When matrices or members of various other associative algebras are multiplied, the product usually depends on the order of the factors.

The language of Clifford algebras ( also called geometric algebras ) provides a complete picture of the spin representations of all the spin groups, and the various relationships between those representations, via the classification of Clifford algebras.

Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures.

* considering a generalized unitary group as an algebraic group, one can take its points over various algebras.

There are various coassociative coproducts under which these algebras are Hopf algebras, for example,

Hence Ω ( X ) is not an arbitrary complete lattice but a complete Heyting algebra ( also called frame or locale-the various names are primarily used to distinguish several categories that have the same class of objects but different morphisms: frame morphisms, locale morphisms and homomorphisms of complete Heyting algebras ).

Though algebras of operators are studied in various contexts ( for example, algebras of pseudo-differential operators acting on spaces of distributions ), the term operator algebra is usually used in reference to algebras of bounded operators on a Banach space or, even more specially in reference to algebras of operators on a separable Hilbert space, endowed with the operator norm topology.

This point of view is elaborated as the philosophy of noncommutative geometry, which tries to study various non-classical and / or pathological objects by noncommutative operator algebras.

In mathematics, quantales are certain partially ordered algebraic structures that generalize locales ( point free topologies ) as well as various multiplicative lattices of ideals from ring theory and functional analysis ( C *- algebras, von Neumann algebras ).

By the mid-4th century however the assembly's judicial functions were largely curtailed, though it always kept a role in the initiation of various kinds of political trial.

The boule coordinated the activities of the various boards and magistrates that carried out the administrative functions of Athens and provided from its own membership randomly selected boards of ten responsible for areas ranging from naval affairs to religious observances.

Both BASICs differed from other dialects on different platforms, in that they allowed the easy creation of fairly demanding multimedia software, with full structured code and many high-level functions to load images, animations, sounds and display them in various ways.

The various concepts relating to functions can also be generalised to binary functions.

The organic part is also composed of various growth factors, the functions of which are not fully known.

* set various password prompts, such as a password for securing access to the BIOS user interface functions itself and preventing malicious users from booting the system from unauthorized peripheral devices.

Partition theory studies various enumeration and asymptotic problems related to integer partitions, and is closely related to q-series, special functions and orthogonal polynomials.

The study of categories is an attempt to axiomatically capture what is commonly found in various classes of related mathematical structures by relating them to the structure-preserving functions between them.

However, many other classification schemes, some of which are completely unrelated, also use colons and other punctuation in various functions.

In various traditions chakras are associated with multiple physiological functions, an aspect of consciousness, a classical element, and other distinguishing characteristics.

The discovery of differential cryptanalysis is generally attributed to Eli Biham and Adi Shamir in the late 1980s, who published a number of attacks against various block ciphers and hash functions, including a theoretical weakness in the Data Encryption Standard ( DES ).

In addition, under Dylan many interfaces can be defined for the same code, for instance the String concatenation method could be placed in both the String interface, and the " concat " interface which collects together all of the different concatenation functions from various classes.

Hormones regulate various human functions, including metabolism, growth and development, tissue function, and mood.

Thus one must be careful with the notation when using these functions, because various reputable references and software packages use different conventions in the definitions of the elliptic functions.

Since classical analog filters are IIR filters, there has been a long history of studying the range of possible transfer functions implementing various of the above desired filter responses in continuous time systems.

The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, various algebraic number fields, p-adic fields, and so forth.

The solution to a fluid dynamics problem typically involves calculating various properties of the fluid, such as velocity, pressure, density, and temperature, as functions of space and time.

The Office provide staff support functions ( such as finance and facilities management ) to the five function Branches and the various Field Divisions.

The gamma function is a component in various probability-distribution functions, and as such it is applicable in the fields of probability and statistics, as well as combinatorics.

The film Avatar depicts a world ( Pandora ) that functions like a single organism, in which various species of earth and sky cooperate with the humanoid population ( the Na ' vi ) to defend the planet against a corporate-military invasion.

Governors-General generally become patrons of various charitable institutions, present honours and awards, host functions for various groups of people including ambassadors to and from other countries, and travel widely throughout Australia.

Mr. Mills had done some figuring on a scrap of paper and given him the various kinds of boards and two-by-fours which, properly handled, would, he had assured him, turn into a workbench.

The rate of oxygen transfer at 1.0-mg. / l. dissolved oxygen concentration and 10-degrees-C for various rotor speeds is given in Table 3.

The word abbot, meaning father, is a title given to the head of a monastery in various traditions, including Christianity.

Although various Ambrosians were given the title of Blessed in recognition of their holiness: Antonio Gonzaga of Mantua, Filippo of Fermo, and Gerardo of Monza, the order was eventually dissolved by Pope Innocent X in 1650.

* Ai ( given name ), various people

* Ai ( given name ), various fictional characters

Similarly, arbitrage affects the difference in interest rates paid on government bonds issued by the various countries, given the expected depreciations in the currencies relative to each other ( see interest rate parity ).

For a detailed discussion of the differences including a more comprehensive table ( several essential tables are given below ) of Biblical scripture for both Testaments and the intertestamental period with regard to canonical acceptance in Christendom's various major traditions, see Wikipedia's article on " Biblical canon ".

Each partnership works jointly by means of various ' calls ' to secure a contract at the highest level deemed advisable by them given their card holdings.

The book is as intriguing for the themes it leaves out as for what it includes: the ark of the covenant, which is given so much importance in the stories of Moses and Joshua, is almost entirely missing, cooperation between the various tribes is limited, and there is no mention of a central shrine for worship or of a high priest ( the office to which Aaron was appointed at the end of the Exodus story ).

The various dates given in the book suggest that Ezekiel was 25 when he went into exile, 30 when he received his prophetic " call ", and 52 at the time of the last vision c. 571.

John Ellis explained that “... there is considerable justice in Matthew Cooper's assertion that the panzer divisions were not given the kind of strategic mission that was to characterize authentic armoured blitzkrieg, and were almost always closely subordinated to the various mass infantry armies .”

The propositional satisfiability problem ( PSAT ), which decides whether a given propositional formula is satisfiable, is of central importance in various areas of computer science, including theoretical computer science, algorithmics, artificial intelligence, hardware design, electronic design automation, and verification.

In the second century Christianity was criticized by the Jews on a various grounds, e. g. that the prophecies of the Hebrew Bible could not have been fulfilled by Jesus, given that he did not have a successful life.

The length and requirements for copyright duration are subject to change by legislation, and since the early 20th century there have been a number of adjustments made in various countries, which can make determining the duration of a given copyright somewhat difficult.

The naming of various substances now known as elements precedes the atomic theory of matter, as names were given locally by various cultures to various minerals, metals, compounds, alloys, mixtures, and other materials, although at the time it was not known which chemicals were elements and which compounds.

) In more prosaic terms, those who hold to the Copenhagen understanding are willing to say that a wave function involves the various probabilities that a given event will proceed to certain different outcomes.

Among the reasons given by those who oppose the use of Common Era notation is that it is selective as other aspects of the Western calendar have origins in various belief systems ( e. g., January is named for Janus ), Style guides for academic texts on religion generally prefer BCE / CE to BC / AD.

However, little headway has been made to institute a playoff tournament, given the entrenched vested economic interests in the various bowls.

A more accurate value of the limit than that given by this simple model requires adjusting for various factors, including electrostatic interactions between the electrons and nuclei and effects caused by nonzero temperature.

The contras ( some references use the capitalized form, " Contras ") is a label given to the various rebel groups opposing the Sandinista Junta of National Reconstruction government in Nicaragua that were active from 1979 through to the early 1990s.

Alternatively, it is identified with the " tree of paradise " of medieval mystery plays that were given on 24 December, the commemoration and name day of Adam and Eve in various countries.