Johnson Scheme
In mathematics, the Johnson scheme, named after Selmer M. Johnson, is also known as the triangular association scheme. It consists of the set of all binary vectors ''X'' of length ''ℓ'' and weight ''n'', such that v=\left, X\=\binom.F. J. MacWilliams and N. J. A. Sloane, ''The Theory of Error-Correcting Codes'', Elsevier, New York, 1978. Two vectors ''x'', ''y'' ∈ ''X'' are called ''i''th associates if dist(''x'', ''y'') = 2''i'' for ''i'' = 0, 1, ..., ''n''. The eigenvalues In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ... are given by : p_\left(k\right)=E_\left(k\right), : q_\left(i\right)=\fracE_\left(k\right), where : \mu_=\frac\binom, and ''E''''k''(''x'') is an Eberlein polynomial defined by : ...
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Selmer M
Selmer may refer to: * Selmer (surname) * Selmer (given name) * Selmer, Tennessee, United States, a town * Selmer group, a group constructed from an isogeny of abelian varieties See also * Conn-Selmer, a manufacturer and distributor of musical instruments * Henri Selmer Paris, a musical instrument manufacturer, associated with Conn-Selmer * Semler Semler is an occupational surname derived from the occupation of baker who bakes '':de:Brötchen, semmels'', i.e., white bread rolls. Notable people with the surname include: *Andrée Sfeir-Semler (born 1953), art historian and gallery owner *Augu ...

Association Scheme
The theory of association schemes arose in statistics, in the theory of design of experiments, experimental design for the analysis of variance. In mathematics, association schemes belong to both algebra and combinatorics. In algebraic combinatorics, association schemes provide a unified approach to many topics, for example combinatorial designs and coding theory, the theory of error-correcting codes. In algebra, the theory of association schemes generalizes the group character, character theory of group representation, linear representations of groups. Definition An ''n''-class association scheme consists of a Set (mathematics), set ''X'' together with a partition of a set, partition ''S'' of ''X'' × ''X'' into ''n'' + 1 binary relations, ''R''0, ''R''1, ..., ''R''''n'' which satisfy: *R_ = \; it is called the identity relation. *Defining R^* := \, if ''R'' in ''S'', then ''R*'' in ''S''. *If (x,y) \in R_, the number of z \in X such that (x,z) \in R_ and (z,y) \ ...
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Eigenvalues
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a constant factor \lambda when the linear transformation is applied to it: T\mathbf v=\lambda \mathbf v. The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor \lambda (possibly a negative or complex number). Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. A linear transformation's eigenvectors are those vectors that are only stretched or shrunk, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or shrunk. If the eigenvalue is negative, the eigenvector's direction is reversed. The ...
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