By applying this general principle, a great number of complex compounds of osmium, ruthenium, iridium, and rhenium, with triphenylphosphine, triphenylarsine, and triphenylstibine have been obtained in this laboratory during the past few years.

By the well ordering principle, if there are positive integers that satisfy a given property, then there is a smallest positive integer that satisfies that property ; therefore, there is a smallest positive integer satisfying the property " not definable in under eleven words ".

By the same principle when the pH is too high, the kidneys excrete bicarbonate ( HCO < sub > 3 </ sub >< sup >-</ sup >) into urine as urea via the Urea Cycle ( aka the Krebs-Henseleit Ornithine Cycle ).

By the mid-19th century, however, the powerful British Empire gave up mercantilism and trade restrictions and introduced the principle of free trade, with few restrictions or tariffs.

By voluntarily stepping down and disbanding his army when the war was won, he permanently established the principle of civilian supremacy in military affairs.

By the superposition principle, the wavepacket at any time t is:

By applying the variational principle to the action

By the same principle, 10 is the least common multiple of-5 and 2 as well.

By Bernoulli's principle, this implies a pressure difference and therefore lift.

By principle, originalists are generally unwilling to defer to precedent when precedent seems to come into conflict with the Constitution.

It is for them alone to point out what we ought to do … By the principle of utility is meant that principle which approves or disapproves of every action whatsoever according to the tendency it appears to have to augment or diminish the happiness of the party whose interest is in question: or, what is the same thing in other words to promote or to oppose that happiness.

By this I mean the principle that, in deciding what is good and what is bad for a given individual, the ultimate criterion can only be his own wants and his own preferences .”

By 1865, the German physicist Rudolf Clausius had shown that this equivalence principle needed amendment.

By analyzing the bank accounts of certain members of the ring, he obtained legal proof of the principle on which the spoils had been divided.

By then the mujtahid would be able to issue his own edicts from the sources of Islamic religious laws: the Qur ' an, the Sunnah, ijmāʻ, and ' aql (" intellect ", rather than the Sunnī principle of qiyas ).

By principle, half of the inhabitants are Jewish Israelis, the rest are Muslim and Christian Arabs.

By weighting utility variations by the marginal utilities, the social welfare function implicit to the Kaldor – Hicks compensation principle is represented by anti-egalitarian, concave indifference curves.

By the equivalence principle gravitational stress-energy will always vanish locally at any chosen point in some chosen frame, therefore gravitational stress-energy cannot be expressed as a non-zero tensor ; instead we have to use a pseudotensor.

By the pigeonhole principle, the probability reaches 100 % when the number of people reaches 367 ( since there are 366 possible birthdays, including February 29th ).

By the Leonid Brezhnev period democratic centralism was described, in the 1977 Soviet Constitution, as a principle for organizing the state: " The Soviet state is organized and functions on the principle of democratic centralism, namely the electiveness of all bodies of state authority from the lowest to the highest, their accountability to the people, and the obligation of lower bodies to observe the decisions of higher ones.

By stressing prosopographical analysis, Syme rejected the force of ideas in politics, dismissing most such invocations of constitutional and political principle as nothing more than " political catchwords.

By examining " hotter " and " cooler " spark plugs of the same manufacturer side by side, the principle involved can be very clearly seen ; the cooler plugs have a more substantial ceramic insulator filling the gap between the center electrode and the shell, effectively allowing more heat to be carried off by the shell, while the hotter plugs have less ceramic material, so that the tip is more isolated from the body of the plug and retains heat better.

By mathematical arguments, the maximum must lie between the two minima.

By studying categories and functors, we are not just studying a class of mathematical structures and the morphisms between them ; we are studying the relationships between various classes of mathematical structures.

* By adding bits to each encoded unit, the redundancy allows both to detect errors in coded data and to correct them based on mathematical algorithms.

* By adding bits to each encoded unit, the redundancy allows both to detect errors in coded data and to correct them based on mathematical algorithms.

By simply rewriting the mathematical expression one can save 50 % of the multiplications required.

By 1901, Peano was at the peak of his mathematical career.

By mathematical analogy: A metasyntactic variable is a word that is a variable for other words, just as in algebra letters are used as variables for numbers.

By revealing ( in modern terms ) that numbers could be irrational, this discovery seems to have provoked the first foundational crisis in mathematical history ; its proof or its divulgation are sometimes credited to Hippasus, who was expelled or split from the Pythagorean sect.

By means of mathematical induction, it is easily proven that the above procedure requires the minimal number of moves possible, and that the produced solution is the only one with this minimal number of moves.

By 1928, Gamow had solved the theory of the alpha decay of a nucleus via tunnelling, with mathematical help from Nikolai Kochin.

By the time of the Surrealist Exhibition of Objects in 1936 a whole range of sub-classifications had been devised — including found objects, readymade objects, perturbed objects, mathematical objects, natural objects, interpreted natural objects, incorporated natural objects, Oceanic objects, American objects and Surrealist objects.

By using counterexamples to show that certain conjectures are false, mathematical researchers avoid going down blind alleys and learn how to modify conjectures to produce provable theorems.

By this Polykleitos meant that a statue should be composed of clearly definable parts, all related to one another through a system of ideal mathematical proportions and balance, no doubt expressed in terms of the ratios established by Pythagoras for the perfect intervals of the musical scale: 1: 2 ( octave ), 2: 3 ( harmonic fifth ), and 3: 4 ( harmonic fourth ).

By the 13th century, Hindu-Arabic numerals were accepted in European mathematical circles ( Fibonacci used them in his Liber Abaci ).

By this change, it will gain in utility, interest and real truth far more than a full compensation for the forfeiture of a fictitious title to mathematical exactness and certainty.

By using eccentrics and epicycles, his geometrical model achieved greater mathematical detail and predictive accuracy than had been exhibited by earlier concentric spherical models of the cosmos.

By assigning " energy values " to the various psychological components of information metabolism, Augustinavičiūtė created a mathematical theory of thinking.

By making simultaneous measurements at the two different frequencies, the resulting data enabled theoretical physicists to verify the mathematical predictions of Albert Einstein's General Theory of Relativity.

By calculating the area of the peak using the mathematical function of integration, the concentration of an analyte in the original sample can be determined.

By staying away from complete mathematical rigor while emphasizing the physical and engineering interpretations of probability, Papoulis's book gained wide popularity.

By the early 1990s, when Second Edition versions of Numerical Recipes ( with code in C, Fortran-77, and Fortran-90 ) were published, it was clear that the constituency for Numerical Recipes was by no means the majority of scientists doing computation, but only that slice that lived between the more mathematical numerical analysts and the larger community using integrated environments.

: Proof: By mathematical induction.

By undermining the traditional role of art and artist, its humor is reflective of a goal to bring life back into art, which Maciunas states in his agenda, “ If man could experience the world, the concrete world surrounding him ( from mathematical ideas to physical matter ) in the same way he experiences art, there would be no need for art, artists and similar ‘ nonproductive ’ elements ”.