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Clifford algebra

In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, is a unital associative algebra. As K-algebras, they generalize the real numbers, complex numbers and several other hypercomplex number systems; the theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing, they are named after the English mathematician William Kingdon Clifford. The most familiar Clifford algebras, the orthogonal Clifford algebras, are referred to as Riemannian Clifford algebras, as distinct from symplectic Clifford algebras. A Clifford algebra is a unital associative algebra that contains and is generated by a vector space V over a field K, where V is equipped with a quadratic form Q: V → K; the Clifford algebra Cℓ is the "freest" algebra generated by V subject to the condition v 2 = Q 1 for all v ∈ V, where the product on the left is that of the algebra, the 1 is its multiplicative identity.

The idea of being the "freest" or "most general" algebra subject to this identity can be formally expressed through the notion of a universal property, as done below. Clifford algebras can be identified by the label Cℓp,q, indicating that the algebra is constructed from p simple basis elements with ei2 = +1, q with ei2 = −1, where R indicates that this is to be a Clifford algebra over the reals—i.e. Coefficients of elements of the algebra are to be real numbers; the free algebra generated by V may be written as the tensor algebra ⊕n≥0 V ⊗... ⊗ V, that is, the sum of the tensor product of n copies of V over all n, so a Clifford algebra would be the quotient of this tensor algebra by the two-sided ideal generated by elements of the form v ⊗ v − Q1 for all elements v ∈ V. The product induced by the tensor product in the quotient algebra is written using juxtaposition, its associativity follows from the associativity of the tensor product. The Clifford algebra has a distinguished subspace V; such a subspace cannot in general be uniquely determined given only a K-algebra isomorphic to the Clifford algebra.

If the characteristic of the ground field K is not 2 one can rewrite this fundamental identity in the form u v + v u = 2 ⟨ u, v ⟩ 1 for all u, v ∈ V, where ⟨ u, v ⟩ = 1 2 is the symmetric bilinear form associated with Q, via the polarization identity. Quadratic forms and Clifford algebras in characteristic 2 form an exceptional case. In particular, if char = 2 it is not true that a quadratic form uniquely determines a symmetric bilinear form satisfying Q = ⟨v, v⟩, nor that every quadratic form admits an orthogonal basis. Many of the statements in this article include the condition that the characteristic is not 2, are false if this condition is removed. Clifford algebras are related to exterior algebras. Indeed, if Q = 0 the Clifford algebra Cℓ is just the exterior algebra ⋀. For nonzero Q there exists a canonical linear isomorphism between ⋀ and Cℓ whenever the ground field K does not have characteristic two; that is, they are isomorphic as vector spaces, but with different multiplications. Clifford multiplication together with the distinguished subspace is richer than the exterior product since it makes use of the extra information provided by Q.

The Clifford algebra is a filtered algebra, the associated graded algebra is the exterior algebra. More Clifford algebras may be thought of as quantizations of the exterior algebra, in the same way that the Weyl algebra is a quantization of the symmetric algebra. Weyl algebras and Clifford algebras admit a further structure of a *-algebra, can be unified as and odd terms of a superalgebra, as discussed in CCR and CAR algebras. Let V be a vector space over a field K, let Q: V → K be a quadratic form on V. In most cases of interest the field K is either the field of real numbers R, or the field of complex numbers C, or a finite field. A Clifford algebra Cℓ is a pair, where A is a unital associative algebra over K and i is a linear map i: V → Cℓ satisfying i2 = Q1 for all v ∈ V, defined by the following universal property: given any unital associative algebra A over K and any linear map j: V → A such that j 2 = Q 1 A for all v ∈ V, there is a unique algebra homomorphism f: Cℓ → A such that the following diagram commutes: In characteristic not 2, the quadratic form Q may be replaced by a s

Karuma

Karuma is a settlement in the Western Region of Uganda. Karuma is in Bunyoro sub-region; the town is 2.7 kilometres, by road, north of Karuma Falls. The town is approximately 11 kilometres, by road, west of Kamdini, on the Lira–Kamdini–Karuma Road; this location is 261 kilometres north of Kampala, Uganda's capital and largest city. The coordinates of the town are 2°15'16.0"N, 32°14'37.0"E. Karuma is where the Lira–Kamdini–Karuma Road connects to the Kampala–Karuma Road and the Karuma–Olwiyo–Pakwach–Nebbi–Arua Road. Murchison Falls National Park List of cities and towns in Uganda List of roads in Uganda Uganda: Government secures $400m World Bank loan for roads Sh4.2b compensation for Karuma power project starts

Let Go of the Stone

"Let Go of the Stone" is a song written by Max D. Barnes and Max T. Barnes, recorded by American country music artist John Anderson, it was released in November 1992 as the fifth single from his album Seminole Wind. The song reached number 7 on the Billboard Hot Country Singles & Tracks chart in 1993. Deborah Evans Price, of Billboard magazine reviewed the song favorably, calling it "a thoughtfully written and exquisitely executed ballad." She goes on to say that Anderson is "at his most persuasive here." The music video was directed by Michael Salomon and premiered on CMT, The Nashville Network and GAC in late 1992. "Let Go of the Stone" debuted at number 59 on the U. S. Billboard Hot Country Singles & Tracks for the week of November 28, 1992. Lyrics of this song at MetroLyrics

Buccal branches of the facial nerve

The buccal branches of the facial nerve, are of larger size than the rest of the branches, pass horizontally forward to be distributed below the orbit and around the mouth. The superficial branches run beneath the skin and above the superficial muscles of the face, which they supply: some are distributed to the procerus, joining at the medial angle of the orbit with the infratrochlear and nasociliary branches of the ophthalmic; the deep branches pass beneath the zygomaticus and the quadratus labii superioris, supplying them and forming an infraorbital plexus with the infraorbital branch of the maxillary nerve. These branches supply the small muscles of the nose; the lower deep branches supply the buccinator and orbicularis oris, join with filaments of the buccinator branch of the mandibular nerve. The facial nerve innervates the muscles of facial expression; the buccal branch supplies these muscles • Puff up cheeks i. Tap with finger over each cheek to detect ease of air expulsion on the affected side • Smile and show teeth Buccal nerve This article incorporates text in the public domain from page 905 of the 20th edition of Gray's Anatomy Anatomy photo:23:06-0104 at the SUNY Downstate Medical Center - "Branches of Facial Nerve" lesson4 at The Anatomy Lesson by Wesley Norman cranialnerves at The Anatomy Lesson by Wesley Norman http://www.dartmouth.edu/~humananatomy/figures/chapter_47/47-5.

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Wonder en is gheen Wonder

Wonder en is gheen Wonder is a popular science magazine of the Flemish skeptical association SKEPP. The paper was founded in 2000 by Tom Schoepen, who served as its editor for its first ten years; the magazine is published four times a year and addresses pseudoscientific as well as science philosophical topics. The title is a reference to the 16th century Flemish mathematician and engineer Simon Stevin's commentary to his famous thought experiment: if something looks strange, it can still have a naturalistic explanation; the subtitle Tijdschrift voor wetenschap en rede was taken from Skeptical Inquirer, the most world-renowned skeptical magazine, published by the Committee for Skeptical Inquiry. As of 2016, the editorial staff is composed as follows: Core staffBart Coenen Cliff Beeckman Johan Braeckman Tim Trachet Luc Vancampenhout Pieter Van Nuffel Wietse WielsEditorial committeeWim Betz Stefaan Blancke Luc Bonneux Maarten Boudry Maxime Darge Geerdt Magiels Ronny Martens Marc Meuleman Pieter Peyskens Griet Vandermassen Frank Verhoft Critical thinking Freethought Skepter Skeptic Skeptical Inquirer Snopes.com The Freethinker The Skeptic The Skeptic's Dictionary Official website

Three-center two-electron bond

A 3-center 2-electron bond is an electron-deficient chemical bond where three atoms share two electrons. The combination of three atomic orbitals form three molecular orbitals: one bonding, one non-bonding, one anti-bonding; the two electrons go into the bonding orbital, resulting in a net bonding effect and constituting a chemical bond among all three atoms. In many common bonds of this type, the bonding orbital is shifted towards two of the three atoms instead of being spread among all three. An example of a 3c–2e bond is the trihydrogen cation H+3; this type of bond is called banana bond. An extended version of the 3c–2e bond model features in cluster compounds described by the polyhedral skeletal electron pair theory, such as boranes and carboranes; these molecules derive their stability from having a filled set of bonding molecular orbitals as outlined by Wade's rules. The monomer BH3 is unstable. A B−H−B 3-center-2-electron bond is formed when a boron atom shares electrons with a B−H bond on another boron atom.

The two electrons in a B−H−B bonding molecular orbital are spread out across three internuclear spaces. In diborane, there are two such 3c-2e bonds: two H atoms bridge the two B atoms, leaving two additional H atoms in ordinary B−H bonds on each B; as a result, the molecule achieves stability since each B participates in a total of four bonds and all bonding molecular orbitals are filled, although two of the four bonds are 3-centre B−H−B bonds. The reported bond order for each B−H interaction in a bridge is 0.5, so that the bridging B−H−B bonds are weaker and longer than the terminal B−H bonds, as shown by the bond lengths in the structural diagram. Three-center, two-electron bonding is pervasive in organotransition metal chemistry. A celebrated family of compounds featuring such interactions as called agostic complexes; this bonding pattern is seen in trimethylaluminium, which forms a dimer Al26 with the carbon atoms of two of the methyl groups in bridging positions. This type of bond occurs in carbon compounds, where it is sometimes referred to as hyperconjugation.

The first stable subvalent Be complex observed contains a three-center two-electron π-bond that consists of donor-acceptor interactions over the C-Be-C core of a Be-carbene adduct. Carbocation rearrangement reactions occur through three-center bond transition states; because the three center bond structures have about the same energy as carbocations, there is virtually no activation energy for these rearrangements so they occur with extraordinarily high rates. Carbonium ions such as ethanium C2H+7 have three-center two-electron bonds; the best known and studied structure of this sort is the 2-Norbornyl cation. Three-center four-electron bond 2-Norbornyl cation Dihydrogen complex