# Special case

reducesspecialspecializedspecific
In logic, especially as applied in mathematics, concept A is a special case or specialization of concept B precisely if every instance of A is also an instance of B but not vice versa, or equivalently, if B is a generalization of A.wikipedia
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### Generalization

generalizedgeneralizegeneralisation
In logic, especially as applied in mathematics, concept A is a special case or specialization of concept B precisely if every instance of A is also an instance of B but not vice versa, or equivalently, if B is a generalization of A. A limiting case is a type of special case which is arrived at by taking some aspect of the concept to the extreme of what is permitted in the general case.
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:

### Limiting case (mathematics)

limiting caselimitinglimiting cases
In logic, especially as applied in mathematics, concept A is a special case or specialization of concept B precisely if every instance of A is also an instance of B but not vice versa, or equivalently, if B is a generalization of A. A limiting case is a type of special case which is arrived at by taking some aspect of the concept to the extreme of what is permitted in the general case.
In mathematics, a limiting case of a mathematical object is a special case that arises when one or more components of the object take on their most extreme possible values.

### Beal conjecture

Beal's conjecture
Hence, Fermat's Last Theorem can be seen as a special case of the Beal conjecture restricted to x = y = z.

### Logic

logicianlogicallogics
In logic, especially as applied in mathematics, concept A is a special case or specialization of concept B precisely if every instance of A is also an instance of B but not vice versa, or equivalently, if B is a generalization of A. A limiting case is a type of special case which is arrived at by taking some aspect of the concept to the extreme of what is permitted in the general case.

### Mathematics

mathematicalmathmathematician
In logic, especially as applied in mathematics, concept A is a special case or specialization of concept B precisely if every instance of A is also an instance of B but not vice versa, or equivalently, if B is a generalization of A. A limiting case is a type of special case which is arrived at by taking some aspect of the concept to the extreme of what is permitted in the general case.

### Specialization (logic)

specialization
In logic, especially as applied in mathematics, concept A is a special case or specialization of concept B precisely if every instance of A is also an instance of B but not vice versa, or equivalently, if B is a generalization of A. A limiting case is a type of special case which is arrived at by taking some aspect of the concept to the extreme of what is permitted in the general case.

### Degeneracy (mathematics)

degeneratedegenerate casedegeneracy
A degenerate case is a special case which is in some way qualitatively different from almost all of the cases allowed.

### Fermat's Last Theorem

Fermat’s Last TheoremLast Theorema long-standing problem

### Hypergeometric function

hypergeometric serieshypergeometric differential equationhypergeometric functions
In mathematics, the Gaussian or ordinary hypergeometric function 2 F 1 (a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases.

### Isosceles triangle

isosceles{ }visoceles triangle
Sometimes it is specified as having exactly two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case.

### Change-making problem

It is a special case of the integer knapsack problem, and has applications wider than just currency.

### Entropy of mixing

entropyentropy change of solutionentropy increase of mixing
In the special case of mixing ideal materials, the final common volume is in fact the sum of the initial separate compartment volumes.

### Special relativity

special theory of relativityrelativisticspecial
The theory is "special" in that it only applies in the special case where the spacetime is "flat", i.e., the curvature of spacetime, described by the energy-momentum tensor and causing gravity, is negligible.

### Electron mobility

mobilitycarrier mobilitymobilities
Electron and hole mobility are special cases of electrical mobility of charged particles in a fluid under an applied electric field.

### Generalized beta distribution

generalized beta distribution of the first kindgeneralized beta distribution of the second kindgeneralized Beta II (GB2)
In probability and statistics, the generalized beta distribution is a continuous probability distribution with five parameters, including more than thirty named distributions as limiting or special cases.

### Circular polarization

circularly polarizedcircularly polarized lightcircular
The other special case is the easier-to-understand linear polarization.

### Law of cosines

cosine lawcosinescosine rule
, and thus the law of cosines reduces to the Pythagorean theorem:

### Leibniz formula for π

Leibniz formula for piLeibniz formula for Madhava–Leibniz series
It is also called the Madhava–Leibniz series as it is a special case of a more general series expansion for the inverse tangent function, first discovered by the Indian mathematician Madhava of Sangamagrama in the 14th century, the specific case first published by Leibniz around 1676.

### Categorical distribution

categoricalcategorical probability distributioncategorical variable
On the other hand, the categorical distribution is a special case of the multinomial distribution, in that it gives the probabilities of potential outcomes of a single drawing rather than multiple drawings.

### Toy theorem

In mathematics, a toy theorem is a simplified instance (special case) of a more general theorem, which can be useful in providing a handy representation of the general theorem, or a framework for proving the general theorem.

### Marginalism

marginalistmarginalmarginalist revolution
Under the special case in which usefulness can be quantified, the change in utility of moving from state S_1 to state S_2 is

### Maxwell's equations in curved spacetime

curved space Maxwell equationscurved spacetime Maxwell equationscurved spacetime Maxwell field equations
For these reasons, it may be useful to think of Maxwell's equations in Minkowski space as a special case, rather than Maxwell's equations in curved spacetimes as a generalization.

### Contraharmonic mean

contraharmonic mean filter
The contraharmonic mean is a special case of the Lehmer mean, L_p, where p = 2.

### Conway–Maxwell–Poisson distribution

Conway-Maxwell-Poisson distribution
It is a member of the exponential family, has the Poisson distribution and geometric distribution as special cases and the Bernoulli distribution as a limiting case.

### Poisson distribution

PoissonPoisson-distributedPoissonian