By the Hilbert Basis Theorem and some elementary properties of Noetherian rings, every affine or projective coordinate ring is Noetherian.

By Chow's theorem, a projective complex manifold is also a smooth projective algebraic variety, that is, it is the zero set of a collection of homogenous polynomials.

By elementary linear algebra, any two Fubini-Study metrics are isometric under a projective automorphism of CP < sup > n </ sup >, so it is common to speak of " the " Fubini-Study metric.

By taking the dual category of which we call we get an exact and fully faithful embedding from our category to an abelian category which has enough projective objects and a cogenerator.

By the Fundamental theorem of projective geometry a reciprocity is the composition of an automorphic function of K and a homography.

By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines through the origin of a complex Euclidean space ( see below for an intuitive account ).

By the same construction, projective spaces can be considered in higher dimensions.

By projective duality, the existence of an ordinary line in a set of non-collinear points in RP < sup > 2 </ sup > is equivalent to the existence of an ordinary point in a nontrivial arrangement of finitely many lines.

By contrast, in the characteristic 0 case every irreducible representation is a direct summand of the regular representation, hence is projective.

By virtue of being a projective test, the results of the HTP are subjective and open to interpretation by the administrator of the exam.

By considering the ball's boundary, a Kleinian group can also be defined as a subgroup Γ of PGL ( 2, C ), the complex projective linear group, which acts by Möbius transformations on the Riemann sphere.

By projective duality, G ( 1, 2 ) is isomorphic to P < sup > 2 </ sup >, so we may give it homogenous coordinates.

By wave-particle duality the effect can be analyzed purely in terms of waves though not as conveniently.

By applying Pontryagin duality, one can see that abelian profinite groups are in duality with locally finite discrete abelian groups.

This follows, in part, from the fact the covariant Hom functor Hom ( N, –): C → Set preserves all limits in C. By duality, the contravariant Hom functor must take colimits to limits.

By Poincaré duality, the homology class of Z is dual to a cohomology class which we will call, and the cap product can be computed by taking the cup product of and α and capping with the fundamental class of X.

By duality, the same is true for join-prime and join-irreducible elements.

By the Serre duality, this is the same as:

By planar duality it became coloring the vertices, and in this form it generalizes to all graphs.

Here it is easy to see that if some linear combination of the x < sub > i </ sub > and 1 with non-zero rational number coefficients is zero, then the coefficients may be taken as integers, and a character χ of the group T other than the trivial character takes the value 1 on P. By Pontryagin duality we have T ′ contained in the kernel of χ, and therefore not equal to T.

By focusing all of one's being on one specific practice, duality is transcended.

By the In-Betweener's own admission, he represents duality itself, specifically in respect to concepts such as good and evil ; reason and emotion ; truth and illusion ; and life and death.

By the duality of networks, it can be proved that the number of essentially series networks is equal to the number of essentially parallel networks.

By the principle of duality, the closure operator satisfies the identities:

By duality, this results also says that any finite nontrvial set of points on the plane has at least three ordinary lines.

By the end of the recording sessions, The Gift was faced with a double album, where two distinctive but complementary environments surfaced ( one more intimate and introspective, the other more euphoric ) and in which duality was the key word.

Jahrhundert ’, Zeitschrift für Musikwissenschaft, iv, 257 – 65 ), who employed it in reference to half of his perceived binarism between that 17th-century dance music which was in fact danced to, and that which was written in dance forms but was actually abstract music intended only for listening ( By extension, this duality applied to the dichotomy between any music for a specific purpose, Gebrauchsmusik, and that for none but the pleasure or edification of listening, Vortragsmusik ).

By a proper choice of coordinate system, the ellipse can be described by the canonical implicit equation

By convention, is ten times the ellipticity of the galaxy, rounded to the nearest integer, where the ellipticity is defined as for an ellipse with semi-major and semi-minor axes of lengths and respectively.

By an effort of historical sympathy we can cast our minds back into the art of a remote past or an alien present, and enjoy the carvings of cavemen and Japanese colour-prints ; ;

By that time, perhaps something better can be done ''.

By supporting the efforts of the many faculty members who are working to attain ever higher standards, the president can encourage faculty leadership.

By sharing the load of important speeches with his colleagues, the president can develop a cadre of able spokesmen who will help to create a public perception of the university as an institution, something more than the lengthened shadow of one man.

By using the appropriate version any one of these factors can be determined for any combination of the other three.

By automobile from New York, for example, you can take a one or two-day tour to Annapolis, Maryland to see the colonial homes and the U.S. Naval Academy ( where you can shoot the dress parade on Wednesdays ) ; ;

Reports Jim Lendrum: `` By studying men on the job, we found that two men -- a carpenter and a helper -- can lay a floor faster than three.

By the use of various weights, data for a force-rate of shear graph can be obtained.

By scrutinizing the flowers, one can also notice that the scale bears one or two tiny warts.

By holding out prospects for external capital assistance, the United States can provide strong incentives to prepare for the concerted economic drive necessary to achieve self-sustaining growth.

By means of geographical isolation and high fertility rates, inbreeding can be fostered and the pattern of isolation from the greater society maintained.

By clarifying fundamental premises in the social sciences, and defining the logical problems emergent at the borderlands of each new scientific discipline, philosophy can offer the sort of distinction that can accelerate growth in human understanding.

By this same combination of the will and the imagination, each one of us can learn to portray permanently the kind of character we would like to be.

By tipping the porter, you can see in the courtyard Borromini's unusual and fascinating trick in perspective.

`` By winning against Bradley, Kentucky and Notre Dame on those teams' home courts, they showed that the home court advantage can be overcome anywhere and that it doesn't take a super team to do it ''.

By the Brønsted-Lowry definition, any compound which can easily be deprotonated can be considered an acid.

By the tenth century two other forms can be distinguished namely Canaanite and Aramaic.

By night, an aardwolf can consume up to 200, 000 harvester termites using its sticky, long tongue.

By knowing the coordinates of an object ( usually given in equatorial coordinates ), the telescope user can use the setting circle to align the telescope in the appropriate direction before looking through its eyepiece.

By its geological origin, islands forming archipelagos can be referred to as oceanic islands, continental fragments and continental islands.

By preserving the value, only one division operation is needed and the higher-order statistics can thus be calculated for little incremental cost.

By having three stations, a total of three different cars can be operated on at the same time, each one at a different stage of its assembly.