Formally the " head of state " can also personally be the " head of government " ( ex officio or by ad hoc cumulation such as an absolute monarch nominating himself ) but otherwise has formal precedence over the Head of Government and other ministers, whether he is their actual political superior ( absolute monarch, executive president ) or rather theoretical or ceremonial in character.

Formally, the theorem is stated as follows: There exist unique integers q and r such that a = qd + r and 0 ≤ r < | d |, where | d | denotes the absolute value of d.

Formally, collective noun forms such as “ a group of people ” are represented by second-order variables, or by first-order variables standing for sets ( which are well-defined objects in mathematics and logic ).

Formally, such genes should increase in frequency when

Formally, an elliptic function is a function meromorphic on for which there exist two non-zero complex numbers and with ( in other words, not parallel ), such that and for all.

Formally, if there exists some B ≥ 0 such that

Formally, cl ( S ) denotes the smallest subset Y of M that contains S such that for each reaction ( A, B )

Formally, the word is applied to persons who are publicly accepted in a recognised capacity, such as professional employment, graduation from a course of study, etc., to give critical commentaries in one or any of a number of specific fields of public or private achievement or endeavour.

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.

Formally, a measurable cardinal is an uncountable cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ.

Formally, a ring is an Abelian group ( R, +), together with a second binary operation * such that for all a, b and c in R,

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, a Hopf algebra is a ( associative and coassociative ) bialgebra H over a field K together with a K-linear map S: H → H ( called the antipode ) such that the following diagram commutes:

Formally, given such that, the corresponding characteristic number is:

Formally, assuming the axiom of choice, cardinality of a set X is the least ordinal α such that there is a bijection between X and α.

Formally a congruence subgroup is one that contains Γ ( N ) for some N, and the least such N is the level or Stufe of the subgroup.

Formally, given a vector field v, a vector potential is a vector field A such that

Formally, knowing the step response of a dynamical system gives information on the stability of such a system, and on its ability to reach one stationary state when starting from another.

Formally, a vertex cover of a graph G is a set C of vertices such that each edge of G is incident to at least one vertex in C. The set C is said to cover the edges of G. The following figure shows examples of vertex covers in two graphs ( and the set C is marked with red ).

Formally, a string is a finite sequence of symbols such as letters or digits.

Formally, a cardinal number κ is λ-unfoldable if and only if for every transitive model M of cardinality κ of ZFC-minus-power set such that κ is in M and M contains all its sequences of length less than κ, there is a non-trivial elementary embedding j of M into a transitive model with the critical point of j being κ and j ( κ ) ≥ λ.

Formally, given a finite set X, a collection C of subsets of X, all of size n, has Property B if we can partition X into two disjoint subsets Y and Z such that every set in C meets both Y and Z.

Formally, we define a bad field as a structure of the form ( K, T ), where K is an algebraically closed field and T is an infinite, proper, distinguished subgroup of K, such that ( K, T ) is of finite Morley rank in its full language.

Even in such technical curricula as engineering, the senior is much more likely than the freshman to choose, as an ideal, liberal education over specific vocational preparation.

The use of abeyance in such instances can allow such an organization to ' settle ' with the party without officially binding its actions in the future, should a new group of decision makers within the organization choose to pursue taking the dispute to court.

So, when a woman considers her breasts deficient in some respect, she might choose to undergo a plastic surgery procedure to enhance them, either to have them augmented or to have them reduced, or to have them reconstructed if she suffered a deformative disease, such as breast cancer.

For this reason, editors choose cutaway shots related to the main action, such as another action or object in the same location.

Often when a fan does not agree with one of the events in a story ( such as the death of a favorite character ) they will choose to ignore the event in question so that their enjoyment of the franchise is not diminished.

At the start, players choose from advances such as pottery, the wheel, and the alphabet to, near the end of the game, nuclear fission and spaceflight.

Although the content of the document file is arranged in a format that the word-processing program understands, the user is able to choose the name and location of the file and provide the bulk of the information ( such as words and text ) that will be stored in the file.

In such cases, cartographers may have to choose between various phonetic spellings of local names versus older imposed, sometimes resented, colonial names.

Bacteria such as E. coli are unable to choose the direction in which they swim, and are unable to swim in a straight line for more than a few seconds due to rotational diffusion.

Those that stayed behind and replanted choose high-yielding varieties like Trebbiano and Sangiovese clones such as the Sangiovese di Romagna from the nearby Romagna region.

Another writer says, " My own position can be summarized in the following terms: just as music is whatever people choose to recognize as such, noise is whatever is recognized as disturbing, unpleasant, or both " ( Nattiez 1990, 47 – 48 ).

To elect means " to choose or make a decision " and so sometimes other forms of ballot such as referendums are referred to as elections, especially in the United States.

) take place at all, with voters free to choose any person at the time of voting — with some possible exceptions such as through a minimum age requirement — in the jurisdiction.

In practice, this means the government remains in power for close to its full term, and choose an election date it calculates to be in its best interests ( unless something special happens, such as a motion of no-confidence ).

Some electrical engineers choose to pursue a postgraduate degree such as a Master of Engineering / Master of Science ( M. Eng ./ M. Sc.

Although unsure as to why these saints should choose such a " sinner " as their patron, Einhard nonetheless set about ensuring they continued to receive a resting place fitting of their honour.

Parents are free to choose any public or special school for their children, although in some urban areas, such as Amsterdam, admissions procedures do exist.

If we choose the shape of the control volume such that all flow in or out occurs perpendicular to its surface, then the flow of matter into the system performs work as if it were a piston of fluid pushing mass into the system, and the system performs work on the flow of matter out as if it were driving a piston of fluid.

They include activities such as parades and galas to choose the Carnival King.

GMs may choose to run a game based on a published game world, with the maps and history already in place ; such game worlds often have pre-written adventures.

Election usually is the constitutional way to choose the head of state of a republic, and some monarchies, either directly through popular election, indirectly by members of the legislature or of a special college of electors ( such as the Electoral College in the United States ), or as an exclusive prerogative.

Prior to Montaigne's time, humanists such as Pico della Mirandola had argued that man was God's greatest creation, made in God's image and able to choose his own nature, but this view was subsequently challenged in Michel de Montaigne's Essais of 1580.

In such cases leadership may choose to deny the risk.

Currently Mac OS X programmers can choose between limited IDEs, including native IDEs like Xcode, older IDEs like CodeWarrior, newer IDEs like MyEclipse, and open-source tools, such as Eclipse and Netbeans.