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Page "Projective geometry" ¶ 23
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Since and coordinates
Since Cartesian coordinates are unique and non-ambiguous, the points of a Cartesian plane can be identified with all possible pairs of real numbers ; that is with the Cartesian product, where is the set of all reals.
Since the actual physical surface of the Earth is too complex for mathematical analysis two levels of abstraction are employed in the definition of these coordinates.
Since there are many different reference ellipsoids the latitude of a feature on the surface is not unique: this is stressed in the ISO standard which states that " without the full specification of the coordinate reference system, coordinates ( that is latitude and longitude ) are ambiguous at best and meaningless at worst ".
Since reference datums can have different radii and different center points, a specific point on the earth can have substantially different coordinates depending on the datum used to make the measurement.
Since its inception, COMUSNAVAK has grown to become responsible for coordinating all Navy activity in the Alaska and Aleutian area, for detailed planning and coordination for the Naval portion of the Joint and Combined EXERCISE Northern Edge, and coordinates high-visibility U. S. Navy ship visits through-out Alaska in support of public relations and recruiting initiatives.
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 a sound is a vibration of a medium ( such as air ), a sound signal associates a pressure value to every value of time and three space coordinates.
Since it is possible to rotate the space-like separated polarizations independently by local perturbations, the phase should not depend on the polarization in appropriately chosen field coordinates.
Since Lagrangian mechanics is based on generalized coordinates, transformations of the coordinates q → Q do not affect the form of Lagrange's equations and, hence, do not affect the form of Hamilton's equations if we simultaneously change the momentum by a Legendre transform into
Since the new and old coordinates are each independent, the following equations must hold
Since the old coordinates and new momenta are each independent, the following equations must hold
Since the new and old coordinates are each independent, the following equations must hold
Since the new and old coordinates are each independent, the following equations must hold
Since, the conversion back to polar coordinates is only possible if.
Since barycentric coordinates are a linear transformation of Cartesian coordinates, it follows that they vary linearly along the edges and over the area of the triangle.
Since the message also gives astronomical coordinates, he also shares it with Saul Shriffer, a fictional scientist who is said to have worked with Frank Drake on Project Ozma.
Since matrix multiplication has no effect on the zero vector ( i. e., on the coordinates of the origin ), rotation matrices can only be used to describe rotations about the origin of the coordinate system.
Since rays in space can be parameterized by three coordinates, x, y, and z and two angles and, as shown at left, it is a five-dimensional function.
Since his first term Pitts has been chairman of the Values Action Team, a subgroup of the Republican Study Committee that coordinates legislation with the Christian right.
Since the Go board is symmetrical with no particular sides, it makes no difference which corner is used as the reference point from which to count coordinates.
Since these are elliptic integrals, the coordinates ξ and η can be expressed as elliptic functions of u.
: Conversely, suppose that φ is a smooth real-valued function on M in a neighborhood of p. Since every smooth function has a local coordinate expression, we may express φ as a function in the coordinates.

Since and are
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 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 we can neither undo nor redo the past, we are limited to the events of today and tomorrow.
Since the recognition of Israel as a nation state, claims are made in many cases which identify the claimant as a member of the new body politic.
Since accurate base maps are necessary for any planning program, the first step taken by the planning division to implement the long-range state plan has been to prepare two series of base maps -- one at a scale of 1 inch to a mile, and the second a series of 26 sheets at a scale of 1 inch to 2000 feet, covering the entire state.
Since conventional methods are insensitive at the low frequencies of these molecular transitions, the paramagnetic resonance method is being used instead.
Since broadcast frequencies are very limited in number, these objectives are to some extent inconsistent in that not all of them can be fully realized, and to the extent that each is realized, there is a corresponding reduction of the possibilities for fullest achievement of the others.
Since the provision of skywave service requires adequate freedom from interference, only class 1, stations are capable of rendering skywave service.
Since the sides are also covered up to the spray rails, they are also rough-sanded in that area.
Since fly cutters are one sided and not balanced, they should be used at the slowest speed available, and fed very slowly to avoid binding.
Since there are almost no schools in the country offering graduate work in interior design this rule cannot at present be observed.
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 practical washing procedures are both lengthy and expensive, a number of laboratory tests have been developed for the numerical evaluation of detergents.
Since the circulating thyroid hormones are the amino acids thyroxine and tri-iodothyronine ( cf. Section C ), it is clear that some mechanism must exist in the thyroid gland for their release from proteins before secretion.
Since emotional reactions in the higher vertebrates depend on individual experience and are aroused in man, in addition, by complex symbols, one would expect that the hypothalamus could be excited from the cortex.
Since Af are distinct prime polynomials, the polynomials Af are relatively prime ( Theorem 8, Chapter 4 ).
Since D and D' are both diagonalizable and they commute, they are simultaneously diagonalizable, and Af is diagonalizable.
Since N and N' are both nilpotent and they commute, the operator Af is nilpotent ; ;
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 choice is by lot each week, the outcomes of different trials are independent.
Since two curves of symbol Af on Q intersect in Af points, it follows that there are Af lines of Af which are tangent to Aj.

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