Formally, the statement that " value decreases over time " is given by defining the linear differential operator as:

Formally, the work is a rondo, with each return of the theme appearing in varied form.

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, if the equilibrium strategy profile under the mechanism maps to the same goods allocation as a social choice function,

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 Lebesgue integral provides the necessary analytic device.

Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero and non-unit x of R can be written as a product ( including an empty product ) of irreducible elements p < sub > i </ sub > of R and a unit u:

Formally, the integral is the inner product of the luminosity function with the light spectrum.

Formally, the idea is to take the integral defining the ( inverse ) Fourier transform

Formally, DFS is an uninformed search that progresses by expanding the first child node of the search tree that appears and thus going deeper and deeper until a goal node is found, or until it hits a node that has no children.

Formally, a statistic s is a measurable function of X ; thus, a statistic s is evaluated on a random variable X, taking the value s ( X ), which is itself a random variable.

Formally known as the Inverurie Parish Church, a split in Church of Scotland over the appointment of ministers in 1843 ( The Disruption ) led to the creation of the West Parish Church ( known locally as " The West Kirk ") which was founded as a Free Church ( a church free from Edinburgh control ) thus causing the Inverurie Parish Church to change its name.

Formally, the algorithm's performance will be a random variable determined by the random bits ; thus either the running time, or the output ( or both ) are random variables.

Formally, the issue is that interfertile " able to interbreed " is not a transitive relation – if A can breed with B, and B can breed with C, it does not follow that A can breed with C – and thus does not define an equivalence relation.

Formally organized vocational programs supported by federal funds allow high school students to gain experience in a field of work which is likely to lead to a full-time job on graduation.

Formally, the set of all context-free languages is identical to the set of languages accepted by pushdown automata ( PDA ).

More rigorously, the divergence of a vector field F at a point p is defined as the limit of the net flow of F across the smooth boundary of a three dimensional region V divided by the volume of V as V shrinks to p. Formally,

* Formally appoints the Prime Minister of Finland selected by the Parliament, and formally appoints the rest of the cabinet ( Council of State ) as proposed by the Prime Minister

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, when working over the reals, as here, this is accomplished by considering the limit as ε → 0 ; but the " infinitesimal " language generalizes directly to Lie groups over general rings.

Formally, these failed when they were rejected by the Church of England's General Synod in 1972 ; conversations and co-operation continued, however, leading in 2003 to the signing of a covenant between the two churches.

Formally, this is a solution of the matrix equation by Jacobi iteration.

Formally it is defined by the equation

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, 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

Limits and colimits in a category C are defined by means of diagrams in C. Formally, a diagram of type J in C is a functor from J to C:

Formally named " United Press Associations " for incorporation and legal purposes, but publicly known and identified as United Press or UP, it was created by the 1907 uniting of three smaller news syndicates by the Midwest newspaper publisher E. W. Scripps.

Formally, most of these people were convicted by NKVD troikas (" triplets ")– special courts martial.

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, 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.

Formally, we are given a set of hypotheses and a set of manifestations ; they are related by the domain knowledge, represented by a function that takes as an argument a set of hypotheses and gives as a result the corresponding set of manifestations.

Formally, a transductive support vector machine is defined by the following primal optimization problem:

Formally, powers with positive integer exponents may be defined by the initial condition

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.