More precisely, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function.

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 technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.

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,

More precisely, they were only able to perform the calculation when the theory contains one less flavor of chiral matter than the number of colors in the special unitary gauge group, because in the presence of fewer flavors an unbroken nonabelian gauge symmetry leads to an infrared divergence and in the case of more flavors the contribution in equal to zero.

More generally, if κ is any infinite cardinal, then a product of at most 2 < sup > κ </ sup > spaces with dense subsets of size at most κ has itself a dense subset of size at most κ ( Hewitt – Marczewski – Pondiczery theorem ).

More generally, Stokes ' theorem applies to oriented manifolds M with boundary.

More recent statements of the theorem are sometimes careful to exclude the equality condition ; that is, the condition is if x ( t ) contains no frequencies higher than or equal to B ; this condition is equivalent to Shannon's except when the function includes a steady sinusoidal component at exactly frequency B.

** More generally, Rademacher's theorem extends the differentiability result to Lipschitz mappings between Euclidean spaces: a Lipschitz map ƒ: U → R < sup > m </ sup >, where U is an open set in R < sup > n </ sup >, is almost everywhere differentiable.

More generally, one can count points of Grassmannian over F: in other words the number of subspaces of a given dimension k. This requires only finding the order of the stabilizer subgroup of one such subspace and dividing into the formula just given, by the orbit-stabilizer theorem.

More examples of the use of the empty product in mathematics may be found in the binomial theorem, factorial, fundamental theorem of arithmetic, birthday paradox, Stirling number, König's theorem, binomial type, difference operator, Pochhammer symbol, proof that e is irrational, prime factor, binomial series, and multiset.

More precisely, the time hierarchy theorem for deterministic Turing machines states that for all time-constructible functions,

More precisely, the theorem states that for any given number of colours c, and any given integers n < sub > 1 </ sub >,..., n < sub > c </ sub >, there is a number, R ( n < sub > 1 </ sub >, ..., n < sub > c </ sub >), such that if the edges of a complete graph of

More precisely, the change of variables formula is stated in the next theorem:

More generally: all uniformly convex Banach spaces are reflexive according to the Milman – Pettis theorem.

More specifically, it implies that it is sufficient to prove the theorem for prime numbers, after which the more general theorem follows.

More generally, while Hurwitz's theorem states that an identity of form,

More generally, one can consider the image, kernel, coimage, and cokernel, which are related by the fundamental theorem of linear algebra.

More generally, Horn clauses are relevant to automated theorem proving by first-order resolution.

More precisely, Gabriel's theorem states that:

More generally, the Chevalley – Shephard – Todd theorem characterizes finite groups whose algebra of invariants is a polynomial ring.

More specifically, the primitive element theorem characterizes those finite degree extensions such that there exists with.

More generally, the theorem fails for equipped with any norm () ( Schwartz 1969, p. 20 ).

More precisely, find necessary and sufficient conditions on the tuple ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >) and ( y < sub > 1 </ sub >, ..., y < sub > n </ sub >) separately, so that there is an element of R with the property that x < sub > i </ sub >· r = y < sub > i </ sub > for all i. If D is the set of all R-module endomorphisms of U, then Schur's lemma asserts that D is a division ring, and the Jacobson density theorem answers the question on tuples in the affirmative, provided that the x's are linearly independent over D.

More precisely, their theorem states that there is no apportionment system that has the following properties ( as the example we take the division of seats between parties in a system of proportional representation ):

More generally, the equipartition theorem states that any degree of freedom x which appears in the total energy H only as a simple quadratic term Ax < sup > 2 </ sup >, where A is a constant, has an average energy of ½k < sub > B </ sub > T in thermal equilibrium.