The most common way to formalize this is by defining a field as a set together with two operations, usually called addition and multiplication, and denoted by + and ·, respectively, such that the following axioms hold ; subtraction and division are defined implicitly in terms of the inverse operations of addition and multiplication :< ref group =" note "> That is, the axiom for addition only assumes a binary operation The axiom of inverse allows one to define a unary operation that sends an element to its negative ( its additive inverse ); this is not taken as given, but is implicitly defined in terms of addition as " is the unique b such that ", " implicitly " because it is defined in terms of solving an equation — and one then defines the binary operation of subtraction, also denoted by "−", as in terms of addition and additive inverse.

For example, the cyclic group of addition modulo n can be obtained from the integers by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class ( known as a congruence class ) as a single entity.

Paul Marino called the addition of this feature " a defining moment for achinima ".

The set of all such formal power series is denoted R « X < sub > I </ sub >», and it is given a ring structure by defining addition pointwise

Angular velocity is a vector defining an addition operation.

In addition to the conventional GATC symbols, there is an expanded letter code to indicate a position within a sequence that may be flexible when defining sequences.

The ambiguity in defining this phenomenon was first recognized by Hans Selye in 1926 who loosely described stress as something that "… in addition to being itself, was also the cause of itself, and the result of itself.

" In addition to criticizing " environmental extremists " and " radical feminism ," he said public morality was a defining issue:

( Here we think of V as a group under its operation of addition, and use the defining representation of GL ( V ) on V to define the semidirect product.

In addition, he showed a preference for a declamatory, homophonic style, which he refined later in his career into a seconda prattica manner influential on Monteverdi, and he also showed a liking for high voices – something which turned out to be a defining characteristic of the music-making at the Este court in Ferrara.

In addition, it lacked a large ENTER key which was seen by many as the defining characteristic of an HP calculator.

In addition to these defining characteristics, Silkies have five toes on each foot.

In addition to bilateral non-aggression pacts, the twentieth century saw multilateral treaties defining entirely new restrictions against going to war.

In addition to defining points and constructs related to points, Euclid also postulated a key idea about points ; he claimed that any two points can be connected by a straight line.

In addition to the way it recapitulates previous modes of royal representation, Helgerson notes a certain affinity between the textual aesthetics of the “ King ’ s Book ” and those of the Counter-Reformation: “ Eikon Basilike drew on a set of culturally conditioned responses against which the new culture of print was defining itself, responses that had previously served Elizabeth and Shakespeare and that even then were serving Counter-Reformation Catholicism.

The 1970s and 1980s brought further change as president Dan C. West oversaw the implementation of significant curricular reforms ( including the adoption of a new core curriculum and the addition of many new non-traditional majors ), the introduction of innovative fundraising techniques ( including the creation of the college ’ s own for-profit development corporation ), the establishment of an international studies program funded by a gift from former board of trustees president Shuford Nichols, and the development of the Scottish heritage program, which had come to be a defining symbol of the college by the twenty-first century.

In addition to the basic function of a calendar — defining and organizing periods of time in a way that allows events to be fixed, ordered and noted relative to each other and some absolute progression — Mesoamerican calendars were also used in religious observances and social rituals, such as for divination.

In addition, one might take the following factors into account when defining a SOA implementation:

The result of the association was the invention of monody, and shortly thereafter, opera ; in addition, the innovations brought to music by the Camerata under the guidance of Bardi were one of the defining characteristics of what we now know as Baroque music.

In addition to the usual problems of defining the novel as a genre, Aphra Behn had written at least one epistolary novel prior to Oroonoko.

The Nordic countries all have the same design ( Iceland, Denmark, Norway, Sweden, Finland, in addition to the autonomous regions of the Faroe Islands and Åland ), a horizontal cross on a single-colored background.

In addition, the legislative assembly of each autonomous community designates one senator, and another for each one million inhabitants.

In addition, two autonomous cities were constituted on the coast of North Africa.

Boroughs have the power to advise the mayor with nonbinding opinions on a large spectrum of topics ( environment, construction, public health, local markets ); in addition they are supplied with an autonomous founding in order to finance local activities.

In addition, miniaturized computer technology and improved batteries allowed the development of a new type of machine — the autonomous robotic vacuum cleaner.

As an autonomous, charitable foundation established in 2000, CFCAS funds research that improves the scientific understanding of processes and predictions, provides relevant science to policy makers and improves understanding of the ways in which these challenges affect human health and the natural environment in addition to strengthening Canada's scientific capacity.

In addition to prefectures, this level also includes autonomous prefectures, leagues, and prefecture-level cities.

In addition, like most Spanish autonomous communities, Catalonia has permanent delegations before international organizations, such as the European Union.

In addition, the same term was and still is, though less frequently than before, applied to an autonomous particular Church within the Catholic Church associated with a particular liturgical tradition.

Serbia, in addition, included two autonomous provinces of Vojvodina and Kosovo and Metohija.

In addition, there was an autonomous faction within the " legal " Lebanese Army called the Army of Free Lebanon.

In 2010, in addition to the 56 million Italians in Italy and 3, 000 in San Marino, Italian-speaking, autonomous groups are found in neighboring countries: about 500, 000 in Switzerland, a large, but undefined population in France, and smaller groups in Slovenia and Croatia, primarily in Istria and Dalmatia.

In addition to this, senators from the autonomous Duchy of Prussia, a fiefdom of the Polish King, were incorporated after having been absent since gaining autonomy in 1466.

There are in addition 23 Miao autonomous counties:

In addition to overseeing the department, the director also serves on an autonomous board known as the Natural Resources Commission, consisting of both government officials and citizen members, which meets at least four times annually to address issues pertaining to the department.

With the addition of a new primary education school, the three units emerged as autonomous components of an integrated education system under the aegis of the University.

In addition to this, rabble. ca is decidedly different from the mainstream media by its participation in events such as the 2010 Making Media Public Conference, which focused on autonomous media, community and alternative media, and media activism.

The South Caucasus region consists of a number of breakaway and autonomous republics in addition to the states of Armenia, Azerbaijan and Georgia recognised by the United Nations.

In addition to the three persons, Irish also has an impersonal form ( also called the " autonomous " form ), which is used in forming passives and can conveniently be translated with " one " or " someone " as the subject.

The complete, four satellite " PQ-Gemini ++" mission group, in addition to validating general satellite systems, will be capable of demonstrating Micro-Space interferometric techniques to characterize orbital differences using only precision range data between ultralight spacecraft in preparation for autonomous rendezvous.

Since 1992, the State Council has opened a number of border cities, and in addition, opened all the capital cities of inland provinces and autonomous regions.

In addition to the autonomous regions of China mentioned above, various other areas of the world are formally described as autonomous regions:

In addition to racial unrest, thorny issues concerning Singapore's rights as an autonomous state further put a dent in relations, such as the failure of a common market to be set up between the Federation and Singapore, and the heavy tax burden placed on Singapore, which was seen as unfair.