If this rule is interpreted as saying that straight-line motion is an indication of zero net force, the rule does not identify inertial reference frames, because straight-line motion can be observed in a variety of frames.

If X and Y are topological vector spaces, the space L ( X, Y ) of continuous linear operators f: X → Y may carry a variety of different possible topologies.

If the opponent is adjusting to the drive serve, the server will throw in any variety of jam serves.

Some students of universal grammar study a variety of grammars to abstract generalizations called linguistic universals, often in the form of " If X holds true, then Y occurs.

If a mayday call cannot be sent because a radio is not available a variety of other distress signals and calls for help can be used.

If all axioms defining a class of algebras are identities, then the class of objects is a variety ( not to be confused with algebraic variety in the sense of algebraic geometry ).

If the plaintiff is successful, judgment will be given in the plaintiff's favor, and a variety of court orders may be issued to enforce a right, award damages, or impose a temporary or permanent injunction to prevent an act or compel an act.

If Vedanta truly epitomises the state of learnedness, in achieving this spiritual progress " the first stage for a layman is the external / material worship ; struggling to rise high, mental prayer is the next stage, but the highest stage is when the divine has been realized " Unity in variety is the scheme of nature, and the Hindu has recognized it and practised eversince the yore through his equanimity to all and universal tolerance ".

#( Betti numbers ) If X is a ( good ) " reduction mod p " of a non-singular projective variety Y defined over a number field embedded in the field of complex numbers, then the degree of P < sub > i </ sub > is the i < sup > th </ sup > Betti number of the space of complex points of Y.

If X is a smooth variety, the two groups are the same.

If it is a smooth affine variety, then all extensions of locally free sheaves split, so group has an alternative definition.

If X is an algebraic variety carrying the Zariski topology, we can define a locally ringed space by taking O < sub > X </ sub >( U ) to be the ring of rational functions defined on the Zariski-open set U which do not blow up ( become infinite ) within U. The important generalization of this example is that of the spectrum of any commutative ring ; these spectra are also locally ringed spaces.

* If I is a prime ideal ( i. e. V is an algebraic variety ), the transcendence degree over K of the field of fractions of A.

If V is a projective variety defined by a homogeneous ideal I, then the values for which A or I appear explicitly in previous definitions must be decreased by one.

If F is algebraically closed, this is equivalent to a curve of genus zero ; however, the field of all real algebraic functions defined on the real algebraic variety x < sup > 2 </ sup >+ y < sup > 2 </ sup > = − 1 is a field of genus zero which is not a rational function field.

If X is the toric variety corresponding to the normal fan of P, then P defines an ample line bundle on X, and the Ehrhart polynomial of P coincides with the Hilbert polynomial of this line bundle.

If X is a projective variety defined by a homogeneous prime ideal I, then the quotient ring

If damage or some form of alteration is made to a hub or die, it is classified a variety.

If K is a field of characteristic p, and if V is an algebraic variety over K of dimension greater than zero, the function field K ( V ) is a purely inseparable extension over the subfield K ( V )< sup > p </ sup > of pth powers ( this follows from condition 2 above ).

If the ground field k is arbitrary and GL ( V ) is considered as an algebraic group, then this construction shows that the Grassmannian is a non-singular algebraic variety.

If X is an affine algebraic variety, and if U is an open subset of X, then K < sub > X </ sub >( U ) will be the field of fractions of the ring of regular functions on U. Because X is affine, the ring of regular functions on U will be a localization of the global sections of X, and consequently K < sub > X </ sub > will be the constant sheaf whose value is the fraction field of the global sections of X.

If X is an algebraic variety over a field k, then over each open set U we have a field extension K < sub > X </ sub >( U ) of k. The dimension of U will be equal to the transcendence degree of this field extension.

* If X is a complex variety, then étale cohomology with finite coefficients is isomorphic to singular cohomology with finite coefficients.

If the circumstances are faced frankly it is not reasonable to expect this to be true.

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.

If A is commutative then the center of A is equal to A, so that a commutative R-algebra can be defined simply as a homomorphism of commutative rings.

If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.

If we define the function f ( n ) = A ( n, n ), which increases both m and n at the same time, we have a function of one variable that dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).

If f is also surjective and therefore bijective ( since f is already defined to be injective ), then S is called countably infinite.

If is any unit vector, the projection of the curl of F onto is defined to be the limiting value of a closed line integral in a plane orthogonal to as the path used in the integral becomes infinitesimally close to the point, divided by the area enclosed.

* If is the norm ( usually noted as ) defined in the square-summable sequence space ℓ < sup > 2 </ sup > ( which also matches the usual distance in a continuous and isotropic cartesian space ), then

* If is the norm ( usually noted as ) defined in the sequence space ℓ < sup >∞</ sup > of all bounded sequences ( which also matches the non-linear distance measured as the maximum of distances measured on projections into the base subspaces, without requiring the space to be isotropic or even just linear, but only continuous, such norm being definable on all Banach spaces ), and is lower triangular non-singular ( i. e., ) then

If the orientation of the tangent relative to some starting position is θ ( s ), then ρ ( s ) is defined by the derivative dθ / ds:

If the axes are named x, y, and z, then the x coordinate is the distance from the plane defined by the y and z axes.

If every term of every definiens must itself be defined, " where at last should we stop?

If a throw lands out of bounds, unless defined by the hole, the thrower has the option of playing from the previous lie, or playing from the approximate spot where the disc crossed into the out-of-bound territory.

If a vector field F with zero divergence is defined on a ball in R < sup > 3 </ sup >, then there exists some vector field G on the ball with F = curl ( G ).

If is defined as the unitary DFT of the vector then

If m and n are natural numbers and f ( x ) is a smooth ( meaning: sufficiently often differentiable ) function defined for all real numbers x in the interval, then the integral

The evidence was compiled by W de Sitter ( 1927 ) who wrote " If we accept this hypothesis, then the ' astronomical time ', given by the earth's rotation, and used in all practical astronomical computations, differs from the ' uniform ' or ' Newtonian ' time, which is defined as the independent variable of the equations of celestial mechanics ".

If one concept is defined by another, and the other is defined by the first, this is known as circular definition, somewhat similar to a circular reasoning: neither offers us any enlightenment about what we wanted to know.

If ' beauty ' is defined as ' aesthetically successful ', one must continue to break down and define the following definition.

If the graph does not contain any cycles ( i. e. it's an acyclic graph ), its girth is defined to be infinity.

# Associativity: If a * b and b * c are defined, then ( a * b ) * c and a * ( b * c ) are defined and equal.

# Identity: If a * b is defined, then a * b * b < sup >− 1 </ sup >

* If a * b is defined, then ( a * b )< sup >− 1 </ sup >

* If G is a topological group and V is a topological vector space, a continuous representation of G on V is a representation ρ such that the application defined by is continuous.

If G is a group and X is a set then a group action may be defined as a group homomorphism h from G to the symmetric group of X.