steps" in mind, but maybe someone can help me out here: 1) Find the eigenvalues of the matrix/operator 2) Find the eigenspaces corresponding to each...
  47. S

    Verifying that the Euler-Lagrange equation uses generalized coordinates

    This is a question that I'm asking myself for my own understanding, not a homework question. I realize that in most derivations of the Euler-Lagrange equations the coordinate system is assumed to be general. However, just to make sure, I want to apply the "brute force" method (as Shankar...
  48. Pythagorean

    Generalized Coordinates: Double Pendulum

    Homework Statement Standard double pendulum setup. A string with mass, connected to a string with a mass, mounted to the ceiling. Given is m1,m2,l1,l2 a) choose a suitable set of coordinates and write a lagrangian function, assuming it swings in a single vertical plane (I did this, using L...
  49. C

    Generalized solutions for the smallest Euclidean norm

    Hi folks, I have to find the generalized solution for the following Ax=y : [1 2 3 4;0 -1 -2 2;0 0 0 1]x=[3;2;1] The rank of A is 3 so there is one nullity so the generalized solution is: X= x+alpha.n (where alpha is a constant , and n represents the nullity) I found the...
  50. S

    Generalized Poisson brackets

    Hi. I've been wondering about the following and haven't made much progress on it. (Note that I've also posted this in the relativity section since the ultimate aim of this is to apply it to canonical relativity but since this is essentially a question about tensors I thought I'd put a copy here...
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