Therefore, in general Euclidean spaces every similarity is an affine transformation, because the Euclidean group E ( n ) is a subgroup of the affine group.

Therefore the symmetry group of a Euclidean space is the Euclidean group and for a Minkowski space it is the Poincaré group.

" Therefore any result in Euclidean geometry that follows from these three postulates will hold in elliptic geometry, such as proposition 1 from book I of the Elements, which states that given any line segment, an equilateral triangle can be constructed with the segment as its base.

Therefore the vectors of the root system are in eight-dimensional Euclidean space: they are described explicitly later in this article.

Therefore a symmetry group of rotational symmetry is a subgroup of E < sup >+</ sup >( m ) ( see Euclidean group ).

Therefore, finding the Euclidean minimum spanning tree as a spanning tree of the complete graph or as a spanning tree of the Delaunay triangulation both take O ( dn < sup > 2 </ sup >) time.

Therefore, the Euclidean shortest path problem may be decomposed into two simpler subproblems: constructing the visibility graph, and applying a shortest path algorithm such as Dijkstra's algorithm to the graph.

Therefore, four loose families of more-efficient light transport modelling techniques have emerged: rasterization, including scanline rendering, geometrically projects objects in the scene to an image plane, without advanced optical effects ; ray casting considers the scene as observed from a specific point-of-view, calculating the observed image based only on geometry and very basic optical laws of reflection intensity, and perhaps using Monte Carlo techniques to reduce artifacts ; and ray tracing is similar to ray casting, but employs more advanced optical simulation, and usually uses Monte Carlo techniques to obtain more realistic results at a speed that is often orders of magnitude slower.

Therefore, even though the relative geometry of the Earth-Sun-Moon system will be nearly identical after a saros, the Moon will be in a different position with respect to the stars.

Therefore, this case is what usually is meant when talking about Hilbert's sixteenth problem in real algebraic geometry.

Therefore, to produce the image which the retina beholds, using projective geometry techniques, it may be convenient to substitute a spherical surface of projection for a plane of projection, i. e., use a " picture sphere " rather than a " picture plane " and place the eye, or " station point ," at the center of the picture sphere.

Therefore distance geometry has immediate relevance where distance values are determined or considered, such as in surveying, cartography and physics.

Therefore, the relative satellite-receiver geometry plays a major role in determining the precision of estimated positions and times.

Therefore, the central atom is predicted to adopt a linear geometry.

Therefore, the predicted geometry is trigonal.

Therefore the multistatic ambiguity function is mostly usefully defined as a function of two-or three-dimensional position and velocity vectors for a given multistatic geometry and transmitted waveform.

Therefore, since geometry is important in screen applications, screen materials should be at least specified by their gain and their total reflectance.

Therefore this is also the geometry of a C < sup > 2 </ sup >/ Z < sub > n </ sub > orbifold in string theory after its conical singularity has been smoothed away by its " blow up " ( i. e., deformation ).

Therefore, for any value of T the number of values of f{t} is equal to the ( finite ) number of tangent points corresponding to the argument T plus an odd number.

Therefore, any measures taken in peacetime which will decrease force requirements in war will contribute greatly to success when hostilities occur.

Therefore, neither merit the beatific vision, nor yet are subjected to any punishment, because they are not guilty of any personal sin although they have not received baptism, so still bear original sin.

Therefore, given any positive integer n, it produces a string with Kolmogorov complexity at least as great as n. The program itself has a fixed length U. The input to the program GenerateComplexString is an integer n. Here, the size of n is measured by the number of bits required to represent n, which is log < sub > 2 </ sub >( n ).

Therefore, any test of a scientific theory must prohibit certain results which will falsify the theory, and expect other specific results which will be consistent with the theory.

Therefore it soon announced and implemented a hostile doctrine against any political dissent, whether inside or outside the party.

Therefore, it is necessary to have the existence of a conflict in public law without any constitutional aspects and no assignment to another jurisdiction.

Therefore, native Caymanians do not have any Amerindian heritage from their own islands ; however, a significant number of Jamaicans have settled in the Cayman Islands over the years, so they and their descendants may have some Amerindian blood via Jamaica.

Therefore, the diagonal of a 4 x 9 lattice does not intersect any other square lattice | lattice points

Therefore strings beginning with C ( or any other capital letter ) would be sorted before strings with lower-case a, b, etc.

Therefore, just as Christianity does not accept that Mosaic Law has any authority over Christians, Judaism does not accept that the New Testament has any religious authority over Jews.

Therefore an optimizer can replace a string copy with a sequence of word moves without any loops, which will be much faster.

Therefore, as outlined in the example above, neither the EPR experiment nor any quantum experiment demonstrates that faster-than-light signaling is possible.

Therefore, General Niazi and Air Marshal Enamul Haque, Commander of Eastern Air Force Command ( EAFC ), were failed to launch any operation in East Pakistan against Indian or its allies.

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.

Therefore, the first and second rules (" always treat the gun as if loaded ", " never let the muzzle cover non-targets ") apply to any gun, even one loaded with blanks.

Therefore, all this must be kept secret-which rules out any approach to the police or other outside agency.

Therefore, any eigenstate of the electron in the hydrogen atom is described fully by four quantum numbers.

Therefore any electron can interchange with another one with no observable physical change of the system at all.

Therefore " Liberty will abolish interest ; it will abolish profit ; it will abolish monopolistic rent ; it will abolish taxation ; it will abolish the exploitation of labour ; it will abolish all means whereby any labourer can be deprived of any of his product.

Therefore, language, in the absence of translation, comprises a barrier to a worldwide community of debate and opinion, although it is also true that media within any given society may be split along class, political or regional lines.

Therefore, before any interrogation begins, the police must advise the suspect that:

Therefore, under independent voting, Af or 0.36 of the couples would cast two votes for picture A, and Af or 0.16 would cast two votes for picture B.

Therefore, the scholar, as he looks at our national folklore of the last 60 years, will be mindful of two facts.

Therefore, the only conceivable alternatives are those represented, on the one hand, by the two at least apparently self-consistent but mutually exclusive positions of Buri and Barth and, on the other hand, by the third but really pseudo position ( analogous to a round square ) of Bultmann.

Therefore, they are considered two forms of antibodies: soluble form or membrane-bound form ( Parham 21-22 ).

Therefore, communication is social interaction where at least two interacting agents share a common set of signs and a common set of semiotic rules.

Therefore, at maximum operation, each potential traffic channel can have two simultaneous data channels available for messages.

Therefore, two main views on Ezekiel are popular in Muslim theology:

Therefore, these two correspond to the elements 1 and 5 of Z < sub > 6 </ sub >, in that order or conversely.

Therefore, the energy eigenstates may be classified by two angular momentum quantum numbers, ℓ and m ( both are integers ).

" Therefore in this early period it was Wycliffe who recognised and formulated one of the two major formal principles of the Reformation — the unique authority of the Bible for the belief and life of the Christian.

Therefore, if the determinant equals zero then the parallelogram has no area, and that will happen when two vectors are on the same line.

Therefore, in a duration of two millennia, there will be 485 leap years.

Therefore, two sources of light which produce the same intensity ( W / m < sup > 2 </ sup >) of visible light do not necessarily appear equally bright.

Therefore, the first three give an idea of the seasons whereas the last two give an idea of the position among the constellations of the full moon.

Therefore, output and income are usually considered equivalent and the two terms are often used interchangeably.

The other triose, the aldose H ( C = O )( CHOH )< sub > 2 </ sub > H ( glyceraldehyde ), has one chiral carbon — the central one, number 2 — which is bonded to groups-H ,-OH ,-C ( OH ) H < sub > 2 </ sub >, and -( C = O ) H. Therefore, it exists as two stereoisomers whose molecules are mirror images of each other ( like a left and a right glove ).

Therefore, if two laws conflict with each other, a court must decide which law applies.

Therefore, to help standardize guidelines, recommendations for water consumption are included in two recent European Food Safety Authority ( EFSA ) documents ( 2010 ): ( i ) Food-based dietary guidelines and ( ii ) Dietary reference values for water or adequate daily intakes ( ADI ).

Therefore, hexadecimal is more commonly used in programming languages today, since two hexadecimal digits exactly specify one byte.

Therefore, two poles rated at the same weight are not necessarily the same stiffness.

Therefore, aperiodic or quasiperiodic structures can be divided into two main classes: those with crystallographic point-group symmetry, to which the incommensurately modulated structures and composite structures belong, and those with non-crystallographic point-group symmetry, to which quasicrystal structures belong.

Therefore, in the 49-seat parliament, all 47 Samoan Members of Parliament are also matai, performing dual roles as chiefs and modern politicians, with the exception of the two seats reserved for non-Samoans.

Therefore, Gauss and Weber chose to encode the alphabet in a binary code, using positive current and negative as the two states.

Therefore, in a right-angled triangle, the two non-right angles total 90 ° ( π / 2 radians ), so each of these angles must be in the range of ( 0 °, 90 °) as expressed in interval notation.