Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds.

Since the energy-momentum tensor T < sub > μν </ sub > is normally understood to be due to concentrations of matter in four-dimensional space, the above result is interpreted as saying that four-dimensional matter is induced from geometry in five-dimensional space.

Since definitions of left and right based on the geometry of the natural environment are unwieldy, in practice, the meaning of relative direction words is conveyed through tradition, acculturation, education, and direct reference.

Since spherical geometry violates the parallel postulate, there exists no such triangle on the surface of a sphere.

Since Euclid, geometry had meant the geometry of Euclidean space of two dimensions ( plane geometry ) or of three dimensions ( solid geometry ).

Since the group of affine geometry is a subgroup of the group of projective geometry, any notion invariant in projective geometry is a priori meaningful in affine geometry ; but not the other way round.

Since projective geometry excludes compass constructions, there are no circles, no angles, no measurements, no parallels, and no concept of intermediacy.

Since A lies between the Euclidean group E and the group of projectivities P, affine geometry is sometimes mentioned in connection with the Erlangen program, which is concerned with group inclusions such as E ⊂ A ⊂ P.

Since the axioms of ordered geometry as presented here include properties that imply the structure of the real numbers, those properties carry over here so that this is an axiomatization of affine geometry over the field of real numbers.

Since the Erlangen program of Klein, the nature of any given geometry has been seen as the connection of symmetry and the content of propositions, rather than the style of development.

Since heat tends to spread through a solid until the body reaches an equilibrium state of constant temperature, Ricci flow may be hoped to produce an equilibrium geometry for a manifold for which the Ricci curvature is constant.

Since then many mathematicians reserve the term geometry for a space together with a group of mappings of that space.

Since this additional check is applied only to geometry found in violation of the Manhattan distance rule, it is not invoked often, so the computational overhead is very small.

Since the geometry of the top of the weir is known and all water flows over the weir, the depth of water behind the weir can be converted to a rate of flow.

Since the molecule has a bent geometry, the result is a dipole across the whole ozone molecule.

Since the number of homology theories has become large ( see: Category: Homology theory ), the terms Betti homology and Betti cohomology are sometimes applied ( particularly by authors writing on algebraic geometry ) to the singular theory, as giving rise to the Betti numbers of the most familiar spaces such as simplicial complexes and closed manifolds.

Since the Supreme Court's decision of that year this is more doubtful ; ;

( Since the time-span of the nation-state coincides roughly with the separate existence of the United States as an independent entity, it is perhaps natural for Americans to think of the nation as representative of the highest form of order, something permanent and unchanging.

Since it is not far from Viareggio, he will visit Puccini's house, as he never fails to do, to pay his respects to the memory of the composer of La Boheme, which he considers one of Puccini's masterpieces.

Since attack serves to stimulate interest in broadcasts, I added to my opening statement a sentence in which I claimed that German youth seemed to lack the enthusiasm which is a necessary ingredient of anger, and might be classified as uninterested and bored rather than angry.

Since the air is a continuum, the network of communication remains intact regardless of the positions or motions of the points ( the people ) in the net.

Since the hazards of poor communication are so great, p can be justified as a habitable site only on the basis of unusual productivity such as is made available by a waterfall for milling purposes, a mine, or a sugar maple camp.

Since the difficulty of drawing the net is great, we will merely discuss it.

Since a civilizational crisis involves also a crisis in private interests and in the ruling class, reaction is normally found among those who feel themselves to be among the ruling class.

Since civilizational change is the most difficult to perceive and analyze, it seldom is given adequate attention.

Since the slogans have little application to reality and are sanctimonious to boot, the applause is faint even in areas of the world where we should expect to find the greatest affection for free government.

Since more is known about Quiney than about any other acquaintance of Shakespeare in Stratford, his career may be followed to its sudden end in 1602.

Since the great flood of these dystopias has appeared only in the last twelve years, it seems fairly reasonable to assume that the chief impetus was the 1949 publication of Nineteen Eighty-Four, an assumption which is supported by the frequent echoes of such details as Room 101, along with education by conditioning from Brave New World, a book to which science-fiction writers may well have returned with new interest after reading the more powerful Orwell dystopia.

Since Laos is of no more purely military value to Moscow itself than it is to Washington, this approach might be expected to head off Mr. Khrushchev for the moment.

Since little is known about autism, and almost nothing has been written for the layman, we'd like to share one experienced mother's comments.

Since the work is done by hand, the only limitation, it is said, `` is that of human conception ''.

Since the apparatus is new, it requires experimentation and changes in technique.

Since the obvious is not always true, the Republican National Committee wisely analyzed its defeat of last autumn and finds that it occurred, as suspected, in the larger cities.

Since he has just shown who is top dog, he may not be ready to receive this highest honor in the gift of the Soviet people.

Since appeals to morality, to humanity, and to sanity have had such small effect, perhaps our last recourse is the deterrent example.

Since the goal of our international planners is a World Government, this Atlantic Community would mark a giant step in that direction for, once American economic autonomy is absorbed, a larger grouping is a question of time.

Since charges are relatively the same, reserving a car before you leave for Europe will assure you of having one on tap when you want it.

Since Af are distinct prime polynomials, the polynomials Af are relatively prime ( Theorem 8, Chapter 4 ).

Since we are considering here relatively small diameter ( 1 - 1.5 inches ) fiber plates, their average thickness can be kept below 1/4 inch and their internal losses may be assumed as 15 percent ( per plate ).

Since gene therapy is relatively new and at an experimental stage, it is an expensive treatment to undertake.

Since then, consistent liberalization and astute economic management has led to the removal of 95 % of all price controls, low unemployment, a positive balance of payments position, a stable exchange rate, a shift of exports from former communist economic bloc markets to Western Europe, and relatively low foreign debt.

Since Fjorgyn is not only the name of a goddess, but the feminine byname for Earth, it is relatively safe to assume that " mær ", in this case, means " daughter ".

Since such substances may affect a person's judgment even after consuming relatively small amounts, zero tolerance is advocated by gun safety teachers.

Since the 2. 4 GHz band is heavily used to the point of being crowded, using the relatively unused 5 GHz band gives 802. 11a a significant advantage.

Since the War Department required a large number of vehicles to be manufactured in a relatively short time, Willys-Overland granted the United States Government a non-exclusive license to allow another company to manufacture vehicles using Willys ' specifications.

Since the emergence of the post-1943 state and after the destruction of the Ottoman Caliphate, national policy has been determined largely by a relatively restricted group of traditional regional and sectarian leaders.

Since then, a whole series of relatively large areas of the brain have been found to be involved in speech processing.

Since combat engineers with mine-clearing equipment can clear a path through a minefield relatively quickly, mines are usually considered effective only if covered by fire.

Since the 12th century, the territory has remained stable and relatively independent of its neighbours ; one of the few German territories for which this is true.

Since dextropropoxyphene produces relatively modest pain relief compared to other opioids but still produces severe respiratory depression at high doses, it is particularly dangerous when abused, as drug users may take dangerously high doses in an attempt to achieve narcotic effects.

Since neodymium coloration depends upon " forbidden " f-f transitions deep within the atom, there is relatively little influence on the color from the chemical environment, so the color is impervious to the thermal history of the glass.

Since the islands are located in the center of the East China Sea relatively close to Japan, China and South-East Asia, the Ryūkyū Kingdom became a prosperous trading nation.

Since cycles typically constitute a relatively small amount of reclaimed space, the collection cycles can be spaced much farther apart than with an ordinary tracing garbage collector.

Since 1848, the Swiss Confederation has been a federal state of relatively autonomous cantons, some of which have a history of confederacy that goes back more than 700 years, arguably putting them among the world's oldest surviving republics.

Since then, changes to the line length have been a relatively common variant, such that Stephen Burt has written: " sestinas, as the form exists today, not require expertise with inherited meter …".

Since 1989 Solidarity has become a more traditional trade union, and had relatively little impact on the political scene of Poland in the early 1990s.

Since the tertiary structure of proteins is an important problem in biochemistry, and since structure determination is relatively difficult, protein structure prediction has been a long-standing problem.

Since all matter and energy gravitate, gravity's effects are cumulative ; by contrast, the effects of positive and negative charges tend to cancel one another, making electromagnetism relatively insignificant on cosmological length scales.

Since the velar region of the roof of the mouth is relatively extensive and the movements of the dorsum

Since wireframe renderings are relatively simple and fast to calculate, they are often used in cases where a high screen frame rate is needed ( for instance, when working with a particularly complex 3D model, or in real-time systems that model exterior phenomena ).