The Clifford algebra Cℓ < sub > n </ sub >( C ) is algebraically isomorphic to the algebra of complex matrices, if is even ; or the algebra of two copies of the matrices, if is odd.

In 1966 / 1967, David Hestenes replaced spinor spaces by the even subalgebra Cℓ < sup >+</ sup >< sub > 1, 3 </ sub > of the Dirac algebra Cℓ < sub > 1, 3 </ sub >.

The Clifford algebra Cℓ < sub > 2, 0 </ sub >( R ) is built up from a basis of one unit scalar, 1, two orthogonal unit vectors, σ < sub > 1 </ sub > and σ < sub > 2 </ sub >, and one unit pseudoscalar.

The even subalgebra Cℓ < sup > 0 </ sup >< sub > 2, 0 </ sub >( R ), spanned by even-graded basis elements of Cℓ < sub > 2, 0 </ sub >( R ), determines the space of spinors via its representations.

As a real algebra, Cℓ < sup > 0 </ sup >< sub > 2, 0 </ sub >( R ) is isomorphic to field of complex numbers C. As a result, it admits a conjugation operation ( analogous to complex conjugation ), sometimes called the reverse of a Clifford element, defined by

The action of an even Clifford element on vectors, regarded as 1-graded elements of Cℓ < sub > 2, 0 </ sub >, is determined by mapping a general vector to the vector

The Clifford algebra Cℓ < sub > 3, 0 </ sub >( R ) is built up from a basis of one unit scalar, 1, three orthogonal unit vectors, σ < sub > 1 </ sub >, σ < sub > 2 </ sub > and σ < sub > 3 </ sub >, the three unit bivectors σ < sub > 1 </ sub > σ < sub > 2 </ sub >, σ < sub > 2 </ sub > σ < sub > 3 </ sub >, σ < sub > 3 </ sub > σ < sub > 1 </ sub > and the pseudoscalar i

In mathematics, the split-complex numbers are members of the Clifford algebra Cℓ < sub > 1, 0 </ sub >( R ) = Cℓ < sup > 0 </ sup >< sub > 1, 1 </ sub >( R ) ( the superscript 0 indicating the even subalgebra ).

Cℓ < sup > 0 </ sup >< sub > 2, 0 </ sub >( R ).

We will denote the Clifford algebra on C < sup > n </ sup > with the standard quadratic form by Cℓ < sub > n </ sub >( C ).

The algebras Cℓ < sub > n </ sub >< sup >±</ sup >( C ) are just the positive and negative eigenspaces of ω and the P < sub >±</ sub > are just the projection operators.

These two isomorphic algebras are each central simple and so, again, isomorphic to a matrix algebra over C. The sizes of the matrices can be determined from the fact that the dimension of Cℓ < sub > n </ sub >( C ) is 2 < sup > n </ sup >.

The Clifford algebra Cℓ < sub > 3 </ sub > has a faithful representation, generated by Pauli matrices, on the spin representation C < sup > 2 </ sup >; further, Cℓ < sub > 3 </ sub > is isomorphic to the even subalgebra of the 3 + 1 Clifford algebra, Cℓ.

When n is even the algebra Cℓ < sub > n </ sub >( C ) is central simple and so by the Artin-Wedderburn theorem is isomorphic to a matrix algebra over C. When n is odd, the center includes not only the scalars but the pseudoscalars ( degree n elements ) as well.

These are subspaces of the Clifford algebra of the form Cℓ ( V, g ) ω, admitting the evident action of Cℓ ( V, g ) by left-multiplication: c: xω → cxω.

Cℓ ( V, g ) ω is a minimal left ideal.

In this construction the representation of the Clifford algebra Cℓ ( V, g ), the Lie algebra so ( V, g ), and the Spin group Spin ( V, g ), all depend on the choice of the orthonormal basis and the choice of the gamma matrices.

One approach seeks to identify the minimal ideals for the left action of Cℓ ( V, g ) on itself.

Given a finite dimensional real quadratic space with quadratic form, the geometric algebra for this quadratic space is the Clifford algebra Cℓ ( V, Q ).

The Clifford algebra Cℓ ( V, Q ) is the " freest " algebra generated by V subject to the condition < ref > Mathematicians who work with real Clifford algebras and prefer positive definite quadratic forms ( especially those working in index theory ) sometimes use a different choice of sign in the fundamental Clifford identity.

In fact, if then the Clifford algebra Cℓ ( V, Q ) is just the exterior algebra Λ ( V ).

A Clifford algebra Cℓ ( V, Q ) is a unital associative algebra over K together with a linear map satisfying for all defined by the following universal property: Given any associative algebra A over K and any linear map such that

It is then straightforward to show that Cℓ ( V, Q ) contains V and satisfies the above universal property, so that Cℓ is unique up to a unique isomorphism ; thus one speaks of " the " Clifford algebra Cℓ ( V, Q ).

The universal characterization of the Clifford algebra shows that the construction of Cℓ ( V, Q ) is functorial in nature.

Below it is shown that distinct Clifford algebras may be algebra isomorphic, as is the case of Cℓ < sub > 2, 0 </ sub >( R ) and Cℓ < sub > 1, 1 </ sub >( R ) which are both isomorphic to the ring of two-by-two matrices over the real numbers.

In physics, the algebra of physical space ( APS ) is the use of the Clifford or geometric algebra Cℓ < sub > 3 </ sub > of the three-dimensional Euclidean space as a model for ( 3 + 1 )- dimensional space-time, representing a point in space-time via a paravector ( 3-dimensional vector plus a 1-dimensional scalar ).

and define Cℓ ( V, Q ) as the quotient algebra

Since W ′ is isotropic, multiplication of elements of W ′ inside Cℓ ( V, g ) is skew.

For nonzero Q there exists a canonical linear isomorphism between Λ ( V ) and Cℓ ( V, Q ) whenever the ground field K does not have characteristic two.

Namely, Cℓ can be considered as a functor from the category of vector spaces with quadratic forms ( whose morphisms are linear maps preserving the quadratic form ) to the category of associative algebras.

Because the clocks had been on average well above sea level, this meant that TAI slowed down, by about 10 < sup >− 12 </ sup >.

Algeria has always been a source of inspiration for different painters who tried to immortalize the prodigious diversity of the sites it offers and the profusion of the facets that passes its population, which offers for Orientalists between the 19 < sup > th </ sup > century and the 20 < sup > th </ sup > century, a striking inspiration for a very rich artistic creation like Eugène Delacroix with his famous painting women of Algiers in their apartment or Etienne Dinet or other painters of world fame like Pablo Picasso with his painting women of Algiers, or painters issued from the Algiers school.

* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.

η < sup > 2 </ sup > ( eta-squared ):

He also offers a conversion table ( see Cohen, 1988, p. 283 ) for eta squared ( η < sup > 2 </ sup >) where 0. 0099 constitutes a small effect, 0. 0588 a medium effect and 0. 1379 a large effect.

When there are only two means to compare, the t-test and the ANOVA F-test are equivalent ; the relation between ANOVA and t is given by F = t < sup > 2 </ sup >.

The Arrhenius definition states that acids are substances which increase the concentration of hydronium ions ( H < sub > 3 </ sub > O < sup >+</ sup >) in solution.

The reason why pHs of acids are less than 7 is that the concentration of hydronium ions is greater than 10 < sup >− 7 </ sup > moles per liter.

An Arrhenius acid is a substance that increases the concentration of the hydronium ion, H < sub > 3 </ sub > O < sup >+</ sup >, when dissolved in water.

The International Time Bureau ( BIH ) began a time scale, T < sub > m </ sub > or AM, in July 1955, using both local caesium clocks and comparisons to distant clocks using the phase of VLF radio signals.

< center ></ center >

< center >

All 128 ASCII characters, including non-printable characters ( represented by their abbreviations ). The 95 ASCII graphic characters are numbered from 20 < sub > hexadecimal | hex </ sub > to 7E < sub > hexadecimal | hex </ sub > ( decimal 32 to 126 ).

The " space " character had to come before graphics to make sorting easier, so it became position 20 < sub > hex </ sub >; for the same reason, many special signs commonly used as separators were placed before digits.

To keep options available for lower case letters and other graphics, the special and numeric codes were arranged before the letters, and the letter " A " was placed in position 41 < sub > hex </ sub > to match the draft of the corresponding British standard.

The @ symbol was not used in continental Europe and the committee expected it would be replaced by an accented À in the French variation, so the @ was placed in position 40 < sub > hex </ sub > next to the letter A.