* 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 positive, then a + h is on the high part of the step, so the secant line from a to a + h has slope zero.

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 one eigenvalue is negative ( i. e., an imaginary frequency ), then the stationary point is a transition structure.

If more than one eigenvalue is negative, then the stationary point is a more complex one, and is usually of little interest.

If we are to keep a right-handed coordinate system, into the page will be in the negative z direction.

If the carriers are holes, replace electron density by the negative of the hole density.

If an evacuated glass tube is equipped with two electrodes and a voltage is applied, the glass opposite of the negative electrode is observed to glow, due to electrons emitted from and travelling perpendicular to the cathode ( the electrode connected to the negative terminal of the voltage supply ).

If F ( r ) represents gravity, it is a negative term proportional to 1 / r < sup > 2 </ sup >, so the net acceleration in r in the rotating frame depends on a difference of reciprocal square and reciprocal cube terms, which are in balance in a circular orbit but otherwise typically not.

If this acceleration is multiplied by the particle mass, the leading term is the centripetal force and the negative of the second term related to angular acceleration is sometimes called the Euler force.

If the air cools and contracts, the divergence is negative and the region is called a sink.

If it is negative, that means that the investment decision would actually lose money even if it appears to generate a nominal profit.

If the vagus nerve on the first heart was stimulated, negative inotropic ( beat amplitude ) and chronotropic ( beat rate ) activity were seen in both hearts.

If one alters a Euclidean space so that its inner product becomes negative in one or more directions, then the result is a pseudo-Euclidean space.

If longer lengths were needed, the unexposed negative rolls could be cemented in a darkroom, but this was largely undesirable by most narrative filmmakers.

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

If an appropriate form of written text is made available before formal schooling begins, reading should be learned inductively, emerge naturally, and with no significant negative consequences.

If a politician, again regardless of party, were to speak outside of this range, he or she would receive negative press or be ignored.

If the inverting input is held at ground ( 0 V ) directly or by a resistor, and the input voltage V < sub > in </ sub > applied to the non-inverting input is positive, the output will be maximum positive ; if V < sub > in </ sub > is negative, the output will be maximum negative.

If predictable operation is desired, negative feedback is used, by applying a portion of the output voltage to the inverting input.

If the input voltages change in the same direction, the negative feedback makes Q3 / Q4 base voltage follow ( with 2V < sub > BE </ sub > below ) the input voltage variations.

If the base current to the transistor is assumed to be zero, the negative feedback forces the transistor to increase its collector-emitter voltage up to 1 V until its base-emitter voltage reaches 0. 625 V ( a typical value for a BJT in the active region ).