Since there is a continual loss of micrometeoritic material in space because of the radiation effects, there must be a continual replenishment: otherwise, micrometeorites would have disappeared from interplanetary space.

Since Af are linearly independent functions and the exponential function has no zeros, these R functions Af, form a basis for the space of solutions.

Since the earth is rotating and the unleveled gyro-stabilized platform is fixed with respect to a reference in space, an observer on the earth will see the platform rotating ( with respect to the earth ).

Since the block headers occupied space, the usable capacity of the drive was dependent on the block size.

Since there are infinitely many vectors in the basis, this is an infinite-dimensional Hilbert space.

Since commercial signage usually has limited physical space, signage is normally in French alone.

Since a maximal ideal in A is closed, is a Banach algebra that is a field, and it follows from the Gelfand-Mazur theorem that there is a bijection between the set of all maximal ideals of A and the set Δ ( A ) of all nonzero homomorphisms from A to C. The set Δ ( A ) is called the " structure space " or " character space " of A, and its members " characters.

Since every compact Hausdorff space is a Tychonoff space, and every subspace of a Tychonoff space is Tychonoff, we conclude that any space possessing a Hausdorff compactification must be a Tychonoff space.

Since the map d restricts to 0 on I < sub > x </ sub >< sup > 2 </ sup > ( the reader should verify this ), d descends to a map from I < sub > x </ sub > / I < sub > x </ sub >< sup > 2 </ sup > to the dual of the tangent space, ( T < sub > x </ sub > M )< sup >*</ sup >.

Since the subarachnoid space around the brain and spinal cord can contain only 135 to 150 ml, large amounts are drained primarily into the blood through arachnoid granulations in the superior sagittal sinus.

Since Cauchy sequences can also be defined in general topological groups, an alternative to relying on a metric structure for defining completeness and constructing the completion of a space is to use a group structure.

Since the corona has been photographated at high resolution in the X-rays by the satellite Skylab in 1973, and then later by Yohkoh and the other following space instruments, it has been seen that the structure of the corona is very various and complex: different zones have been immediately classified on the coronal disc

Since traditional " Euclidean " space never reaches infinity, the projective equivalent, called extended Euclidean space, must be formed by adding the required ' plane at infinity '.

Since work is defined as a force acting through a distance ( a length of space ), energy is always equivalent to the ability to exert pulls or pushes against the basic forces of nature, along a path of a certain length.

Since Euclidean space is a metric space it is also a topological space with the natural topology induced by the metric.

Since this implies that one particle is communicating with the other instantaneously across space, i. e. faster than light, this is the " paradox ".

Since the poles of a meromorphic function are isolated, there are at most countably many.

Since the indeterminacy is relatively harmless, and has meromorphic continuation everywhere, there is a sense in which the properties of Z ( s ) do not essentially depend on it.

Since any vector v is a linear combination ∑ v < sup > j </ sup > e < sub > j </ sub > of its components, df is uniquely determined by df < sub > p </ sub >( e < sub > j </ sub >) for each j and each p ∈ U, which are just the partial derivatives of f on U. Thus df provides a way of encoding the partial derivatives of f. It can be decoded by noticing that the coordinates x < sup > 1 </ sup >, x < sup > 2 </ sup >,... x < sup > n </ sup > are themselves functions on U, and so define differential 1-forms dx < sup > 1 </ sup >, dx < sup > 2 </ sup >, ..., dx < sup > n </ sup >.

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 the center of U ( n ) is a 1-dimensional abelian normal subgroup of U ( n ), the unitary group is not semisimple.

Since we are assuming circular motion, let r ( t ) = R · u < sub > r </ sub >, where R is a constant ( the radius of the circle ) and u < sub > r </ sub > is the unit vector pointing from the origin to the point mass.

Since at a minimum the starting code address vector must always be valid on reset, systems commonly included some nonvolatile memory ( e. g. ROM ) starting at address zero to contain the vectors and bootstrap code.

Since the diagonal vector has a vertical component, the drive phase continues to provide some support against gravity and can be regarded as an extension of the support phase.

Since this construction works for any vector space V, we conclude that T is a functor from K-Vect to K-Alg.

Since the physicist's concept of force has a direction and a magnitude, it may be seen as a vector.

Since the typical vector operation would involve loading a small set of data into the vector registers and then running several operations on it, the vector system of the new design had its own separate pipeline.

Since this is an electromagnetic wave each electric field vector has a corresponding, but not illustrated, magnetic field vector that is at a right angle to the electric field vector and proportional in magnitude to it.

Since the propagation vector is proportional to the photon's momentum, the transverse propagation direction must remain the same in both regions.

Since the Poynting vector only occurs in Poynting's theorem as a divergence ∇ • S, the Poynting vector S is arbitrary to the extent that one can add a curl of a field F to S,

Since elements of W are necessarily elements of V, axioms 1, 2 and 5-8 of a vector space are satisfied.

Since then, the supercomputer market has focused much more on massively parallel processing rather than better implementations of vector processors.

Since a vector field is quite difficult to visualize at first, in elementary physics one may instead visualize this field with field lines.

Since the intensity or irradiance, I, of light is proportional to the square of the electric-field vector, the ellipticity becomes:

Since all these values are distinct, this is usually done using a bit vector.

However, the more difficult proof of the converse ( see below ) makes use of the vector space structure: Since both of the factors are vector spaces over Q, the tensor product can be taken over Q.

Since version 3. 0, Leveller includes vector shapes, which can be used for both modeling and coloration.

The set of ( p < sub > i </ sub >, n < sub > i </ sub >) is thus a sampling of a continuous vector field V. The implicit function f is found by integrating the vector field V. Since not every vector field is the gradient of a function, the problem may or may not have a solution: the necessary and sufficient condition for a smooth vector field V to be the gradient of a function f is that the curl of V must be identically zero.