However, there will be several terms of the form x < sup > n − 2 </ sup > y < sup > 2 </ sup >, one for each way of choosing exactly two binomials to contribute a y.

However, if both parents are non-citizens then the parents must take the extra step of choosing Latvian citizenship for their child — who is automatically entitled, but for whom citizenship is not automatic ( neither granted nor imposed ).

However, by 1956 the term " lesbian " had such a negative meaning that the DOB refused to use it as a descriptor, choosing " variant " instead.

However, five captures can, so players should consider captures when choosing moves.

However, there is no evidence that she consciously pressured him to do so, though she was an effective manager of his career and often collaborated with him in choosing subjects.

However, because the queens are all alike, and that no two queens can be placed on the same square, the candidates are all possible ways of choosing of a set of 8 squares from the set all 64 squares ; which means 64 choose 8

However, the book lacks a finale to match that description ; its actual ending focuses more on Zarathustra recognizing that his legacy is beginning to perpetuate, and consequently choosing to leave the higher men to their own devices in carrying his legacy forth.

However, planning permission was not granted by the local council and Yarm School is no longer planning to move, instead choosing to renovate and improve the current site.

However, it might be that in addition to choosing for poetic skill, they are also inclined to choose people with whom they have values in common, either about poetry or about other matters, resulting in a continuous stream of prizewinning white female poets.

However, Nintendo did not initiate any legal proceedings against either company, choosing rather to pressure them with warning of legal action should they not take appropriate reaction themselves.

However, seven months later ITV changed its mind, choosing instead to close the nearby studios dedicated to the production of Emmerdale and transferring production of this programme to Kirkstall Road which would be upgraded for HDTV production.

However, the Mongol leader Kitbuqa did not retreat, choosing instead to continue to fight, until he was eventually killed by veteran Mamluk warrior Jamal al-Din Akoush al-Shamsy.

" However, any of these can be changed to one of seven different angles ( Top, Bottom, Left, Right, Front, Back, or 3D ) by right-clicking anywhere in the viewing area and choosing " View Angle.

However, this force was spread out piecemeal over a 170-mile front, whereas the Japanese Army could concentrate its efforts on any single point of its choosing.

However, he argues, choosing manipulation negates, in its means, the democratic ideal.

However, Great Khan Mongke had died in late 1259, requiring Hulagu to return Karakorum to engage in choosing the next Great Khan.

However, the party had abandoned much of the egalitarian aspects associated with conventional Communist Party practice, instead choosing to pursue a mixed economy.

However, unlike Stojaković, Bodiroga declined the offer to play in the NBA, choosing instead to remain in Europe.

However a delay in choosing the paint in turn inevitably delays buying the paint which, although it may not subsequently mean any delay to the entire project, does mean that choosing the paint has no ' free float ' attached to it-despite having no free float of its own the choosing of the paint is involved with a path through the network which does have ' total float '.

However he was unsuccessful in his bid for the Speakership, with the House instead choosing Labour MP Michael Martin for the role.

However, the persons elected to the Assembly at the 2003 Assembly Election were called together on 15 May 2006 under the Northern Ireland Act 2006 for the purpose of electing a First Minister and Deputy First Minister and choosing the members of an Executive ( before 25 November 2006 ) as a preliminary to the restoration of devolved government.

However, Husayn refused, choosing to abide by the treaty between Hassan ibn Ali and Muawiyah, which could not be broken at that time.

However, polynomials with integer coefficients that are irreducible over the field can be reducible over a finite field.

However, it is possible to prove that any NFA is reducible to an equivalent DFA.

However the Turing machine model only provides an answer to the question of what computability of functions means and, with interactive tasks not always being reducible to functions, it fails to capture our broader intuition of computation and computability.

However, there exist linkless graphs that are not YΔY reducible, such as the apex graph formed by connecting an apex vertex to every degree-three vertex of a rhombic dodecahedron.

However, a degree of independence is also asserted of emergent properties, so that they are not identical to, or reducible to, or predictable from,

However, this tends to be complicated by the fact that this requires finding the roots of a high degree polynomial, and there can be multiple roots that represent local maxima.

However, one can obtain an explicit formula for majority of polynomial size using a sorting network of Ajtai, Komlós, and Szemerédi.

However for certain sets of such polynomials it may happen that for only finitely many combinations all polynomial functions take the value zero.

However, this is not a polynomial time algorithm because the number of digits in the solution may be as large as √ n, far larger than a polynomial in the number of digits in the input value n.

However, a quantum computer can compute the answer to some of these problems in polynomial time ; one well-known example of this is Shor's factoring algorithm.

However, probably the most important work on polynomial curves and sculptured surface was done by Pierre Bézier ( Renault ), Paul de Casteljau ( Citroen ), Steven Anson Coons ( MIT, Ford ), James Ferguson ( Boeing ), Carl de Boor ( GM ), Birkhoff ( GM ) and Garibedian ( GM ) in the 1960s and W. Gordon ( GM ) and R. Riesenfeld in the 1970s.

However, in 1702 Leibniz said that no polynomial of the type x < sup > 4 </ sup > + a < sup > 4 </ sup > ( with a real and distinct from 0 ) can be written in such a way.

However, if the set of numbers ( called the knapsack ) is superincreasing — that is, each element of the set is greater than the sum of all the numbers before it — the problem is ' easy ' and solvable in polynomial time with a simple greedy algorithm.

However in expressions that involve terms with other types of factors than constants and powers of variables, the notion of constant term cannot be used in this sense, since that would lead to calling " 4 " the constant term of, whereas substituting 0 for x in this polynomial makes it evaluate to 13. so

) However some polynomial equations, of arbitrarily high degree, are solvable with such operations.

However, since the splitting field of a quintic polynomial has at most 5!

However, the quotient group ( isomorphic to itself ) is not an abelian group, and so is not solvable, so it must be that the general polynomial of the fifth degree has no solution in radicals.

However, since the result is on the general polynomial, it does say that a general " quintic formula " for the roots of a quintic using only a finite combination of the arithmetic operations and radicals in terms of the coefficients is impossible.

However, certain classes of puzzles, such as those in which each row or column has only one block of cells and all cells are connected, may be solved in polynomial time by transforming the problem into an instance of 2-satisfiability.

However, one can combine these remainders recursively to reduce the cost, using the following trick: if we want to evaluate modulo two polynomials and, we can first take the remainder modulo their product, which reduces the degree of the polynomial and makes subsequent modulo operations less computationally expensive.

However, this assumption does leave out finite energy states like solitons which can't be generated by a polynomial of fields smeared by test functions because a soliton, at least from a field theoretic perspective, is a global structure involving topological boundary conditions at infinity.

However, Vieta created many innovations: the binomial formula, which would be taken by Pascal and Newton, and the link between the roots and coefficients of a polynomial, called Vieta's formula.

However, a polynomial in several variables may be regarded as a polynomial in only " the last " variable, but with coefficients being polynomials in the others.

However, the directions on links are a vital detail to retain as one recurses through a polynomial calculation.

However, more sophisticated top-down parsers have been created by Frost, Hafiz, and Callaghan which do accommodate ambiguity and left recursion in polynomial time and which generate polynomial-sized representations of the potentially exponential number of parse trees.

However, when G is part of the input but H is fixed, it can be solved in polynomial time.

However the polynomial will satisfy the criterion for p after substitution of for x: this gives