* Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) =

* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

If this limit exists, then it may be computed by taking the limit as h → 0 along the real axis or imaginary axis ; in either case it should give the same result.

If h is negative, then a + h is on the low part of the step, so the secant line from a to a + h is very steep, and as h tends to zero the slope tends to infinity.

If K is a subset of ker ( f ) then there exists a unique homomorphism h: G / K → H such that f = h φ.

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.

If the minibus exceeds 100 km / h, the beeping will turn into a sustained tone.

If a spatial extra dimension is of radius R, the invariant mass of such standing waves would be M < sub > n </ sub > = nh / Rc with n an integer, h being Planck's constant and c the speed of light.

If c, h, and e were all changed so that the values they have in metric ( or any other ) units were different when we looked them up in our tables of physical constants, but the value of α remained the same, this new world would be observationally indistinguishable from our World.

If the areas of the two parallel faces are A < sub > 1 </ sub > and A < sub > 3 </ sub >, the cross-sectional area of the intersection of the prismatoid with a plane midway between the two parallel faces is A < sub > 2 </ sub >, and the height ( the distance between the two parallel faces ) is h, then the volume of the prismatoid is given by ( This formula follows immediately by integrating the area parallel to the two planes of vertices by Simpson's rule, since that rule is exact for integration of polynomials of degree up to 3, and in this case the area is at most a quadratic in the height.

: If we are given h ( x )= g ( f < sub > 1 </ sub >( x ), ..., f < sub > m </ sub >( x ) )

⟨ H ⟩, be the group generated by H. Then the word problem in H < sup >*</ sup > is solvable: given two words h, k in the generators H of H < sup >*</ sup >, write them as words in X and compare them using the solution to the word problem in G. It is easy to think that this demonstrates a uniform solution the word problem for the class K ( say ) of finitely generated groups that can be embedded in G. If this were the case the non-existence of a universal solvable word problem group would follow easily from Boone-Rogers.

If h has a fixed ( non-zero ) value instead of approaching zero, then the right-hand side of the above equation would be written

If ~ is a unary operation, then h (~ x ) = ~ h ( x ).

If * is a binary operation, then h ( x * y ) = h ( x ) * h ( y ).

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 the rake angle **yc of the knife is high enough and the friction angle **yt between the front of the knife and the back of the chip is low enough to give a positive value for Af, the resultant vector R will lie above the plane of the substrate.

If there is refraction at a collective spherical surface, or through a thin positive lens, O ' 2 will lie in front of O ' 1 so long as the angle u2 is greater than u1 ( under correction ); and conversely with a dispersive surface or lenses ( over correction ).

If the eigenvalues are all positive, then the frequencies are all real and the stationary point is a local minimum.

If a " positive list " approach were taken, only species evaluated and approved for the positive list would be permitted in trade, thus lightening the review burden for member states and the Secretariat, and also preventing inadvertent legal trade threats to poorly known species.

To make a figure larger or smaller is equivalent to multiplying the Cartesian coordinates of every point by the same positive number m. If ( x, y ) are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates

If the electrode has a positive potential with respect to the SHE, then that means it is a strongly reducing electrode which forces the SHE to be the anode ( an example is Cu in aqueous CuSO < sub > 4 </ sub > with a standard electrode potential of 0. 337 V ).

If ΔH is positive, the reaction is endothermic, that is heat is absorbed by the system due to the products of the reaction having a greater enthalpy than the reactants.

Axiom 1: If a property is positive, then its negation is not positive.

Axiom 4: If a property is positive, then it is necessarily positive

Theorem 1: If a property is positive, then it is consistent, i. e., possibly exemplified.

: If a property A entails a property B ( ie in every possible world if an object has property A it must also have property B ), and if A is positive, B must also be positive.

: Axiom 4: If a property A is positive, then it is so in every possible world.

If p is a non-zero real number, we can define the generalized mean with exponent p ( or power mean with exponent p ) of the positive real numbers as:

If X is a positive random variable and q > 0 then for all ε > 0

If the pH of the buffer is above the pI of the protein being run, the protein will migrate to the positive pole ( negative charge is attracted to a positive pole ).

If the pH of the buffer is below the pI of the protein being run, the protein will migrate to the negative pole of the gel ( positive charge is attracted to the negative pole ).

If it is 0, then the number is positive.

If a large, b-bit number is the product of two primes that are roughly the same size, then no algorithm has been published that can factor in polynomial time, i. e., that can factor it in time O ( b < sup > k </ sup >) for some constant k. There are published algorithms that are faster than O (( 1 + ε )< sup > b </ sup >) for all positive ε, i. e., sub-exponential.

If μ is positive, the origin will be shifted to the right, and if μ is negative, it will be shifted to the left.