* 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, 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 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, 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.

; Electronics: Formally a fixture in radio shacks, now used for assembly and repair of all sorts of electronic equipment including communication, computer, and home entertainment items.

Formally, Aff ( V ) is naturally isomorphic to a subgroup of, with V embedded as the affine plane, namely the stabilizer of this affine plane ; the above matrix formulation is the ( transpose of ) the realization of this, with the ( n × n and 1 × 1 ) blocks corresponding to the direct sum decomposition.

Formally, HPGCC is a cross-compiler ; it compiles code for the ARM-based HP calculators, but runs on a PC rather than the target system.

Formally, Korea's governmental FTC ( Fair Trade Commission: 공정거래위원회 ; 公正去來委員會 ) announces a limited Chaebol list every year as size of industrial assets ( not including financial companies ).

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, 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, a deterministic algorithm computes a mathematical function ; a function has a unique value for any given input, and the algorithm is a process that produces this particular value as output.

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

Formally, a squeeze preserves the hyperbolic metric expressed in the form ; in a different coordinate system.

Bābak Khorram-Din ( Formally known as " Pāpak " meaning " Young Father ") (, alternative spelling: Bābak Xorramdin ; 795, according to some other sources 798 — January 838 ) was one of the main PersianArmenian historian Vardan Areweltsʻi, ca.

Formally described, it alternates slow sections in a modal F with faster sections, " Neue Kraft fühlend " ( with renewed strength ), in D. The slow sections each have two elements, ( 1 ) a passage reminiscent of the opening of the first movement in which the instruments overlap each other with a brief motive ; ( 2 ) a chorale, the actual song.

Formally, a Markov partition is used to provide a finite cover for the smooth system ; each set of the cover is associated with a single symbol, and the sequences of symbols result as a trajectory of the system moves from one of the covering sets to another.

Formally, citizen science has been defined as " the systematic collection and analysis of data ; development of technology ; testing of natural phenomena ; and the dissemination of these activities by researchers on a primarily avocational basis ".

Formally, at least, the European Commission has primarily a monitoring role ; in practice, however, there is considerable scope for it to help set the policy agenda and persuade reluctant Member States to implement agreed policies.

Formally, let P be a poset ( partially ordered set ), and let F be a filter on P ; that is, F is a subset of P such that:

Formally ;

Formally, a biased graph Ω is a pair ( G, B ) where B is a linear class of circles ; this by definition is a class of circles that satisfies the theta-graph property mentioned above.

The slaves never shared in their profits, while they did share, in a very real sense, in the profits of the slave-owners: they were fed, clothed, doctored, and so forth ; ;

The latter's real descendents were unable to take root in England ; ;

Thus, many a creativity-oriented aspirant for a career in architecture, drama, or journalism, resigns himself to a real estate business ; ;

The Domina sounds real enough, if we could only trust the conditions under which we learn of its use ; ;

But this is not the real problem ; ;

He gave Mr. Jack a real going-over ; ;

This is the real protagonist of the drama ; ;

It never seemed real ; ;

a review of the philosophy of Communist leaders by Ted Slack, another real estate agent who became interested as a philosophy major at the University of Miami ; ;

The same argument proves that no subfield of the real field is algebraically closed ; in particular, the field of rational numbers is not algebraically closed.

Numbers such as π that are not algebraic are said to be transcendental ; almost all real and complex numbers are transcendental.

If one quantizes a real scalar field, then one finds that there is only one kind of annihilation operator ; therefore, real scalar fields describe neutral bosons.

* In arithmetic, addition and multiplication of real numbers are associative ; i. e.,

He was extremely skeptical of Peter Bartholomew's discovery in Antioch of the Holy Lance, especially because he knew such a relic already existed in Constantinople ; however, he was willing to let the Crusader army believe it was real if it raised their morale.

* Given any topological space X, the continuous real-or complex-valued functions on X form a real or complex unitary associative algebra ; here the functions are added and multiplied pointwise.

The field, then, is ripe for new Southerners to step to the fore and write of this twentieth-century phenomenon, the Southern Yankeefication: the new urban economy, the city-dweller, the pains of transition, the labor problems ; ;

For example, there are persons who are in physical science, in the field of mineralogy, trained in crystallography, who use only X-rays, applying only the powder technique of X-ray diffraction, to clay minerals only, and who have spent the last fifteen years concentrating on the montmorillonites ; ;

In a symposium and general exploration of the field of Personal Perception and Interpersonal Behavior the discussion does not touch upon this aspect of the subject, with one possible exception ; ;

He had come to learn that a reputation for peculiarity allowed mere field officers a certain leeway at Court ; ;

In support of the emphasis placed by the Department of Defense on instruction in emergency medical care, the Medical Illustration Service developed casualty simulation kits and rescue breathing manikins which are being field tested ; ;

Mossberg's latest contribution to the field is the Model 500 ( from $73.50 ) ; ;

Many studies indicate that elementary-school children's interests cover the whole field of science ; ;

he hunts folklore for the very `` field and stream '' reasons that many persons hunt game ; ;

Indeed, it is in the field of transportation that Congress has most frequently granted employers exemption from the anti-trust laws ; ;

In Greece, there was since the 19th century a science of the folklore called laographia ( laography ), in the form of " a science of the interior ", although theoretically weak ; but the connotation of the field deeply changed after World War II, when a wave of Anglo-American anthropologists introduced a science " of the outside ".

The same kind of work that Einstein did, the same kind of work, using the same methods ; but in a much broader field, much more close to human relationships.

Louisa May Alcott, who was ten years old at the time, later wrote of the experience in Transcendental Wild Oats ( 1873 ): " The band of brothers began by spading garden and field ; but a few days of it lessened their ardor amazingly.

Alfred's military reorganisation of Wessex consisted of three elements: the building of thirty fortified and garrisoned towns ( burhs ) along the rivers and Roman roads of Wessex ; the creation of a mobile ( horsed ) field force, consisting of his nobles and their warrior retainers, which was divided into two contingents, one of which was always in the field ; and the enhancement of Wessex's seapower through the addition of larger ships to the existing royal fleet.