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 30 July the Greek army was outnumbered by the counter-attacking Bulgarian army, which attempted to encircle the Greeks in a Cannae-type battle, by applying pressure on their flanks.

By applying the chain rule, the last expression becomes:

By twice applying the transformation above from the stationary to the rotating frame, the absolute acceleration of the particle can be written as:

By 1912 and 1913, there are beginning to be many films from many American companies that rely on applying novel decoration to the story, rather than supplying any twists to the drama itself to sustain interest.

By applying voltage to G, one can control I < sub > D </ sub >.

By applying his construction to the sequence of real algebraic numbers, Cantor produces a transcendental number.

By placing the molecules in wells in the gel and applying an electric field, the molecules will move through the matrix at different rates, determined largely by their mass when the charge to mass ratio ( Z ) of all species is uniform, toward the ( negatively charged ) cathode if positively charged or toward the ( positively charged ) anode if negatively charged.

By applying the variational principle to the action

By charging the right amount of water and applying heat, the system can be brought to any point in the gas region of the phase diagram.

By using this arrangement, many common operations can be implemented by uniformly applying the same operation to each pixel independently.

By applying Pontryagin duality, one can see that abelian profinite groups are in duality with locally finite discrete abelian groups.

By darkening areas of texture maps corresponding to corners, joints and recesses, and applying them via self-illumination or diffuse mapping, a radiosity-like effect of patch interaction could be created with a standard scanline renderer ( cf.

By an offshore drilling contractor effectively applying HSE Management in its organization, it can optimize risk to achieve levels of residual risk that are tolerable.

By applying such principles in an obvious way ( using repeated applications of modus ponens ), Holmes is able to infer from his observation that " the sides of Watson's shoes are scored by several parallel cuts " that:

" By this, James meant that truth is a quality, the value of which is confirmed by its effectiveness when applying concepts to practice ( thus, " pragmatic ").

By applying sprezzatura to his speech and everything else he does, the courtier appears to have grazia and impresses his audience, thereby achieving excellence and perfection.

By applying a bias voltage to a " gate " terminal, the channel is " pinched ", so that the electric current is impeded or switched off completely.

By applying his operational calculus to the analysis of this network he discovered ( 1887 ) what has become known as the Heaviside condition.

By repeatedly applying the definition of conditional probability:

By applying a single spatial model on an entire domain, one makes the assumption that is a stationary process.

By applying the inverse Fourier transform, we can write:

By applying Mauss's theory to data such as Mead's, Lévi-Strauss proposed what he called alliance theory.

By its standard practices of applying good scientific methods, mainstream is distinguished from pseudoscience as a demarcation problem and specific types of inquiry are debunked as junk science, cargo cult science and scientific misconduct etc.

By applying conventional cost-estimating techniques, he concluded that it would cost more than a billion US Dollars to develop this distribution by conventional proprietary means.

By constructing a calibration curve of the ID of species ' pairs with known divergence times in the fossil record, the data could be used as a molecular clock to estimate the times of divergence of pairs with poorer or unknown fossil records.

By the mid-17th century, the linguistic divergence between the Ukrainian and Russian languages was so acute that there was a need for translators during negotiations for the Treaty of Pereyaslav, between Bohdan Khmelnytsky, head of the Zaporozhian Host, and the Russian state.

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

where n is the outward unit normal to the boundary of V. By the divergence theorem,

By applying a molecular clock to the amount of mitochondrial DNA sequence divergence between the two, it was estimated that Troglodytes pacificus and Troglodytes troglodytes last shared a common ancestor approximately 4. 3 million years ago, long before the glacial cycles of the Pleistocene, thought to have promoted speciation in many avian systems inhabiting the boreal forest of North America.

By 1848, divergence of practice and belief led to the development of two separate branches.

By issue # 87, direct reference to the plot divergence was not required.

By the Helmholtz theorem a vector is described completely by its divergence and curl.

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.