More precisely, Ural – Altaic came to subgroup Finno-Ugric and Samoyedic as " Uralic " and Turkic, Mongolic, and Tungusic as " Altaic ", with Korean sometimes added to Altaic, and less often Japanese.

More precisely, if S

More precisely, a binary operation on a non-empty set S is a map which sends elements of the Cartesian product S × S to S:

More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other.

More precisely, objects can be reachable in only two ways:

More precisely, it tries to classify problems that can or cannot be solved with appropriately restricted resources.

More precisely:

More generally, to insist that all evidence converge precisely with no deviations would be naïve falsificationism, equivalent to considering a single contrary result to falsify a theory when another explanation, such as equipment malfunction or misinterpretation of results, is much more likely.

More precisely, the positions of the pixels and subpixels on the screen must be exactly known to the computer to which it is connected.

( More precisely, this is true of the sines of the angles.

More importantly, however, he went on to maintain the same high level of performance throughout the season, kicking precisely 100 goals for the year to become the first player to top the ton since Richmond's Jack Titus in 1940.

" More precisely, it is " the quantitative analysis of actual economic phenomena based on the concurrent development of theory and observation, related by appropriate methods of inference.

More precisely, a full moon occurs when the geocentric apparent ( ecliptic ) longitudes of the Sun and Moon differ by 180 degrees ; the Moon is then in opposition with the Sun .< ref >

More precisely, all known FFT algorithms require Θ ( N log N ) operations ( technically, O only denotes an upper bound ), although there is no known proof that better complexity is impossible.

More precisely, because of the relation between spin and statistics, a particle containing an odd number of fermions is itself a fermion: it will have half-integer spin.

More precisely, and technically, a Feynman diagram is a graphical representation of a perturbative contribution to the transition amplitude or correlation function of a quantum mechanical or statistical field theory.

More precisely, he showed that a random graph on vertices, formed by choosing independently whether to include each edge with probability has, with probability tending to 1 as goes to infinity, at most cycles of length or less, but has no independent set of size Therefore, removing one vertex from each short cycle leaves a smaller graph with girth greater than in which each color class of a coloring must be small and which therefore requires at least colors in any coloring.

More precisely, a groupoid G is:

( More precisely, the nodes are spherical harmonics that appear as a result of solving Schrödinger's equation in polar coordinates.

More precisely, the square of x is not invertible because it is impossible to deduce from its output the sign of its input.

More precisely, the law states that alleles of different genes assort independently of one another during gamete formation.

More precisely, it returns a new list whose elements are in reverse order compared to the given list.

More precisely, these proofs have to be verifiable in polynomial time by a deterministic Turing machine.

More precisely, it is the instant when the Moon and the Sun have the same ecliptical longitude.

More informally, there is an algorithm that halts if u =

: More typically, solving a game means providing an algorithm that secures a win for one player, or a draw for either, against any possible moves by the opponent, from the beginning of the game.

More commonly, only the name of the library ( and not the path ) is stored in the executable, with the operating system supplying a method to find the library on-disk based on some algorithm.

More commonly, a function is defined by a formula, or ( more generally ) an algorithm — a recipe that tells how to compute the value of f ( x ) given any x in the domain.

More specific refinements to the algorithm can help to generate mazes that are harder to solve.

More often than not, a particular algorithm can be specifically tailored to a particular problem, yielding better performance than a generic algorithm.

More specifically, the time for this algorithm is.

More specifically, a trait with ' good ' permanence will be reasonably invariant over time with respect to the specific matching algorithm.

More formally, a language L belongs to UP if there exists a two input polynomial time algorithm A and a constant c such that

More specifically, the running time for testing whether H is a minor of G in this case is O ( n < sup > 3 </ sup >), where n is the number of vertices in G and the big O notation hides a constant that depends superexponentially on H. Thus, by applying the polynomial time algorithm for testing whether a given graph contains any of the forbidden minors, it is possible to recognize the members of any minor-closed family in polynomial time.

More engineer-minded cranks may claim to have invented a magic compression algorithm or a perpetual motion / free energy machine.

More modern variations of tomography involve gathering projection data from multiple directions and feeding the data into a tomographic reconstruction software algorithm processed by a computer.

More complicated algorithms include the Luhn algorithm ( 1954 ), which captures 98 % of single digit transposition errors ( it does not detect 90 ↔ 09 ), while more sophisticated is the Verhoeff algorithm ( 1969 ), which catches all single digit substitution and transposition errors, and many ( but not all ) more complex errors.

) More commonly, however, the expected value ( mean or average ) of the sampled values is chosen ; this is a Bayes estimator that takes advantage of the additional data about the entire distribution that is available from Bayesian sampling, whereas a maximization algorithm such as expectation maximization ( EM ) is capable of only returning a single point from the distribution.

More precisely, the algorithm outputs an estimate for, the number of marked entries, with the following accuracy:.

More specifically, it is an open question whether a nesting depth of more than 1 is required, and if so, whether there is an algorithm to determine the minimum required star height.

More specifically, the greedy algorithm provides a factor 1 + log | V | approximation of a minimum dominating set, and show that no algorithm can achieve an approximation factor better than c log | V | for some c > 0 unless P = NP.

More specifically, a polynomial-time approximation algorithm for domatic partition with the approximation factor for a constant would imply that all problems in NP can be solved in slightly super-polynomial time.

More advanced algorithms, such as Arnoldi's algorithm and the Lanczos algorithm, save this information and use the Gram – Schmidt process or Householder algorithm to reorthogonalize them into a basis spanning the Krylov subspace corresponding to the matrix.