The generalization of electric potential to this case is described below.

In this way, chemical potential is a generalization of " potentials " in physics such as gravitational potential.

If magnetic monopoles are included into the theory, then a generalization to include magnetic charge is obtained by replacing by in the metric and including the term in the electromagnetic potential.

A generalization of the concept of a transversal would be a set that just has a non-empty intersection with each member of C. An example of this would be a Bernstein set, which is defined as a set that has a non-empty intersection with each set of C, but contains no set of C, where C is the collection of all perfect sets of a topological Polish space.

The B-tree is a generalization of a binary search tree in that a node can have more than two children.

The general theory for Lie groups deals with semidirect products of the two types, by means of general results called Mackey theory, which is a generalization of Wigner's classification methods.

Of any two related concepts, such as A and B, A is considered a " generalization " of concept B if and only if:

The generalization to more than two goods consists of modelling y as a composite good.

The carpet is a generalization of the Cantor set to two dimensions ( another is Cantor dust ).

In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other.

Much of what can be said about the Bernoulli process can also be generalized to more than two outcomes ( such as the process for a six-sided die ); this generalization is known as the Bernoulli scheme.

Assume two matrices are to be multiplied ( the generalization to any number is discussed below ).

The current algorithm with the lowest known exponent k is a generalization of the Coppersmith – Winograd algorithm that has an asymptotic complexity of O ( n < sup > 2. 3727 </ sup >) thanks to Vassilevska Williams .< ref > The original algorithm was presented by Don Coppersmith and Shmuel Winograd in 1990, has an asymptotic complexity of O ( n < sup > 2. 376 </ sup >).</ ref > This algorithm, and the Coppersmith-Winograd algorithm on which it is based, are similar to Strassen's algorithm: a way is devised for multiplying two k × k-matrices with fewer than k < sup > 3 </ sup > multiplications, and this technique is applied recursively.

There are two variations of this generalization.

Affine geometry can be viewed as the geometry of affine space, of a given dimension n, coordinatized over a field K. There is also ( in two dimensions ) a combinatorial generalization of coordinatized affine space, as developed in synthetic finite geometry.

However, bisexuals are also stigmatized in other ways: two classifications of negative stereotypes about them center on the belief that bisexuality does not exist and the generalization that bisexual people are promiscuous.

These two directions of generalization can be combined in the inverse function theorem for Banach manifolds.

It is a generalization of the fermion and boson concept, in which the operation of exchanging two identical particles may cause a global phase shift but cannot affect observables.

An algebraic structure is an entropic algebra if every two operations satisfy a generalization of the medial identity.

At the beginning is the well-known generalization of Euclid I. 47, then follow various theorems on the circle, leading up to the problem of the construction of a circle which shall circumscribe three given circles, touching each other two and two.

* Centerpoint ( geometry ), a generalization of the median to two or more dimensions

Dualism is the interdependence between opposing elements, the generalization that two opposing-complementary forces are found in all things.

In other words, the higher-dimensional, multi-player, n-in-a-row generalization of game of tic-tac-toe cannot end in a draw, no matter how large n is, no matter how many people c are playing, and no matter which player plays each turn, provided only that it is played on a board of sufficiently high dimension H. By a standard strategy stealing argument, one can thus conclude that if two players alternate, then the first player has a winning strategy when H is sufficiently large, though no constructive algorithm for obtaining this strategy is known.

This theorem is a generalization of Pappus's ( hexagon ) theorem – Pappus's theorem is the special case of a degenerate conic of two lines.

Moreover, it has been proved that solving the new proposed optimization problem ( GL1-SVM ) gives smaller error penalty and enlarges the margin between two support vector hyper-planes, thus possibly gives better generalization capability of SVM than solving the traditional L1-norm SVM.

In geometry, polytope means the generalization to any dimension of the sequence: polygon in two dimensions, polyhedron in three dimensions, and polychoron in four dimensions.

Kleinberg realized that this generalization implies two different classes of important web pages, which he called " hubs " and " authorities ".

A topological generalization of Radon's theorem states that, if ƒ is any continuous function from a ( d + 1 )- dimensional simplex to d-dimensional space, then the simplex has two disjoint faces whose images under ƒ are not disjoint.

A similar situation exists between the functional classes FP and # P. By a generalization of Ladner's theorem, there are also problems in neither FP nor # P-complete as long as FP ≠ # P. As in the decision case, a problem in the # CSP is defined by a set of relations.

* Enterprise data model: this type of data model according to Simsion ( 2005 ) can contain up to 50 to 200 entity classes, which results from specific " high level of generalization in data modeling ".

There is a generalization of the theory of Chern classes, where ordinary cohomology is replaced with a generalized cohomology theory.

Smale's classification of immersions of spheres as the homotopy groups of Stiefel manifolds, and Hirsch's generalization of this to immersions of manifolds being classified as homotopy classes of maps of frame bundles are much further-reaching generalizations, and much more involved, but similar in principle – immersion requires the derivative to have rank k, which requires the partial derivatives in each direction to not vanish and to be linearly independent, and the resulting analog of the Gauss map is a map to the Stiefel manifold, or more generally between frame bundles.

When the logical system is equipped with this order, it is possible to restrict the scope of a generalization such as " All ravens are black " so that it applies to ravens only and not to non-black things, since the order privileges ravens over non-black things.

Leibniz picks up on the generalization used by Locke and adopts a less rigid approach: clearly there is no perfect correspondence between words and things, but neither is the relationship completely arbitrary, although he seems vague about what that relationship might be.

We can ask, after Stake, ‘ does the story have ‘ naturalistic generalization ’?’ meaning that it brings ‘ felt ’ news from one world to another and provides opportunities for the reader to have vicarious experience of the things told.

: If we are justified in assuming that the scope of the generalization ' All ravens are black ' can be restricted to ravens, then this means that we have some outside information which we can rely on concerning the factual situation.

By generalization it means the instantaneous spreading of an idea or ability to the remainder of a population once a certain portion of that population has heard of the new idea or learned the new ability by some unknown process currently beyond the scope of science.

#: In the full-entry model information is stored redundantly at all relevant levels in the taxonomy, which means that it operates, if at all, with minimal generalization.

No generalization of these results to spaces of more than three dimensions has so far been found possible.

The concept of generalization has broad application in many related disciplines, sometimes having a specialized context-meaning.

Although not unchallenged, validity generalization has broad acceptance with regard to many selection instruments ( e. g. cognitive ability tests, job knowledge tests, work samples, and structured interviews ) across a broad range of jobs.

There are several algorithms that identify noisy training examples and removing the suspected noisy training examples prior to training has decreased generalization error with statistical significance.

It is a generalization of the statement that if the span of n vectors has dimension p, then p of those vectors span the space: one can choose a spanning set that is a subset of the vectors.

where is the initial probability that has the predicate ; is the number of objects which have been examined ( according to the available evidence ); is the number of examined objects which turned out to have the predicate, and is a constant which measures resistance to generalization.

It has also been proposed that the poverty of the stimulus problem can be largely avoided, if we assume that children employ similarity-based generalization strategies in language learning, generalizing about the usage of new words from similar words that they already know how to use.

Kripke semantics has a straightforward generalization to logics with more than one modality.

Fermat's little theorem also has a nice generalization in finite fields.

Another generalization of curvature relies on the ability to compare a curved space with another space that has constant curvature.

Intuitively, a good separation is achieved by the hyperplane that has the largest distance to the nearest training data point of any class ( so-called functional margin ), since in general the larger the margin the lower the generalization error of the classifier.

Orthogonality is a concept that has undergone a huge amount of generalization in mathematics, science, and engineering over the many centuries of time since the Ancient Babylonians, Ancient Egyptians, and Ancient Greeks.

Much of this generalization has happened since the beginning of the 16th Century, and much of it has involved the concepts of mathematical functions and calculus.

Another generalization states that a faithfully flat morphism locally of finite type with X quasi-compact has a quasi-section, i. e. there exists affine and faithfully flat and quasi-finite over X together with an X-morphism.

Research has used highly selected individuals, case studies, or very small numbers of subjects, making generalization difficult.

Recursion theory in mathematical logic has traditionally focused on relative computability, a generalization of Turing computability defined using oracle Turing machines, introduced by Turing ( 1939 ).

This is because the dog has made a generalization.

Piero Sraffa ( 1960 ) has provided the complete generalization of the simultaneous method for classical and marxian analysis.

* A generalization next was introduced as the Fortuin – Kasteleyn random cluster model, which has many connections with the Ising model and other Potts models.

Stimulus generalization is said to occur if, after a particular CS has come to elicit a CR, another test stimulus elicits the same CR.

# Gauss – Bonnet theorem The integral of the Gauss curvature on a compact 2-dimensional Riemannian manifold is equal to 2πχ ( M ) where χ ( M ) denotes the Euler characteristic of M. This theorem has a generalization to any compact even-dimensional Riemannian manifold, see generalized Gauss-Bonnet theorem.

A generalization of the Bohr model discusses a hydrogen-like ions ; that is, an atom with atomic number Z that has only one electron, such as C < sup > 5 +</ sup >.