Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory: the idea of symmetry, as exemplified by Galois through the algebraic notion of a group ; geometric theory and the explicit solutions of differential equations of mechanics, worked out by Poisson and Jacobi ; and the new understanding of geometry that emerged in the works of Plücker, Möbius, Grassmann and others, and culminated in Riemann's revolutionary vision of the subject.

In the case that n = 2 ( two-body problem ), the situation is much simpler than for larger n. Various explicit formulas apply, where in the more general case typically only numerical solutions are possible.

In two's complement arithmetic, attempts to divide the smallest signed integer by are attended by similar problems, and are handled with the same range of solutions, from explicit error conditions to undefined behavior.

However, Hamilton's equations usually don't reduce the difficulty of finding explicit solutions.

In numerical analysis, the Runge – Kutta methods () are an important family of implicit and explicit iterative methods for the approximation of solutions of ordinary differential equations.

Weaker theorems, qualitatively speaking, but with explicit constants, can actually be applied, in conjunction with machine computation, to prove that lists of solutions ( suspected to be complete ) are actually the entire solution set.

While explicit solutions can be found for equations that are quadratic, cubic, and quartic in y, the same is not in general true for quintic and higher degree equations, such as

A passive DSS is a system that aids the process of decision making, but that cannot bring out explicit decision suggestions or solutions.

This category is referred to as " posterior articulation of preferences ," implying that the DM ’ s involvement starts posterior to the explicit revelation of " interesting " solutions ( see for example Karasakal and Köksalan, 2009 ).

The Thue equation is soluble effectively: there is an explicit bound on the solutions x, y of the form where constants C < sub > 1 </ sub > and C < sub > 2 </ sub > depend only on the form ƒ.

Even the answers to real questions began with vague hints, so a player could choose to stop short of getting explicit solutions to the puzzles.

Only the simplest differential equations admit solutions given by explicit formulas ; however, some properties of solutions of a given differential equation may be determined without finding their exact form.

Well known examples of explicit vacuum solutions include:

Given the difficulty of constructing explicit small families of solutions, much less presenting something like a " general " solution to the Einstein field equation, or even a " general " solution to the vacuum field equation, a very reasonable approach is to try to find qualitative properties which hold for all solutions, or at least for all vacuum solutions.

, if and are source density and circulation density, respectively, of the electromagnetic induction fields and calculated as usual from and by the equations and The explicit solutions for and – unique, if all quantities vanish sufficiently fast at infinity – are known as retarded potentials.

If there is no explicit general solution, computers can use slope fields ( even if they aren ’ t shown ) to numerically find graphical solutions.

Steffen practices what he describes as solutions-based journalism, that is, the explicit goal of his work at Worldchanging is to highlight the possible solutions to what the editorial team sees as the planet's most pressing problems, rather than to spread news of those problems or critiques of their causes.

Wheeler did not exhibit explicit geon solutions to the vacuum Einstein field equation, a gap which was partially filled by Brill and Hartle in 1964 by the Brill-Hartle geon.

They found that phonics instruction must be systematic ( following a sequence of increasingly challenging phonics patterns ) and explicit ( teaching students precisely how the patterns worked, e. g., " this is b, it stands for the / b / sound ").

Virtual team leaders need to spend more time than conventional team counterparts being explicit about expectations, because the patterns of behaviour and dynamics of interaction are unfamiliar.

Synthetic and analytic phonics approaches both generally involve explicit, carefully sequenced instruction that teach a large body of phonics patterns.

Similarly in D version 1, it is assumed that the programmer knows what to do with delegates and local variables, as their references will be invalid after return from its definition scope ( local variables are on the stack )-this still permits many useful functional patterns, but for complex cases needs explicit heap allocation for variables.

Similar patterns were found in other countries and in 1960 Paul Samuelson and Robert Solow took Phillips ' work and made explicit the link between inflation and unemployment: when inflation was high, unemployment was low, and vice-versa.

The increased attention in recent years on spatial dynamics has highlighted the need for new quantitative methods that can analyze patterns, determine the importance of spatially explicit processes, and develop reliable models.

Searches may be explicit or may use wildcards / patterns or terms to be excluded.

They provide different ways ( patterns ) to remove explicit references in the concrete classes from the code that needs to instantiate them.

While " most attributes such as micro-residue distribution patterns and micro-wear will develop similarly on points used to tip spears, darts or arrows " and " explicit tests for distinctions between thrown spears and projected arrows have not yet been conducted " the researchers find " contextual support " for the use of these points on arrows: a broad range of animals were hunted, with an emphasis on taxa that prefer closed forested niches, including fast moving, terrestrial and arboreal animals.

The site employs a proven technique called " Direct Teaching ", which emphasizes explicit teaching of skills such as letter-sound associations, spelling patterns and vocabulary words.

Both use introspective, subjectivist methods of depth psychology, attempting to disclose or make explicit the ‘ archetypal patterns ’ of prelogical experience and culture ” ( Aldrich 1953: 153 ).

The Join Java language can express virtually all published concurrency patterns without explicit recourse to low-level monitor calls.

In 2006, following concerns expressed by the committee that the wording of the Irish Constitution does not allow the State to intervene in cases of abuse other than in very exceptional cases, the Irish government undertook to amend the constitution to make a more explicit commitment to children's rights.

The following year William N. Pendleton, Lee's artillery chief, claimed in the same venue that Longstreet disobeyed an explicit order to attack at sunrise on July 2.

The Black – Scholes model of the market for a particular stock makes the following explicit assumptions:

To be more explicit, the following universal property can be used.

Public policy increasingly targeted the Roma on the explicit basis of race following the Law for the Fight Against Gypsies, Vagrants and the Workshy.

The triangle numbers are given by the following explicit formulas:

In the cases of one, two, and three variables they are called unary, binary, and ternary and have the following explicit form:

Unlike most national newspapers, the Daily Star has limited articles on politics and has rarely shown clear support for any specific party or leader ; although in the run-up to the 2010 general election the newspaper printed several articles which hinted that it wanted to see Labour and Gordon Brown voted out of power, while at the same time it offering no explicit support for the Conservatives or their leader David Cameron – who ultimately became Prime Minister following the formation a new coalition government with the Liberal Democrats when his party failed to gain an overall majority in parliament.

Bellmer gave up doll-making and spent the following decades creating erotic drawings, etchings, sexually explicit photographs, paintings, and prints of pubescent girls.

This view has been disputed, with documents uncovered from the Soviet archives revealing that the North Vietnamese invasion of 1970 was launched at the explicit request of the Khmer Rouge following negotiations with Nuon Chea.

The earliest post was more explicit, saying it contained the following:

) are used to indicate that only the first portion of a known reference is given explicitly, with broad reference to the following passages which logically follow in sequence to the explicit reference.

Future New Zealand, which was not represented in parliament, was a " secularised " evolution of the Christian Democrats, following the same basic principles as the Christian Democrats, but abandoning the explicit religious connection.

< span id = ch2 ></ span > The following chapter, The Laws of Robotics, included stories that were written after the explicit formulation of the three laws, however both stories include elements that place them outside the Foundation universe.

* 1975 – Noboru Tanaka's A Woman Called Sada Abe was released, for a Japan-only audience, but was overshadowed internationally by its more explicit successor that came out the following year.

* 1976 – Nagisa Oshima's In the Realm of the Senses which was widely banned following its release, for explicit scenes of sex and nudity.

Accordingly, profits are only charged to corporation tax if they fall within one of the following, and are not otherwise exempted by an explicit provision of the Taxes Acts:

Examples of explicit data collection include the following:

The base language provides the following functions only but extensions are discussed in Henderson's book for the explicit support of lazy evaluation and non-deterministic programming.

Brice accepted, and proved a shrewd businessman ; Foster would later name Brice “ the most remarkable man ever met the most successful borrower ever saw .” Ignoring Foster ’ s explicit instructions for his work on the business proceedings in New York and instead following his instincts, Brice returned with a shocking profit of $ 40, 000, and earned the trust and friendship of the governor.

Although consensual sex acts between adults can be shown in explicit detail in R18 pornographic works, the following is prohibited:

Shepherd's followers claim that the Covenant of Works between Adam and God in the Garden of Eden was not originally part of covenant theology, following John Murray's observation that a covenant of works at creation does not receive explicit mention in early confessions such as the French Confession ( 1559 ), the Scots Confession ( 1560 ), the Belgic Confession ( 1561 ), the Thirty-Nine Articles ( 1562 ), the Heidelberg Catechism ( 1563 ), and the Second Helvetic Confession ( 1566 ).

In the novel the Prime Minister is ultimately forced from office following a catastrophic nuclear accident at an experimental nuclear plant that he had pushed for during his role as Secretary of State for Energy during a previous government ( the most explicit parallel between Harry Perkins and Tony Benn in the novel ).