A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which xy does not always equal yx ; or more generally an algebraic structure in which one of the principal binary operations is not commutative ; one also allows additional structures, e. g. topology or norm to be possibly carried by the noncommutative algebra of functions.

When I is a principal ideal αO < sub > K </ sub > then N ( I ) is equal to the absolute value of the norm to Q of α, for α an algebraic integer.

The industry norm can be argued to be anything between 93 %- 96 % when a UPS is in full operational mode, and to reach these figures companies often put their UPS in an ideal scenario.

of a number field K contains an integral ideal of norm not exceeding a certain bound, depending on K, called Minkowski's bound: the finiteness of the class number of an algebraic number field follows immediately.

The government-sponsored image of the conversion process emphasises psychological persuasion and a variety of " soft-sell " techniques ; this is the " ideal norm " in regime reports, according to Tong.

This result gives a bound, depending on the ring, such that every ideal class contains an ideal norm less than the bound.

If a is an ideal of O < sub > K </ sub >, other than the zero ideal we denote its norm by Na.

Hence this norm of an ideal is always a positive integer.

In commutative algebra, the norm of an ideal is a generalization of a norm of an element in the field extension.

When the less complicated number ring is taken to be the ring of integers, Z, then the norm of a nonzero ideal I of a number ring R is simply the size of the finite quotient ring R / I.

By convention, the norm of the zero ideal is taken to be zero.

The norm of an ideal can be used to bound the norm of some nonzero element by the inequality

Ushering in the modern recording era, he challenged our perception and processes of how recordings could be made, explored the possibilities of modern recording and exploited the potential for the popularity of classical music-and all this while setting standards of artistic achievement, integrity, risk-taking, and of the aesthetic ideal that became our new norm.

A six-partner group marriage ( three male, three female ) is considered the ideal norm in the alien society described in the novel ; the main characters are in a five-partner group marriage, and much of the dramatic tension hinges on there being more than one candidate for the sixth position.

The previous norm, the mortise lock, is a more complex device, and its higher manufacturing cost as well as its more labor intensive installation make the bored cylindrical lock an ideal substitute, both in price and functionality.

* Ideal norm, the ideal theoretic generalization of the field norm

This norm diverges only minimally from the original ideal of " one letter, one sound "; that is, it accepts only minor allophonic variation.

Namely, Γ ( I ) is a subgroup of the group of norm 1 elements in the quaternion algebra generated as an associative algebra by the generators i, j and relations.

generated by an element s and if a is an element of L of relative norm 1, then there exists b in L such that

The notion of weak convergence defines a topology on H and this is called the weak topology on H. In other words, the weak topology is the topology generated by the bounded functionals on H. It follows from Schwarz inequality that the weak topology is weaker than the norm topology.

The ideal norm of δ < sub > K </ sub > is equal to the ideal of Z generated by the field discriminant D < sub > K </ sub > of K.

Some investigators have found a parallelism between remissions and return of the sympathetic reactivity of the hypothalamus to the normal level as indicated by the Mecholyl test and, conversely, between clinical impairment and increasing deviation of this test from the norm.

Every Hilbert space X is a Banach space because, by definition, a Hilbert space is complete with respect to the norm associated with its inner product, where a norm and an inner product are said to be associated if for all x ∈ X.

If the norm of a Banach space satisfies this identity, the associated inner product which makes it into a Hilbert space is given by the polarization identity.

Given a ket of norm 1, the orthogonal projection onto the subspace spanned by is

The algebra multiplication and the Banach space norm are required to be related by the following inequality:

* The set of real ( or complex ) numbers is a Banach algebra with norm given by the absolute value.

* The quaternions form a 4-dimensional real Banach algebra, with the norm being given by the absolute value of quaternions.

Indeed, in those cases where no clear consensus exists on a given norm, the drafting of criminal law by the group in power to prohibit the behaviour of another group may seem to some observers an improper limitation of the second group's freedom, and the ordinary members of society have less respect for the law or laws in general — whether the authorities actually enforce the disputed law or not.

Even some women of the literate elite, for whom Confucianism was quite explicitly the norm, were able to flourish by living their lives according to that model.

By the time of Henry IV, bargaining by the peasants for the benefit of the group was the norm.

Following World War II the duration of civil wars grew past the norm of the pre-19th century, largely due to weakness of the many postcolonial states and the intervention by major powers on both sides of conflict.

For instance, together with the spectral radius formula, it implies that the C *- norm is uniquely determined by the algebraic structure:

Though some new cable car systems were still being built, by 1890 the cheaper to construct and simpler to operate electrically-powered trolley or tram started to become the norm, and eventually started to replace existing cable car systems.

Moral and scientific thinking together do not suffice to understand human existence, so a further set of categories, governed by a norm of " authenticity ", is necessary to understand human existence.

Instead, education officials across the world believe that by setting clear, achievable, higher standards, aligning the curriculum, and assessing outcomes, learning can be increased for all students, and more students can succeed than the 50 percent who are defined to be above or below grade level by norm referenced standards.

Finally, one can use the norm to define a metric ( or distance function ) on R < sup > n </ sup > by

The period between Henry's accession and the birth of Eleanor's youngest son was turbulent: Aquitaine, as was the norm, defied the authority of Henry as Eleanor's husband ; attempts to claim Toulouse, the rightful inheritance of Eleanor's grandmother and father, were made, ending in failure ; the news of Louis of France's widowhood and remarriage was followed by the marriage of Henry's son ( young Henry ) to Louis ' daughter Marguerite ; and, most climactically, the feud between the King and Thomas Becket, his Chancellor, and later Archbishop of Canterbury.

During the past five hundred years, however, probably due to Confucian influence, inheritance by sons — but not always, or even most often, the eldest son — has been the norm.

Regional bus lines have been regulated by the provincial administration to protect old transit companies, leading to cartel situations like TLO in the Turku region, but strong regional regulating bodies, like the Helsinki Regional Transport Authority ( HSL / HRT ), whose routes are put out to tender exist as well and will become the norm after the transitionary period during the 2010s.

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure ( e. g. inner product, norm, topology, etc.

He also sought to break from the class-conscious conventions of the 1930s and early 40s, when top hat and tails or tuxedos were the norm, by dancing in casual or everyday work clothes, so as to make his dancing more relevant to the cinema-going public.

After the Peace of Westphalia which ended the war in favour of nations deciding their own religious allegiance, Absolutism became the norm of the continent, while parts of Europe experimented with constitutions foreshadowed by the English Civil War and particularly the Glorious Revolution.

Later hymnody in the Western church introduced four-part vocal harmony as the norm, adopting major and minor keys, and came to be led by organ and choir.