where γ < sup >*</ sup > is the conjugate of γ, and the product is Clifford multiplication.

where is the commutator, is exterior derivative, is the dual 1 form corresponding to under the metric ( i. e. with lowered indices ) and is Clifford multiplication.

This is not obvious from the right-hand side of Kosmann's local expression, as the right-hand side seems to depend on the metric through the spin connection ( covariant derivative ), the dualisation of vector fields ( lowering of the indices ) and the Clifford multiplication on the spinor bundle.

: for all tangent vectors X, where is the spinor covariant derivative, is Clifford multiplication and is a constant, called the Killing number.

This article uses the (+) sign convention for Clifford multiplication so that

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

John Fogerty, Doug Clifford, and Stu Cook ( all born 1945 ) met at Portola Junior High School in El Cerrito, California and began playing instrumentals and " juke box standards " together under the name The Blue Velvets.

Junior high buddies Doug Clifford and Stu Cook continued to work together following the demise of Creedence Clearwater Revival both as session players and members of the Don Harrison Band.

But Arlington's support of the court policy was entirely subordinate to personal interests ; and after the appointment of Clifford in November 1672 to the treasurership, his jealousy and mortification, together with his alarm at the violent opposition aroused in parliament, caused him to veer over to the other side.

He was joined shortly thereafter ( June 1946 ) by Clifford Shull, and together they established the basic principles of the technique, and applied it successfully to many different materials, addressing problems like the structure of ice and the microscopic arrangements of magnetic moments in materials.

1598-The English attacked a second time San Juan, this time led by George Clifford who wanted to clean England's honor which had been " dirtied " by Sir Francis Drake's defeat, he organized the greatest squadron ever put together by the English against Spain.

* The Real World Muppets-A spoof of The Real World that follows Clifford, Rizzo, Bobo, Bill the Bubble Guy, and Darci living together in one house.

The two work together at a partner's desk, Sidney suffering from writer's block but Clifford busily typing page after page of a new play that he keeps suspiciously under lock and key.

Leiserson coauthored the standard algorithms textbook Introduction to Algorithms together with Thomas H. Cormen, Ronald Rivest, and Clifford Stein.

It is also mentioned that he and Clifford are secretly playing the accordion and the ukulele together.

During the next decade ` Abdu ' l-Bahá would be in constant communication with Bahá ' ís around the world, helping them to teach the religion ; the group included May Ellis Bolles in Paris, Englishman Thomas Breakwell, American Herbert Hopper, French Hippolyte Dreyfus, Susan Moody, Lua Getsinger, and American Laura Clifford Barney.

However, the Cabal Ministry they formed can hardly be seen as such ; the Scot Lauderdale was not much involved in English governance at all, while the Catholic ministers of the Cabal ( Clifford and Arlington ) were never much in sympathy with the Protestants ( Buckingham and Ashley ).

He had a small, basic golf facility installed at Camp David, and became close friends with the Augusta National Chairman Clifford Roberts, inviting Roberts to stay at the White House on several occasions ; Roberts, an investment broker, also handled the Eisenhower family's investments.

By late 1166, and the birth of her final child, however, Henry's notorious affair with Rosamund Clifford had become known, and her marriage to Henry appears to have become terminally strained.

A geometric algebra is the Clifford algebra of a vector space over the field of real numbers endowed with a quadratic form.

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 case could be solved when Clifford Stoll, a scientist working as a system administrator, found ways to log the attacks and to trace them back ( with the help of many others ).

Money-Coutts commissioned and provided him with librettos for the opera Henry Clifford and for a projected trilogy of Arthurian operas.

A widely related story, attributed to Richard ( Prophet ) Jennings was that Davis, while in Detroit playing at the Blue Bird club as a guest soloist in Billy Mitchell's house band along with Tommy Flanagan, Elvin Jones, Betty Carter, Yusef Lateef, Barry Harris, Thad Jones, Curtis Fuller and Donald Byrd stumbled into Baker's Keyboard Lounge out of the rain, soaking wet and carrying his trumpet in a paper bag under his coat, walked to the bandstand and interrupted Max Roach and Clifford Brown in the midst of performing Sweet Georgia Brown by beginning to play My Funny Valentine, and then, after finishing the song, stumbled back into the rainy night.

As Mitchell was about to graduate from the Washington Seminary in 1918, she met and fell in love with another Harvard student, a young army lieutenant, Clifford West Henry, who was chief bayonet instructor at Camp Gordon from May 10 until the time he set sail for France on July 17.

The anthropologist Clifford Geertz defined religion as a " system of symbols which acts to establish powerful, pervasive, and long-lasting moods and motivations in men by formulating conceptions of a general order of existence and clothing these conceptions with such an aura of factuality that the moods and motivations seem uniquely realistic.

* 1946 – Clark Clifford and George Elsey, military advisers to U. S. President Harry S. Truman, present him with a top-secret report on the Soviet Union that first recommends the containment policy.

In 1998 a team of students at Union College in Schenectady, New York, with their political science professor Clifford Brown, undertook a project to document Northup's historic narrative.

Specifically, spinors are objects associated to a vector space with a quadratic form ( like Euclidean space with the standard metric or Minkowski space with the Lorentz metric ), and are realized as elements of representation spaces of Clifford algebras.

In detail, if V is a finite-dimensional complex vector space with nondegenerate bilinear form g, the Clifford algebra is the algebra generated by V along with the anticommutation relation.

The Lie algebra is embedded as a Lie subalgebra in equipped with the Clifford algebra commutator as Lie bracket.

In general, a quick check of the Clifford relations reveals that even-graded elements conjugate-commute with ordinary vectors:

In this way, the spinor representations are identified with certain subspaces of the Clifford algebra itself.

After the cartoonist Clifford K. Berryman illustrated the President with a bear, a toy maker heard the story and asked Roosevelt if he could use his name on a toy bear.

Furthermore, a group impersonating Fleetwood Mac ( which later became Stretch ) was touring the United States with encouragement from the band's manager, Clifford Davis.

* Henry II of England begins an affair with Rosamund Clifford.

These are the defining relations of a Clifford algebra over a pseudo-orthogonal 4-d space with metric signature (+−−−).

Dr. Clifford would not identify the ranch, calling that " privileged information.

The definition of a Clifford algebra endows it with more structure than a " bare " K-algebra: specifically it has a designated or privileged subspace that is isomorphic to V. Such a subspace cannot in general be uniquely determined given only a K-algebra isomorphic to the Clifford algebra.

The second approach is to construct a vector space using a distinguished subspace of V, and then specify the action of the Clifford algebra externally to that vector space.

Sandman said: `` The announcement that Sen. Clifford Case Aj, has decided to spend all his available time campaigning for Mr. Mitchell is a dead giveaway.

This expression is related to the development of Bessel functions in terms of the Bessel – Clifford function.

He used this rather disparaging term in his 1830 novel Paul Clifford: He is certainly a man who bathes and ‘ lives cleanly ’, ( two especial charges preferred against him by Messrs. the Great Unwashed ).

Bulwer-Lytton's name lives on in the annual Bulwer-Lytton Fiction Contest, in which contestants think-up terrible openings for imaginary novels, inspired by the first line of his novel Paul Clifford: It was a dark and stormy night ; the rain fell in torrents — except at occasional intervals, when it was checked by a violent gust of wind which swept up the streets ( for it is in London that our scene lies ), rattling along the housetops, and fiercely agitating the scanty flame of the lamps that struggled against the darkness.

Moreover, the algebra generated by the three matrices is isomorphic to the 3-dimensional Euclidean real Clifford Algebra.

Clifford Cocks, an English mathematician working for the UK intelligence agency GCHQ, described an equivalent system in an internal document in 1973, but given the relatively expensive computers needed to implement it at the time, it was mostly considered a curiosity and, as far as is publicly known, was never deployed.

He was apparently of above average height: according to Clifford Brewer he was but his remains have been lost since at least the French Revolution, and his exact height is unknown.

The contemporary historians Clifford Brown and Carol Wilson believe it is likely that he died of natural causes.

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.

It therefore has a unique irreducible representation ( also called simple Clifford module ), commonly denoted by Δ, whose dimension is 2 < sup > k </ sup >.

Irreducible representations over the reals in the case when V is a real vector space are much more intricate, and the reader is referred to the Clifford algebra article for more details.

The most typical type of spinor, the Dirac spinor, is an element of the fundamental representation of the complexified Clifford algebra, into which the spin group Spin ( p, q ) may be embedded.

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.

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