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 advanced questions involve the topology of the curve and relations between the curves given by different equations.

More than one century after Euler's paper on the bridges of Königsberg and while Listing introduced topology, Cayley was led by the study of particular analytical forms arising from differential calculus to study a particular class of graphs, the trees.

More generally, if F is a subset of the algebraic dual space, then the initial topology of X with respect to F, denoted by σ ( X, F ), is the weak topology with respect to F.

More generally, if a family of seminorms Q defines the topology on Y, then the seminorms p < sub > q, x </ sub > on L ( X, Y ) defining the strong topology are given by

More generally, Euclidean n-space R < sup > n </ sup > with addition and standard topology is a topological group.

More information about the topology can be exchanged using opaque LSA carrying type-length-value elements.

More explicitly, an injective continuous map f: X → Y between topological spaces X and Y is a topological embedding if f yields a homeomorphism between X and f ( X ) ( where f ( X ) carries the subspace topology inherited from Y ).

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. Notations used for boundary of a set S include bd ( S ), fr ( S ), and ∂ S.

More specifically, it is in general topology that basic notions are defined and theorems about them proved.

More transparent proofs rely on the mathematical machinery of algebraic topology, and these lead to generalizations to higher-dimensional spaces.

More specifically, in topology, compact Hausdorff topological spaces can be reconstructed from the Banach algebra of functions on the space ( Gel ' fand-Neimark ).

More formally, in algebraic topology and differential topology a line bundle is defined as a vector bundle of rank 1.

More generally, one can define a sheaf for any Grothendieck topology on a category in a similar way.

More narrowly, one can require that the map f: N → M be an inclusion ( one-to-one ), in which we call it an injective immersion, and define an immersed submanifold to be the image subset S together with a topology and differential structure such that S is a manifold and the inclusion f is a diffeomorphism: this is just the topology on N, which in general will not agree with the subset topology: in general the subset S is not a submanifold of M, in the subset topology.

More general formulations of the theorem exist that give necessary and sufficient conditions for a family of functions from a compactly generated Hausdorff space into a uniform space to be compact in the compact-open topology.

More abstractly, the " pay-off set " ( i. e., the set of all plays in which the angel wins ) is a closed set ( in the natural topology on the set of all plays ), and it is known that such games are determined.

More complex panels use a Bus network topology where the wire basically is a data loop around the perimeter of the facility, and has " drops " for the sensor devices which must include a unique device identifier integrated into the sensor device itself ( e. g. iD biscuit ).

More precisely, the images of open sets in the Zariski topology are again open.