Because these two definitions can be transformed simply by into the other, some formulae have this alternatingly (- 1 )< sup > n </ sup >- term and others not depending on the context, but it is not possible to decide in favor of one of these definitions to be the correct or appropriate or natural one ( for the abstract Bernoulli numbers ).

Similarly, a mathematician does not restrict his study of numbers to the integers ; rather he considers more abstract structures such as rings, and in particular number rings in the context of algebraic number theory.

In this context, vectors are abstract entities which may or may not be characterized by a magnitude and a direction.

The associated combinatorial structure is called an abstract simplicial complex, in which context the word “ simplex ” simply means any finite set of vertices.

In this context, " almost surely " is a mathematical term with a precise meaning, and the " monkey " is not an actual monkey, but a metaphor for an abstract device that produces a random sequence of letters and symbols ad infinitum.

As defined in linguistics, a sentence is an abstract entity — a string of words divorced from non-linguistic context — as opposed to an utterance, which is a concrete example of a speech act in a specific context.

In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism.

For this reason, people will often distinguish the meanings of the two terms in this context, using " image " for the abstract categorical concept and " range " for the elementary function-theoretic concept.

In this context, internal language applies to the study of syntax and semantics in language on the abstract level ; as mentally represented knowledge in a native speaker.

In other fields it is medium-neutral and refers to any use of language in context ( in contrast with the abstract form sentence ).

Despite this he was able to view France's foreign affairs with a more abstract nature, taking in the wider European context.

A rangatira is a chief, the nominalising suffix-tanga makes the word an abstract noun referring to the quality or attributes of chieftainship, and the addition of intensifier tino in this context means the phrase can be translated as ' highest chieftainship ' ( Kawharu, 1989, p. 314 ), the intention of which was to ' emphasize to a chief the Queen ’ s intention to give the complete control according to their customs ' ( ibid., p. 319 ).

Sometimes it is mentioned in this context that various difficult combinatorial problems ( in fact, any NP-complete problem ) can be converted to Go-like problems on a sufficiently large board ; however, the same is true for other abstract board games, including chess and minesweeper, when suitably generalized to a board of arbitrary size.

In current context, in scientific fields, a homunculus may refer to any scale model of the human body that, in some way, illustrates physiological, psychological, or other abstract human characteristics or functions.

Lave and Wenger ( 1991 ) argue that learning should not be viewed as simply the transmission of abstract and decontextualised knowledge from one individual to another, but a social process whereby knowledge is co-constructed ; they suggest that such learning is situated in a specific context and embedded within a particular social and physical environment.

Nussbaum accepts Catharine MacKinnon's critique of abstract liberalism, assimilating the salience of history and context of group hierarchy and subordination, but concludes that this appeal is rooted in liberalism rather than a critique of it.

Generally the terms “ equality of opportunity ” and “ equal opportunity ” are interchangeable, with occasional slight variations: “ equality of opportunity ” has more of a sense of being an abstract political concept, while “ equal opportunity ” is sometimes used as an adjective, usually in the context of employment regulations, to identify an employer, a hiring approach, or law.

Demonstratives are employed for spatial deixis ( using the context of the physical surroundings of the speaker and sometimes the listener ) and for discourse deixis ( including abstract concepts ) where the meaning is dependent on something other than the relative physical location of the speaker, for example whether something is currently being said or was said earlier.

In very abstract terms, general position is a discussion of generic properties of a configuration space ; in this context one means properties that hold on the generic point of a configuration space, or equivalently on a Zariski-open set.

A key aspect of the Cartan connection point of view is to elaborate this notion in the context of principal bundles ( which could be called the " general or abstract theory of frames ").

Since a formalist model usually states what is to be maximized in terms of preferences, which often but not necessarily include culturally expressed value goals, it is deemed to be sufficiently abstract to be capable of explaining human behavior in any context.

Although these later works have been sometimes called " philosophy ", it has been argued that there is no abstract or theoretical context to justify such a description.

Numerous games have narrative elements which give a context to an event in a game, make the activity of playing it less abstract and enhance its entertainment value, although narrative elements are not always clearly present or present at all.

From about 1955 he started to work on sheaf theory and homological algebra, producing the influential " Tôhoku paper " ( Sur quelques points d ' algèbre homologique, published in 1957 ) where he introduced Abelian categories and applied their theory to show that sheaf cohomology can be defined as certain derived functors in this context.

The mathematical definition of an elementary function, or a function in elementary form, is considered in the context of differential algebra.

Any field may be used as the scalars for a vector space, which is the standard general context for linear algebra.

The notion of a homomorphism can be given a formal definition in the context of universal algebra, a field which studies ideas common to all algebraic structures.

Actually, in the algebraic setting ( at least within the context of universal algebra ) this extra condition on isomorphisms is automatically satisfied.

The term was introduced by Benjamin Peirce in the context of elements of an algebra that remain invariant when raised to a positive integer power, and literally means "( the quality of having ) the same power ", from idem + potence ( same + power ).

A vector may also be multiplied, or re-scaled, by a real number r. In the context of conventional vector algebra, these real numbers are often called scalars ( from scale ) to distinguish them from vectors.

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.

In probability theory, the sigma algebra often represents the set of available information, and a function ( in this context a random variable ) is measurable if and only if it represents an outcome that is knowable based on the available information.

( Sometimes the subscripts on f are taken off when it is clear from context which algebra your function is from ) For example, if e is a constant ( nullary operation ), then h ( e < sub > A </ sub >) = e < sub > B </ sub >.

In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.

The general notion of a congruence relation can be given a formal definition in the context of universal algebra, a field which studies ideas common to all algebraic structures.

* Tropical analysis ( or idempotent analysis ) – analysis in the context of the semiring of the max-plus algebra where the lack of an additive inverse is compensated somewhat by the idempotent rule A + A = A.

The real numbers also play an important role in algebra as a field, but in this context is rarely referred to as a line.

Since the join operation ∨ in a Boolean algebra is often written additively, it makes sense in this context to denote ring addition by ⊕, a symbol that is often used to denote exclusive or.

Thus additive categories can be seen as the most general context in which the algebra of matrices makes sense.

The use of both terms in the modern context of multilinear algebra is a specific example of corresponding notions in category theory.

From a very young age he was an outstanding figure in the school of Henri Cartan, working on algebraic topology, several complex variables and then commutative algebra and algebraic geometry, in the context of sheaf theory and homological algebra techniques.

Since the definition of the spectrum does not mention any properties of B ( X ) except those that any such algebra has, the notion of a spectrum may be generalised to this context by using the same definition verbatim.

In 1957, Artin wrote a book on geometric algebra an insightful development of the classical groups in a Kleinian context.