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Moreover ( and more embarrassingly, although this is essentially trivial ), mathematicians usually write the dual entity not at the first place, as the physicists do, but at the second one, and they don't use the *- symbol, but an overline ( which the physicists reserve to averages ) to denote conjugate-complex numbers, i. e. for scalar products mathematicians usually write

It is generally believed that C *- algebras were first considered primarily for their use in quantum mechanics to model algebras of physical observables.

Another active area of research is the program to obtain classification, or to determine the extent of which classification is possible, for separable simple nuclear C *- algebras.

Electrical conductivity does occur in graphene, because the π and π *- like bands overlap, making it a semimetal, with partly filled bands, fulfilling the other requirement for conductivity.

The majority of the efforts have looked at embedding spatial database indices such as the Space Filling Curves ( SFCs ) including the Hilbert curves, Z-curves, k-d tree, MX-CIF Quad tree and R *- tree for managing, routing, and indexing of complex Grid resource query objects over DHT networks.

Various generalizations of the Cauchy – Schwarz inequality exist in the context of operator theory, e. g. for operator-convex functions, and operator algebras, where the domain and / or range of φ are replaced by a C *- algebra or W *- algebra.

We now give an operator theoretic proof for the Cauchy – Schwarz inequality which passes to the C *- algebra setting.

Theorem ( Modified Schwarz inequality for 2-positive maps ) For a 2-positive map Φ between C *- algebras, for all a, b in its domain,

*- graphy ( suffix ; Latin for " to write " or " draw ")

Computer scientists and mathematicians often pronounce it as star ( as, for example, in the A * search algorithm or C *- algebra ).

The array of referents for this word, together with the masculine and feminine alternations it exhibits, suggests that the word is an adjective, with an adjectival *- n-suffix used to form adjectives in many Centum Indo-European languages ( cf.

Cohomology theories have been defined for many different objects such as topological spaces, sheaves, groups, rings, Lie algebras, and C *- algebras.

Several different such patterns have been discerned, but the commonest one, by a wide margin, is e / o / zero alternation found in a majority of roots, in many verb and noun stems, and even in some affixes ( the genitive singular ending, for example, is attested as *- es, *- os, and *- s ).

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Furthermore this approach led to a noncommutative K-theory for C *- algebras.

*- ite ( Ann Arbor → Ann Arborite, Austin → Austinite, Brisbane → Brisbanite ( also " Brisbanian "), Chennai → Chennaite, Dallas → Dallasite, Delhi → Delhite, Denver → Denverite, Dunedin → Dunedinite, Irmo → Irmite, Israel → Israelite ( also " Israeli ", depending on the usage ; see below ), Karachi → Karachiite, Moscow → Muscovite ( also Latin " Muscovia "), New Hampshire → New Hampshirite, Perth → Perthite, Ruskin → Ruskinite, Seattle → Seattleite, Seoul → Seoulite, Springfield → Springfieldianite, Spokane → Spokanite, Tokyo → Tokyoite, Vancouver → Vancouverite, Wisconsin → Wisconsinite, Wyoming → Wyomingite ), Wenham → Wenhamite mostly for cities.

*- ish ( Åland → Ålandish, Britain → British, Cornwall → Cornish, Denmark → Danish, England → English, Finland → Finnish, Ireland → Irish, Kurdistan → Kurdish, Poland → Polish, Scotland → Scottish, Spain → Spanish, Sweden → Swedish, Turkey → Turkish ), mostly for countries

*- i ( Afghanistan → Afghanistani, Azawad → Azawadi, Azerbaijan → Azerbaijani, Bahrain → Bahraini, Bangladesh → Bangladeshi, Bengal → Bengali, Bihar → Bihari, Dagestan → Dagestani, Desh → Desi, Gujarat → Gujarati, Hyderabad → Hyderabadi, Iraq → Iraqi, Israel → Israeli ( in the Modern State of Israel ), Kashmir → Kashmiri, Kazakhstan → Kazakhstani ( also " Kazakh "), Kuwait → Kuwaiti, Nepal → Nepali, Oman → Omani, Pakistan → Pakistani, Punjab → Punjabi, Qatar → Qatari, Rajasthan → Rajasthani, Sindh → Sindhi, Somalia → Somali ( not Somalian ), Tajikistan → Tajikistani ( also " Tajik "), United Arab Emirates → United Arab Emirati, Uzbekistan → Uzbekistani ( also " Uzbek "), Yemen → Yemeni, Yerevan → Yerevani ), mostly for Middle Eastern and South Asian locales and in Latinate names for the various people that ancient Romans encountered ( e. g. Allemanni, Helvetii )

*- iot ( e ) ( Corfu → Corfiot, Cyprus → Cypriot, Phanar → Phanariote ), especially for Greek locations.

Many properties of the transpose hold for general *- algebras:

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49. 70 % of residents only have level one or two qualifications ( at least 1 GCSE pass ( A *- G ) or equivalent ), and 18. 25 % have at least level three qualiffications ( at least 2 A-levels ( A-E ) or 4 AS levels ( A-E ) or equivalent ).

The construction of Gelfand – Naimark representation depends only on the GNS construction and therefore it is meaningful for any Banach *- algebra A having an approximate identity.

*- ul-Mulk ( or ul-Molk ):-of the kingdom ; e. g. Malik Usman Khan, who served the Sultan of Gujarat as Governor of Lahore, received the title of Zubdat ul-Mulk ' best of the kingdom ' as a hereditary distinction, which was retained as part of the style of his heirs, the ruling Diwans ( only since 1910 promoted to Nawab ) of Palanpur.

Moreover, an element of a C *- algebra is positive if and only if there is some in such that.

This tells us that a unital C *- subalgebra M of B ( H ) is a von Neumann algebra if, and only if,, and that if not, the von Neumann algebra it generates is.

Another essential ingredient in Mackey's work was the assignment of a Borel structure to the dual object of a locally compact group ( specifically a locally compact separable metric group ) G. One of Mackey's important conjectures, which was eventually solved by work of James Glimm on C *- algebras was that G is type I ( meaning that all its factor representations are of type I ) if and only if the Borel structure of its dual is a standard Borel space.

The decreased overlap of rectangles in the R *- tree is one of the key benefits over the traditional R-Tree ( this is also a consequence of the other heuristics used, not only the subtree choosing ).

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A *- representation π of A on a Hilbert space H is irreducible if, and only if, there is no closed subspace K different from H and

*- adja-just or only ( e. g. " adja ra man kaw magsinaoy saoy " You are just interrupting )

If A is a C *- algebra, an element W in A is a partial isometry if and only if

It asks whether every C *- algebra that has only one irreducible representation up to unitary equivalence is isomorphic to the algebra of compact operators on some Hilbert space.

Also, the corresponding etymon for *- kel is only found in Chuvash ( meaning ' house, shelter ') and is not extant in the Oghuzic branch of the language family.

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# a subcategory having subobjects and direct sums, such that the C *- algebra of endomorphisms of the monoidal unit contains only scalars.

Stroud High School has consistently achieved a GCSE Level 2 threshold ( the equivalent of 5 + A *- C ) of 100 %.

School pupils are now on track and almost 90 % of pupils are on track to receive 5 A *- C grades.

Over 1000 pupils attend The King's School, of whom approximately 400 are in the Sixth form, for which there is a minimum examination qualification for entry of seven A *- C grades at GCSE level, of which three must be at grade B or above.

From 2006 to the present, The King's School has been the top-performing state school in the Peterborough local authority area for GCSE and A-Level results, with 91 %+ of students achieving 5 or more passes at GCSE grades A *- C.

Culcheth High School opened in 1931, and received a ' good ' rating by Ofsted ( 2009 ), with ' The Class of 2011 ' attaining the best results the school has produced so far, with 77 % of pupils attaining 5 A *- C GCSEs including English and Maths ( an increase of 3 % on 2010 ).

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