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Page "Finite difference" ¶ 12
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If and h
* 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 )( 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 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 ~ is a unary operation, then h (~ x ) = ~ h ( x ).
If * is a binary operation, then h ( x * y ) = h ( x ) * h ( y ).

If and has
`` If Blue Throat has his way he'll keep us all cooped up in here for days '', he said.
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 the man on the sidewalk is surprised at this question, it has served as an exclamation.
If any one of them has any power to veto the Secretary General's decisions the nature of the organization will have changed.
If this new Bible does not increase in significance by repeated readings throughout the years, it will not survive the ages as has the King James Version.
If the case is thus determined by us to be domestic, the court has no jurisdiction.
If this capacity had not failed them, they would see that their enemy has made a disastrous miscalculation.
If the new Soviet series has followed the general pattern of previous Russian tests, the shots were roughly half fission and half fusion, meaning a fission yield of 30 to 40 megatons thus far.
If the superintendents do not receive more cooperation from Handlers, it has been suggested that licensed Judges also be qualified to judge this Class.
If your state has no provisions for the numbering of pleasure boats, you must apply for a number from the U.S. Coast Guard for any kind of boat with mechanical propulsion rated at more than 10 horsepower before it can be used on Federal waterways.
If design head has a deep cavity, clay lid will be quite thick at this point ; ;
If you run into excess plug fouling on one truck, check to be sure that the rig has a thermostat.
If we manage to keep track of a Bombus queen after she has left her feeding place, we may discover the snug little hideout which she has fixed up for herself when she woke up from her winter sleep.
If cell Af has not previously been assigned as the information cell of a form in the text-form list, it is now assigned as the information cell of Af.
If of the founders of glottochronology Swadesh has escaped our steady plodding, and Lees has repudiated his own share in the founding, that is no reason why we should swerve.
If the transferor has substantial assets other than the claim, it seems reasonable to assume no corporation would be willing to acquire all of its properties in the dim hope of collecting a claim for refund of taxes.
If the patient can perceive figure kinesthetically when he cannot perceive it visually, then, it would seem, the sense of touch has immediate contact with the spatial aspects of things in independence of visual representations, at least in regard to two dimensions, and, as we shall see, even this much spatial awareness on the part of unaided touch is denied by the authors.
If the distant patron of the suburban branch has been frightened away from downtown by traffic problems, however, the city store can only pressure the politicians to do something about the highways or await the completion of the federal highway program.
If a statement has been assigned an address in the index word area
If she has not had such experiences, the female's normal adolescent degree of indecision will be compounded.
If the change, at first sight, seems minor, we may recall that it took the Italian painters about two hundred years to make an analogous change, and the Italian painters, by universal consent, were the most brilliant group of geniuses any art has seen.
If Barnett doesn't call a special session in 1961, it will be the first year in the last decade that the Legislature has not met in regular or special session.
If the church has followed the plan of cultivation of prospects and carried through a program of membership preparation as outlined earlier in this book, the process of assimilation and growth will be well under way.

If and fixed
If the net force on some body is directed always toward some fixed point, the center, then there is no torque on the body with respect to the center, and so the angular momentum of the body about the center is constant.
If x is held fixed, then the Bessel functions are entire functions of α.
If A is a fixed element of a ring ℜ, the first additional relation can also be interpreted as a Leibniz rule for the map given by B ↦.
If G is a group, and g is a fixed element of G, then the conjugation map
If a rotating frame is chosen so that just the angular position of an object is held fixed, more complicated motion, such as elliptical and open orbits, appears because the centripetal and centrifugal forces will not balance.
If each pixel on the display actually contains three rectangular subpixels of red, green, and blue, in that fixed order, then things on the screen that are smaller than one full pixel in size can be rendered by lighting only one or two of the subpixels.
If a display does not have the type of fixed pixels that ClearType expects, text rendered with ClearType enabled actually looks worse than type rendered without it.
If someone sees something completely different than another, our language prevents us from acknowledging the other's point of view by limiting our perception to fixed states.
:" If a period fixed by weeks, months, and years does not commence from the beginning of a week, month, or year, it ends with the ending of the day which proceeds the day of the last week, month, or year which corresponds to that on which it began to commence.
Tensor products: If C denotes the category of vector spaces over a fixed field, with linear maps as morphisms, then the tensor product defines a functor C × C → C which is covariant in both arguments.
If an equation can be put into the form f ( x ) = x, and a solution x is an attractive fixed point of the function f, then one may begin with a point x < sub > 1 </ sub > in the basin of attraction of x, and let x < sub > n + 1 </ sub > = f ( x < sub > n </ sub >) for n ≥ 1, and the sequence
( If great wealth is not diluted, the positions in society tend to be much more fixed and opportunities to make an individual success are lower.
# If G is any group acting smoothly on the manifold M, then it acts on the vector fields, and the vector space of vector fields fixed by the group is closed under the Lie bracket and therefore also forms a Lie algebra.
If two elements form more than one compound between them, then the ratios of the masses of the second element which combine with a fixed mass of the first element will be ratios of small whole numbers.
If the meteoroid maintains a fixed orientation for some time, without tumbling, it may develop a conical " nose cone " or " heat shield " shape.
If the revenue the firm is receiving is greater than its total variable cost ( R > VC ) then the firm is covering all variable cost plus there is additional revenue (" contribution "), which can be applied to fixed costs.
If the arity of the operators is fixed, the result is a syntax lacking parentheses or other brackets that can still be parsed without ambiguity.
If the SEM is equipped with a cold stage for cryo-microscopy, cryofixation may be used and low-temperature scanning electron microscopy performed on the cryogenically fixed specimens.
At least two assumptions are necessary for the validity of the standard model: first, that supply and demand are independent ; and second, that supply is " constrained by a fixed resource "; If these conditions do not hold, then the Marshallian model cannot be sustained.
If a traveling wave has a fixed shape that repeats in space or in time, it is a periodic wave.
If an inconsistency is encountered, the program can immediately halt, so that the bug can be located and fixed.
If the management ethos is to reward the number of bugs fixed, then some developers may quickly write sloppy code knowing they can fix the bugs later and be rewarded for it, whereas careful, perhaps " slower " developers do not get rewarded for the bugs that were never there.
If the waveform is a pure sine wave, the relationships between peak-to-peak, peak, mean, and RMS amplitudes are fixed and known, as they are for any continuous periodic wave.
If there is a fixed point, we can take that as the origin, and the affine transformation reduces to a linear transformation.

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