Suggest the following twenty-first-century amendment: By moving the term `` Republic '' to lower case, substituting the modern phrase, `` move ahead '' for the stodgy `` keep '', and by using the Postmaster's name on every envelope ( in caps, of course, with the `` in spite '' as faded as possible ), the slogan cannot fail.

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

By using instruments of gradually increasing size, the vagina is gently, and with minimum pain at each stage, taught to yield to an object of the appropriate shape.

By 1827 Laplace was using least

By comparing the form of several yupanas, researchers found that calculations were based using the Fibonacci sequence 1, 1, 2, 3, 5 and powers of 10, 20 and 40 as place values for the different fields in the instrument.

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

By using the palette to mask the top area of the display and taking care about when it changes mode it can shift the continuous graphics at the bottom of the display down in two pixel increments because the internal display counter is not incremented on blank scanlines during non-continuous graphics modes.

By using the formula

By the Roman period brass was being deliberately produced from metallic copper and zinc minerals using the cementation process and variations on this method continued until the mid 19th century.

By comparison to biotechnology, bioengineering is generally thought of as a related field with its emphasis more on higher systems approaches ( not necessarily altering or using biological materials directly ) for interfacing with and utilizing living things.

By the late 1920s, New York writers other than Fitz Gerald were starting to use " Big Apple " and were using it outside of a horse-racing context.

By the mid-1980s at least 500 courses in business ethics reached 40, 000 students, using some twenty textbooks and at least ten casebooks along supported by professional societies, centers and journals of business ethics.

By 1000 BC ancient civilizations were using technologies that formed the basis of the various branches of chemistry such as ; extracting metal from their ores, making pottery and glazes, fermenting beer and wine, making pigments for cosmetics and painting, extracting chemicals from plants for medicine and perfume, making cheese, dying cloth, tanning leather, rendering fat into soap, making glass, and making alloys like bronze.

By using custom formatting and load / save routines ( sometimes included in third-party DOSes, see below ), all of the mechanically possible 40 tracks could be used.

By the early 1990s, the performance of microprocessor-based computers reached the point that real-time generation of computer music using more general programs and algorithms became possible.

By using the commutator as a Lie bracket, every associative algebra can be turned into a Lie algebra.

By a strange twist of their respective fates, this foursome ends up partnering together, using their unique talents to become a ragtag team of bounty hunters, although their fortunes as such are at best mixed.

By 1970, a calculator could be made using just a few chips of low power consumption, allowing portable models powered from rechargeable batteries.

By reducing the number of tracks used ( and thus the capacity ), Commodore could further reduce cost-in contrast to the double-density drives used e. g. in IBM PCs of the day which saved 180 KB on one side ( by using a 40-track format ).

By the time of the Vietnam War, IDs spelled out the broad religious choices such as PROTESTANT and CATHOLIC, rather than using initials, and also began to show individual denominations such as " METHODIST " or " BAPTIST.

By 1695 he was back in England, now formally using the name " Defoe ", and serving as a " commissioner of the glass duty ", responsible for collecting taxes on bottles.

By contrast, non-digital ( or analog ) systems represent information using a continuous function.

By using doublespeak in advertisements, aspirin production rose by almost 50 percent from over 23 million pounds in 1960 to over 35 million pounds in 1970.

By the 12th to 13th centuries A. D., monks were preparing illuminated manuscripts on vellum and parchment in monasteries throughout Europe and were using lead styli to draw lines for their writings and for the outlines for their illuminations.

By direct substitution, the solution to this equation can be readily shown to be the scalar Green's function, which in the spherical coordinate system ( and using the physics time convention ) is:

By using the scoring matrix ( substitution matrix ) to score the comparison of each residue pair, there are 20 ^ 3 possible match scores for a 3-letter word.

By the time of the rifle's introduction, rifle design had moved on to using small-calibre smokeless powder cartridges, which allowed bullets to be propelled at much higher velocities without as much smoke or residue.

By the residue theorem we have that the integral about C is the product of 2πi and the sum of the residues.

By spraying a surface with layers of these atoms under high temperature, this wetting layer residue is produced on the surface.

By the 1930s the residue of the farm was surrounded by houses, and was open to the public, with tea rooms and an exhibition of objects related to the dairy industry.

has no zero in F. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed.

By a theorem of Gelfand and Naimark, given a B * algebra A there exists a Hilbert space H and an isometric *- homomorphism from A into the algebra B ( H ) of all bounded linear operators on H. Thus every B * algebra is isometrically *- isomorphic to a C *- algebra.

By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.

By setting K = ker ( f ) we immediately get the first isomorphism theorem.

* By the convolution theorem, Fourier transforms turn the complicated convolution operation into simple multiplication, which means that they provide an efficient way to compute convolution-based operations such as polynomial multiplication and multiplying large numbers.

By " satisfactory " one would mean at least the equivalent of Plancherel theorem.

By the fundamental theorem of arithmetic, every positive integer has a unique prime factorization.

By the Glivenko – Cantelli theorem, if the sample comes from distribution F ( x ), then D < sub > n </ sub > converges to 0 almost surely.

By this theorem, once a star's chemical composition and its position on the main sequence is known, so too is the star's mass and radius.

By Gödel's incompleteness theorem, Peano arithmetic is incomplete and its consistency is not internally provable.

By Rice's theorem, the 1-halting problem is undecidable.

By the corollary to the recursion theorem, there is an index such that returns.

By Liouville's theorem, Hamiltonian flows preserve the volume form on the phase space.

By the fundamental theorem, we may replace the new set by the old set subject to a unitary transformation.

By the first isomorphism theorem, the image of A under ƒ is a substructure of B isomorphic to the quotient of A by this congruence.

By the central limit theorem, this distribution approaches the normal distribution as n increases.

By analogy to the prior and posterior probability terms in Bayes ' theorem, Bayes ' rule can be seen as Bayes ' theorem in odds form.

By the extreme value theorem, a continuous function on a closed interval must attain its minimum and maximum values at least once.

By the rank-nullity theorem, a system of n vectors in k dimensions ( where all dimensions are necessary ) satisfies a ( p = n − k )- dimensional space of relations.

By Tychonoff's theorem we have that is compact since is, so the closure of in is a compactification of.

By Stone's representation theorem every Boolean ring is isomorphic to a field of sets ( treated as a ring with these operations ).

By induction, Hilbert's basis theorem establishes that, the ring of all polynomials in n variables with coefficients in, is a Noetherian ring.

By Euler's rotation theorem, we may replace the vector with where is a 3x3 rotation matrix and is the position of the particle at some fixed point in time, say t = 0.

By the divergence theorem Gauss's law can alternatively be written in the differential form: