For nearby astronomical objects ( such as stars in our galaxy ) luminosity distance D < sub > L </ sub > is almost identical to the real distance to the object, because spacetime within our galaxy is almost Euclidean.

For much more distant objects the Euclidean approximation is not valid, and General Relativity must be taken into account when calculating the luminosity distance of an object.

For example, in three-dimensional complex Euclidean space,

For any subset A of Euclidean space R < sup > n </ sup >, the following are equivalent:

For a set P of points in the ( d-dimensional ) Euclidean space, a Delaunay triangulation is a triangulation DT ( P ) such that no point in P is inside the circum-hypersphere of any simplex in DT ( P ).

For over two thousand years, the adjective " Euclidean " was unnecessary because no other sort of geometry had been conceived.

For example a Euclidean straight line has no width, but any real drawn line will.

For example, a smooth manifold is a Hausdorff topological space that is locally diffeomorphic to Euclidean space.

For illustration, the Euclidean algorithm can be used to find the greatest common divisor of a = 1071 and b = 462.

( For example, the scientific progress of general relativity demonstrates that philosophers be wrong to pronounce a priori that space should have a Euclidean nature.

For any natural number n, an " n-sphere ," often written as S < sup > n </ sup >, is the set of points in ()- dimensional Euclidean space which are at a fixed distance r from a central point of that space, where r is, as before, a positive real number.

For example, in Z < sup > n </ sup > with Euclidean metric, a sphere of radius r is nonempty only if r < sup > 2 </ sup > can be written as sum of n squares of integers.

For example, the real line is Tychonoff under the standard Euclidean topology.

For example, the minimum spanning tree of the graph associated with an instance of the Euclidean TSP is a Euclidean minimum spanning tree, and so can be computed in expected O ( n log n ) time for n points ( considerably less than the number of edges ).

For any natural number n, an n-sphere of radius r is defined as the set of points in ( n + 1 )- dimensional Euclidean space which are at distance r from a central point, where the radius r may be any positive real number.

For any natural number n, an n-sphere of radius r is defined as the set of points in ( n + 1 )- dimensional Euclidean space that are at distance r from some fixed point c, where r may be any positive real number and where c may be any point in ( n + 1 )- dimensional space.

For S a subset of a Euclidean space, x is a point of closure of S if every open ball centered at x contains a point of S ( this point may be x itself ).

For instance, the general linear group GL ( n, R ) of all invertible n-by-n matrices with real entries can be viewed as a topological group with the topology defined by viewing GL ( n, R ) as a subset of Euclidean space R < sup > n × n </ sup >.

For example, in the " game " of Euclidean geometry ( which is seen as consisting of some strings called " axioms ", and some " rules of inference " to generate new strings from given ones ), one can prove that the Pythagorean theorem holds ( that is, you can generate the string corresponding to the Pythagorean theorem ).

For instance, bending without stretching or tearing a page of paper gives an isometric embedding of the page into Euclidean space because curves drawn on the page retain the same arclength however the page is bent.

For example, it follows that any closed oriented Riemannian surface can be C < sup > 1 </ sup > isometrically embedded into an arbitrarily small ε-ball in Euclidean 3-space ( there is no such C < sup > 2 </ sup >- embedding since from the formula for the Gauss curvature an extremal point of such an embedding would have curvature ≥ ε < sup >- 2 </ sup >).

For example, Euclidean geometry without the parallel postulate is incomplete ; it is not possible to prove or disprove the parallel postulate from the remaining axioms.

For a subset S of Euclidean space R < sup > n </ sup >, the following two statements are equivalent:

For example, Euclidean space is an example of a flat space, and Minkowski space is an example of a flat space-time.

" For Steiner, the human capacity for rational thought would allow individuals to comprehend spiritual research on their own and bypass the danger of dependency on an authority.

For a period after World War I, Steiner was extremely active and well known in Germany, in part because he lectured widely proposing social reforms.

For a period after World War I, Steiner was extremely active as a lecturer on social reform.

For Steiner, nature becomes self-conscious in the human being.

For Steiner, the cosmos is permeated and continually transformed by the creative activity of non-physical processes and spiritual beings.

For a memoir on " Certain cubic and biquadratic problems " the Royal Academy of Sciences of Berlin awarded him the Steiner prize.

For example, Vivi is designated as a black mage and is the only character who can use black magic, and Steiner is a knight and is the only character who can use sword skills.

For each temperament Steiner described less and more mature forms: the introspective and sensitive melancholic may be sullen and self-absorbed but can also become a sympathetic helper or a deep thinker.

For the Steiner tree problem, 31 is the number of possible Steiner topologies for Steiner trees with 4 terminals.

For N = 3, solution is given by a Steiner point located at the Fermat point of the triangle formed by the given points.

For general N, the Euclidean Steiner tree problem is NP-hard, and hence it is not known whether an optimal solution can be found by using a polynomial-time algorithm.

Team 3D would later regain the IWGP titles on October 18, 2009 at TNA Bound For Glory by defeating Beer Money, Booker T and Scott Steiner, and the British Invasion to capture their 24th Tag Team Championship.

For a vineyard to be considered biodynamic the wine-grower must use the nine biodynamic preparations, as described in 1924 by Rudolf Steiner.

For example, the Swordsworn were absorbed by House Davion, and the Stormhammers were absorbed by House Steiner.

For Steiner, the animal, vegetable, and mineral kingdoms represent " cast off " elements of the human entity from previous rounds.

For example, Max Steiner quotes the song in the opening scene of his late 1930s score to Gone with the Wind as a down-beat nostalgic instrumental to set the scene and Ken Burns makes use of instrumental versions in his 1990 Civil War documentary.

For them only a little more needed to be learned, and then all physical knowledge could be neatly sorted, packaged and put in the inventory to be drawn on for the solution of any human problem.

For those communities which have financial difficulties in effecting adjustments, there are a number of alternatives any one of which alone, or in combination with others, would minimize if not even eliminate the problem.

For, if so, the path leads through a complex process of parliamentary diplomacy which adds still another dimension to the problem.

For example, for the problem Af, 10 from 25 equals 15, then 6 from 15 equals 9.

For example, the officials of Poughkeepsie town ( township ) where the project is located think highly of it because it simplifies their snow clearing problem.

For outdoor signs and displays, where the problem of weathering resistance is no longer a factor, the choice of plastics is almost unlimited.

For example, child welfare experience abounds with cases in which the parental request for substitute care is precipitated by a crisis event which is meaningfully linked with a fundamental unresolved problem of family relationships.

For the purposes of this discussion, the problem of relative prices is encompassed in these two variables, since GNP includes other prices.

For the first time in history, the U.S. has produced a society in which less than one-tenth of the people turn out so much food that the Government's most embarrassing problem is how to dispose inconspicuously of 100 million tons of surplus farm produce.

For example, if for a given problem size a parallelized implementation of an algorithm can run 12 % of the algorithm's operations arbitrarily quickly ( while the remaining 88 % of the operations are not parallelizable ), Amdahl's law states that the maximum speedup of the parallelized version is times as fast as the non-parallelized implementation.

For the frequent case of propositional logic, the problem is decidable but Co-NP-complete, and hence only exponential-time algorithms are believed to exist for general proof tasks.

For instance, the P-38 Lightning with its thick high-lift wing had a particular problem in high-speed dives that led to a nose-down condition.

For example, the second edition of the popular introductory textbook, An Outline of Money, devoted the last three of its ten chapters to questions of foreign exchange management and in particular the ' problem of balance '.

" For instance ," said Dr. Joseph Merlino, Senior Editor of the book Freud at 150: 21st Century Essays on a Man of Genius, " that ... I'm a cross-dresser and I don't want to keep it confined to my circle of friends, or my party circle, and I want to take that to my wife and I don't understand why she doesn't accept it, or I take it to my office and I don't understand why they don't accept it, then it's become a problem because it's interfering with my relationships and environment.

For instance, because the 10 in the problem represents ten seconds, the expression f ′( 10 ) represents the change in pressure at a height of ten seconds, which is nonsense.

" For Adorno and Horkheimer, this posed the problem of how to account for the apparent persistence of domination in the absence of the very contradiction that, according to traditional critical theory, was the source of domination itself.

For example, primality testing is the problem of determining whether a given number is prime or not.

For example, consider the problem of primality testing.

For this reason, complexity theory addresses computational problems and not particular problem instances.

For each problem, the first person to solve it will be awarded $ 1, 000, 000 by the CMI.

For each problem, the Institute had a professional mathematician write up an official statement of the problem, which will be the main standard by which a given solution will be measured against.

For example, consider the problem of finding an integer x such that

For instance, Winter cites the example of a patient with a persistent headache supposedly tracing the problem to a doctor saying " Take him now " during the preclear's birth.

For example, the problem " given two numbers x and y, does x evenly divide y?