Here we define some basic observational concepts, like angles and coordinates, defined in geodesy ( and astronomy as well ), mostly from the viewpoint of the local observer.

An example of the three parameters that specify a spatial rotation are the roll, pitch and yaw angles used to define the orientation of an aircraft.

Euclid uses right angles in definitions 11 and 12 to define acute angles ( those smaller than a right angle ) and obtuse angles ( those greater than a right angle ).

A Riemannian metric ( tensor ) makes it possible to define various geometric notions on a Riemannian manifold, such as angles, lengths of curves, areas ( or volumes ), curvature, gradients of functions and divergence of vector fields.

The two angles introduced will always add up to π ; we therefore define our Dehn invariant so that multiples of angles of π give a net contribution of zero.

Each of the above listed items can be looked at from different angles, which leads to a myriad of ways to look at ( or define ) the digital divide.

Then we define its Euler angles as:

Though is possible to define the angles by geometry, normally the intrinsic rotations equivalence or the extrinsic rotations equivalence are used instead.

The reason to define a special function for the exsecant is similar to the rationale for the versine: for small angles < var > θ </ var >, the sec (< var > θ </ var >) function approaches one, and so using the above formula for the exsecant will involve the subtraction of two nearly equal quantities and exacerbate roundoff errors.

( In particular, the exponential map can be used to define abstract index groups.

These categories surely have some objects that are " special " in a certain way, such as the empty set or the product of two topologies, yet in the definition of a category, objects are considered to be atomic, i. e., we do not know whether an object A is a set, a topology, or any other abstract concept – hence, the challenge is to define special objects without referring to the internal structure of those objects.

Knowledge Discovery Metamodel uses Meta-Object Facility to define an XMI interchange format between tools that work with existing software and an abstract interface for the next-generation assurance and modernization tools.

When Polish notation is used as a syntax for mathematical expressions by interpreters of programming languages, it is readily parsed into abstract syntax trees and can, in fact, define a one-to-one representation for the same.

But Jean Piaget, who would better define himself as constructivist, considers structuralism as " a method and not a doctrine " because for him " there exists no structure without a construction, abstract or genetic "

It prescribes that software designers should define formal, precise and verifiable interface specifications for software components, which extend the ordinary definition of abstract data types with preconditions, postconditions and invariants.

Many authors in abstract algebra and universal algebra define an epimorphism simply as an onto or surjective homomorphism.

* Formal groups are used to define an abstract group law using formal power series

The inheritance mechanism in object-oriented programming can be used to define an abstract class as the common interface.

In object-oriented programming theory, abstraction involves the facility to define objects that represent abstract " actors " that can perform work, report on and change their state, and " communicate " with other objects in the system.

This definition is too abstract for practical measurement, so a device known as a current balance is used to define the ampere operationally.

In negative theology, it is accepted that the Divine is ineffable, an abstract experience that can only be recognized or remembered — that is, human beings cannot describe in words the essence of the perfect good that is unique to the individual, nor can they define the Divine, in its immense complexity, related to the entire field of reality, and therefore all descriptions if attempted will be ultimately false and conceptualization should be avoided ; in effect, it eludes definition by definition:

Applications of chain complexes usually define and apply their homology groups ( cohomology groups for cochain complexes ); in more abstract settings various equivalence relations are applied to complexes ( for example starting with the chain homotopy idea ).

The meta-language actually adopted (" Meta-IV ") is used to define major portions of PL / 1 ( as given in ECMA 74-interestingly a " formal standards document written as an abstract interpreter ") in BEKIČ 74 .»

In abstract algebra, one uses homology to define derived functors, for example the Tor functors.

MOF only provides a means to define the structure, or abstract syntax of a language or of data.

Template method's abstract class may also define hook methods that may be overridden by subclasses.

Common positions on the other hand, define the approach that the EU takes on a certain matter of geographical or thematic nature, and define in the abstract the general guidelines to which the national policies of Member states must conform.

The specification is summarized in the abstract as follows: " This specification provides a model and grammar for representing the structure of information resources used to define topics, and the associations ( relationships ) between topics.

The test consisted of 30 items ranging from the ability to touch one's nose or ear, when asked, to the ability to draw designs from memory and to define abstract concepts, and varying in difficulty.

It is possible to define abstract homotopy groups for simplicial sets.

If X and Y are algebraic structures of some fixed type ( such as groups, rings, or vector spaces ), and if the function f from X to Y is a homomorphism, then ker f will be a subalgebra of the direct product X × X. Subalgebras of X × X that are also equivalence relations ( called congruence relations ) are important in abstract algebra, because they define the most general notion of quotient algebra.

Operational semantics may define an abstract machine ( such as the SECD machine ), and give meaning to phrases by describing the transitions they induce on states of the machine.

If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.

The real line R with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it.

* Consider the set of all functions from the real number line to the closed unit interval, and define a topology on so that a sequence in converges towards if and only if converges towards for all.

Assuming we can define at least one nonreal complex number, however, a complex number is definable if and only if both its real part and its imaginary part are definable.

Using the diagonal argument, it is possible to define a real number x, which is not equal to G ( n ) for any n. This means that there is a language L ' that defines x, which is undefinable in L.

The standard way to do this, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product.

Clemence's 1948 proposal did not adopt a correction of this kind in terms of mean solar time: instead, the same numbers were used as in Newcomb's original uncorrected formula ( 1 ), but now in a reverse sense, to define the time and time scale implicitly, based on the real position of the Sun:

The essential distinction between the frequentists and the non-frequentists is, I think, that the former, in an effort to avoid anything savouring of matters of opinion, seek to define probability in terms of the objective properties of a population, real or hypothetical, whereas the latter do not.

If p is a non-zero real number, we can define the generalized mean with exponent p ( or power mean with exponent p ) of the positive real numbers as:

A complex Lie group is defined in the same way using complex manifolds rather than real ones ( example: SL < sub > 2 </ sub >( C )), and similarly one can define a p-adic Lie group over the p-adic numbers.

It is used throughout real analysis, in particular to define Lebesgue integration.

Moreover, these positions do not define the meaning of " real ".

In classical real analysis, one way to define a real number is as an equivalence class of Cauchy sequences of rational numbers.

For f a real polynomial in x, and for any a in such an algebra define f ( a ) to be the element of the algebra resulting from the obvious substitution of a into f. Then for any two such polynomials f and g, we have that ( fg ) ( a )

We assume that A is an m-by-n matrix over either the real numbers or the complex numbers, and we define the linear map f by f ( x ) = Ax as above.

The SNOBOL4 variant of the language supports a number of built-in data types, such as integers and limited precision real numbers, strings, patterns, arrays, and tables ( associative arrays ), and also allows the programmer to define additional data types and new functions.

These derivations form a real vector space if we define addition and scalar multiplication for derivations by

One way to rigorously define the delta function is as a measure, which accepts as an argument a subset A of the real line R, and returns δ ( A )

In analysis the infimum or greatest lower bound of a subset S of real numbers is denoted by inf ( S ) and is defined to be the biggest real number that is smaller than or equal to every number in S. If no such number exists ( because S is not bounded below ), then we define inf ( S ) = −∞.

If S is empty, we define inf ( S ) = ∞ ( see extended real number line ).

It is possible to extend the definition of the negative binomial distribution to the case of a positive real parameter r. Although it is impossible to visualize a non-integer number of “ failures ”, we can still formally define the distribution through its probability mass function.