Formally the Prince-Electors elected a King of the Romans, who was elected in Germany but became Holy Roman Emperor only when crowned by the Pope.

Formally, Gaussian curvature only depends on the Riemannian metric of the surface.

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, 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 the euro is the only legal tender in Büsingen ; however, the Swiss franc is far more popular in everyday use as shopping is generally done in nearby Schaffhausen instead of across the border ( s ) in Germany.

Formally, though, the name only survives in Finland and Sweden, where provinces bear the name.

Formally, Mbenga – Mbuti Pygmy music consists of at most only four parts, and can be described as an, " ostinato with variations ," or similar to a passacaglia, in that it is cyclical.

Formally, the complexity class MA is the set of decision problems that can be decided in polynomial time by an Arthur – Merlin protocol where Merlin's only move precedes any computation by Arthur.

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, a system is said to be observable if, for any possible sequence of state and control vectors, the current state can be determined in finite time using only the outputs ( this definition is slanted towards the state space representation ).

Formally, the definition only requires some invertibility, so we can substitute for Q any matrix M whose eigenvalues do not include − 1.

Formally, if is a limit ordinal, then a set is closed in if and only if for every < math >

Formally Roman Kalyn is now the only member of the group, but de-facto GreenJolly exists in name only.

Formally, if and only if, for any given, there exists an such that

Formally, a number n is colossally abundant if and only if there is an ε > 0 such that for all k > 1,

Formally, a natural number n is called highly abundant if and only if for all natural numbers m < n,

Formally this mission was declared to be only in pursuit of the Russian vessels and ammunition taken to Anzali by the White Russian counter-revolutionary General Denikin, who had been given asylum by British forces in Anzali.

Formally known as Crown Point Airport, the airport was renamed the Arthur Napoleon Raymond Robinson International Airport on May 19, 2011 in honour of the only Prime Minister and President from the island of Tobago.

Formally speaking, the opportunity was extended to all classes to join China's project on its Leninist terms.

Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base.

Formally, their designation is the letter Ž and the number.

Formally, they were part of the non-resident sections, but in fact constituted a separate division, largely autonomous in their activities.

Formally, according to the Constitution, citizens of Turkmenistan have the right to set up political parties and other public associations, acting within the framework of the Constitution and laws, and public associations and groups of citizens have the right to nominate their candidates in accordance with the election law.

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, 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.

Formally, her possessions and privileges ( including the government ) passed to her husband Robert Hathaway upon their marriage ( 1929 ) until his death ( 1954 ).

By 1912, the Social Democrats, with an explicitly anti-antisemitic program, were the largest party in the German Reichstag, and the Progressives ran very strongly as well ... Formally, at least, the Jews had been fully emancipated with the establishment of the German Empire, although they were kept out of certain influential occupations, enjoyed extraordinary prosperity ... Germans intermarried with Jews: in the 1930s some 50, 000 Jews were living in mixed German-Jewish marriages, so at least 50, 000 Germans, and presumably parts of their families, had familial contact with the Jews.

Formally the local Rabobanks are the mother organisation of Rabobank Nederland, their central organisation.

Formally titled ' An Act to protect all Persons in the United States in their Civil Rights, and furnish the Means of their vindication, the Act declared that people born in the United States and not subject to any foreign power are entitled to be citizens, without regard to race, color, or previous condition of slavery or involuntary servitude.

Formally, Chrétien and Martin have remained publicly respectful of each other ; while much of the verbal sparring was between their supporters.

Formally called the Final Declaration of the Stresa Conference, its aim was to reaffirm the Locarno Treaties and to declare that the independence of Austria " would continue to inspire their common policy ".

Formally, this means classifying finitely generated groups with their word metric up to quasi-isometry.

Formally, Siddharta continued their work without a bass player.

Formally, a redistribution of all agents ’ initial endowments of goods among all agents in the system is a core allocation if no group of agents can redistribute their own initial endowments among themselves so as to improve the satisfaction of someone in the group without impairing that of any other in the group.

Formally, they have always been considered Orthodox Christians, but most of them followed shamanism and animism as a part of their culture.

Formally, this meant that the whole peloton finished out of time limits, but the referees understandably used a rule saying that they could give clemency to any group of more than 20 % of the stage's starting riders, officially citing the weather conditions as their reason to do so.

However, shortly after this positive result, Kurt Gödel published On Formally Undecidable Propositions of Principia Mathematica and Related Systems ( 1931 ), showing that in any sufficiently strong axiomatic system there are true statements which cannot be proved in the system.

Formally neutral and reluctant to become involved with the great powers except as a last resort, Kuwait turned to the United States, the Soviet Union and Britain for naval protection of its tanker fleet after twenty-one ships were attacked in the gulf since late 1986.

Formally, a mortgage lender ( mortgagee ), or other lien holder, obtains a termination of a mortgage borrower ( mortgagor )' s equitable right of redemption, either by court order or by operation of law ( after following a specific statutory procedure ).

Formally introduced into Malaysia sometime after the end of the Confrontation, angklung found immediate popularity.

Formally nicknamed the " Pacemaker " by a staffer at Peterbilt after an in-house name contest in 1969 ( the winner got a color TV ).

Formally abolished in 1918 after the Russian Revolution, the academy was renamed several times.

Formally trained specialists recognized by the American Dental Association undergo a minimum of 2 – 3 years full-time rigorous education program after dental school graduation.

Formally, the Speaker ranks second in the protocol, after the President of Finland.

Formally known as the " International Copyright Act of 1891 ", but more commonly referred to as the " Chace Act " after Sen. Jonathan Chace of Rhode Island.

Formally called The Commissioner's Cup, it was renamed The Sasser Cup after former Commissioner George F. " Buddy " Sasser.