Formally, a binary operation on a set S is called associative if it satisfies the associative law:

Formally, a topological space X is called compact if each of its open covers has a finite subcover.

Formally, oxidation state is the hypothetical charge that an atom would have if all bonds to atoms of different elements were 100 % ionic.

Formally, a set S is called finite if there exists a bijection

Formally, if M is a set, the identity function f on M is defined to be that function with domain and codomain M which satisfies

Formally we mean that is an ideal if it satisfies the following conditions:

Formally, if d is the dimension of the parameter, and n is the number of samples, if as and as, then the model is semi-parametric.

Formally, in the finite-dimensional case, if the linear map is represented as a multiplication by a matrix A and the translation as the addition of a vector, an affine map acting on a vector can be represented as

Formally, a decision problem is P-complete ( complete for the complexity class P ) if it is in P and that every problem in P can be reduced to it by using an appropriate reduction.

If R is a ring, let R denote the ring of polynomials in the indeterminate X over R. Hilbert proved that if R is " not too large ", in the sense that if R is Noetherian, the same must be true for R. Formally,

Formally, a hypothesis is compared against its opposite or null hypothesis (" if I release this ball, it will not fall to the floor ").

Formally, the sequence of partial sums of some infinite summation converges if for every fixed power of X the coefficient stabilizes: there is a point beyond which all further partial sums have the same coefficient.

Formally, a function ƒ is real analytic on an open set D in the real line if for any x < sub > 0 </ sub > in D one can write

Formally, if is an open subset of the complex plane, a point of, and is a holomorphic function, then is called a removable singularity for if there exists a holomorphic function which coincides with on.

Formally, two variables are inversely proportional ( or varying inversely, or in inverse variation, or in inverse proportion or in reciprocal proportion ) if one of the variables is directly proportional with the multiplicative inverse ( reciprocal ) of the other, or equivalently if their product is a constant.

Informally, G has the above presentation if it is the " freest group " generated by S subject only to the relations R. Formally, the group G is said to have the above presentation if it is isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R.

Formally, if there exists some B ≥ 0 such that

Formally, if is any non-zero polynomial, it must be writable as.

Formally, if we write F < sub > Δ </ sub >( x ) to mean the f-polynomial of Δ, then the h-polynomial of Δ is

Formally, a cardinal κ is defined to be weakly compact if it is uncountable and for every function f: < sup > 2 </ sup > →

Formally speaking, the strategy pattern defines a family of algorithms, encapsulates each one, and makes them interchangeable.

Formally, a market is complete with respect to a trading strategy,, if there exists a self-financing trading strategy, such that at any time, the returns of the two strategies, and are equal.

Formally it was part of Arles, but in reality it was under the influence of the French king.

Formally, this means symmetry under a sub-group of the Euclidean group of isometries in two or three dimensional Euclidean space.

Formally, under Henry IV, Arundel served twice as Lord Chancellor, first in 1399 and again from 1407 to 1410.

Formally, the case where only a subset of parameters is defined is still a composite hypothesis ; nonetheless, the term point hypothesis is often applied in such cases, particularly where the hypothesis test can be structured in such a way that the distribution of the test statistic ( the distribution under the null hypothesis ) does not depend on the parameters whose values have not been specified under the point null hypothesis.

* General Headquarters United Nations Command ( Korea ) ( UNC ) — Formally activated 10 July 1950, before then Allied forces were formally under American operational control.

Formally, the city remained under Italian sovereignty until September 15, 1947 ( Paris Peace Treaties ).

Formally designated as the Manhattan Engineer District, it refers specifically to the period of the project from 1941 – 1946 under the control of the U. S. Army Corps of Engineers, under the administration of General Leslie R. Groves.

Formally, one inserts under the trace for each degree of freedom

Formally a vassal of the Persians, he secured the northern borders by subjugating the Caucasian mountaineers, and brought the adjacent western and southern Georgian lands under his control.

Formally, the differential appearing under the integral behaves exactly as a differential: thus, the integration by substitution and integration by parts formulae for Stieltjes integral correspond, respectively, to the chain rule and product rule for the differential.

Dick Smith Electronics has also long been known for its " home brand " range of electronics which fall under the brand name " Dick Smith " ( Formally called DSE, a commonly used abbreviation of Dick Smith Electronics ).

Formally, the 1772 Constitution was adopted by the Parliament ( Riksdag ) on 21 August 1772, but this took place as members of the Parliament and the Privy Council were under threat by the royal garrison on order by King Gustav III outside of Stockholm Palace, where the Parliament and Privy Council were assembled in different parts of the palace.

Formally the Greyhound Cafe, " The Entry " opened its door in March 1980 under the club Sam's as a venue that catered to more local bands.

Formally under the command of Poniatowski was Major General Józef Orłowski, the commander of the fortress of Kamieniec Podolski.

Formally, for any dimension, the orientation of the image of an object under a direct isometry with respect to that object is the linear part of that isometry.

Formally, the problem of supervised pattern recognition can be stated as follows: Given an unknown function ( the ground truth ) that maps input instances to output labels, along with training data assumed to represent accurate examples of the mapping, produce a function that approximates as closely as possible the correct mapping.

Formally, a coalgebra over a field K is a vector space C over K together with K-linear maps Δ: C → C ⊗ C and ε: C → K such that

Formally, sending X to and acting as the identity on maps is a functor ( indeed, an involution ).

Formally speaking, a collation method typically defines a total order on a set of possible identifiers, called sort keys, which consequently produces a total preorder on the set of items of information ( items with the same identifier are not placed in any defined order ).

Formally, this sharing of dynamics is referred to as universality, and systems with precisely the same critical exponents are said to belong to the same universality class.

Formally, a detailed balanced process is the one on which the probability to be in the state x and transit to x ' must be the same as to be in the state x ' and transit to x.

Formally, the Secretary-General is given the same rank as a High Commissioner or ambassador.

Formally nominated on December 2, 1902, Holmes was unanimously confirmed by the United States Senate on December 4, receiving his commission the same day.

Formally, the situation has the characteristics of a prisoner's dilemma game that is played repeatedly between the same partners, and in this case " defection "-jumping the queue-would not be advantageous.

Formally nominated on December 11, 1851, Curtis was confirmed by the United States Senate on December 20, 1851, and received his commission the same day.

Formally nominated on January 3, 1820, Bland was confirmed by the United States Senate on January 5, 1820, and received his commission the same day.

Formally nominated on July 13, 1867, Blatchford was confirmed by the United States Senate three days later, receiving his commission the same day.