Applying Leibniz's Law to the Ship of Theseus problem, S2 is the same as S1 if, and only if, S2 and S1 have all the same properties and relations.

Applying this same sensibility to photography, Alfred Stieglitz later stated it this way: " Atmosphere is the medium through which we see all things.

Applying the same techniques to studying the Hudson Gorge that he had used in the Galápagos, Beebe encountered a surprising variety of sea animals, many of which had previously been thought to be exclusive to the tropics.

Applying different sampling methods will lead to different sets of individuals being observed for the same area of interest, and the species diversity of each set may be different.

Applying a result of linear algebra to both orthogonality relations (| C < sub > i </ sub >| is always positive ), we find that the number of conjugacy classes is greater than or equal to the number of inequivalent irreducible representations ; and also at the same time less than or equal to.

Applying the same operators to the right hand side gives

Applying the same approach of exhaustive research and writing narrative based on objective fact, Freeman completed the first two volumes, titled Young Washington, in 1948.

Applying the same techniques ( which involved gradual immunization of horses and sheep by administering small doses of venoms, and then extracting, purifying and freeze-drying the antibody portion from the blood of injected animals ), Vital Brazil and his coworkers were able to discover the first sera against two species of scorpions ' ( 1908 ) and spiders ' ( 1925 ) venoms.

Applying the same requirements to all students may not be the most effective teaching method.

Applying different species delimitations will lead to different species richness values for the same set of individuals.

Applying multiple, successive gaussian blurs to an image has the same effect as applying a single, larger gaussian blur, whose radius is the square root of the sum of the squares of the blur radii that were actually applied.

Applying these figures to the formula we get the equation: Af.

Applying Lagrange's formula with the observation that Ω • r ( t ) = 0 at all times,

Applying the completeness theorem to this result, gives the existence of a model of T where the formula C < sub > T </ sub > is false.

Applying the formula ( AB )< sup > t </ sup >

Applying the above formula to the 25-meter-diameter antennas often used in radio telescope arrays and satellite ground antennas at a wavelength of 21 cm ( 1. 42 GHz, a common radio astronomy frequency ), yields an approximate maximum gain of 140, 000 times or about 50 dBi ( decibels above the isotropic level ).

Applying both sides to e < sub > j </ sub >, the result on each side is the jth partial derivative of f at p. Since p and j were arbitrary, this proves the formula (*).

Applying this estimate to any even power V < sup > k </ sup > of V and using the Künneth formula shows that the eigenvalues of Frobenius on the middle cohomology of a variety V of any dimension d satisfy

Applying the formula at the top shows this to be about 386 cents.

Applying the formula, we get:

Applying the formula above we obtain:

:- Applying the aforementioned substitution ratios in the formula: ( of hydrogens * factor )/ hydrogens * 1 ) + ( secondary hydrogens * 3. 8 ) + ( tertiary hydrogens * 5 )

Applying the Lagrange interpolation formula to do quadratic interpolation on the inverse of f yields

Applying Ito's formula, we get:

Applying gravel to the roads improved their condition markedly, but asphalt and concrete were not used in local roads until around 1930 ( the year West Harrisville Road was paved ).

Applying the results of local studies to the global scale are complicated by the role the ocean's circulation plays in different ocean regions.

Applying ice to weak muscle groups characteristically leads to improvement in strength of those muscles.

Applying the spiritual warfare theme explored in This Means War !, Petra formed " Prayer Warrior " groups across the country — bands of youth groups to pray for each other.

Applying Birkhoff's theorem, this is sufficient to tell us that the groups form a variety, and so it should be defined by a collection of identities.

Applying sound changes apparent in other Sogdian words and inherent in Indo-European he traces the development of * Suγδa from Skuda, " archer ," as follows: Skuda > * Sukuda by anaptyxis > * Sukuδa > * Sukδa ( syncope ) > * Suγδa ( assimilation ).

Applying this idea to corrugated boxboard was a straightforward development when the material became available in the early twentieth century.

Applying SEA to development co-operation provides the environmental evidence to support more informed decision making, and to identify new opportunities by encouraging a systematic and thorough examination of development options.

Applying this idea to corrugated boxboard was a straightforward development when the material became available around the turn of the twentieth century.

Applying the lessons of processor obsolescence learned during the Lockheed Martin F-22 Raptor, the Lockheed Martin F-35 Lightning II planned for processor upgrades during development, and switched to the more widely supported C ++ programming language.

Applying EVM in fast-changing work environments is, in fact, an area of project management research.

Applying adaptive management in a conservation project or program involves the integration of project / program design, management, and monitoring to systematically test assumptions in order to adapt and learn.

Applying a delta-v at the LEO of only 0. 78 km / s more ( 3. 20-2. 42 ) would give the rocket the escape speed, while at the geostationary orbit a delta-v of 1. 46 km / s is needed for reaching the escape speed of this circular orbit.

Applying the method to hydrogen, they found that the moment of a proton was 2. 785 ± 0. 02 nuclear magnetons, and not 1 as predicted by the current theory, while that of a deuteron was 0. 855 ± 0. 006 nuclear magnetons.

Applying the fundamental theorem of calculus twice gives

Applying the supertranspose twice to a supermatrix X gives

The Cholesky decomposition is commonly used in the Monte Carlo method for simulating systems with multiple correlated variables: The correlation matrix is decomposed, to give the lower-triangular L. Applying this to a vector of uncorrelated samples, u, produces a sample vector Lu with the covariance properties of the system being modeled.

Applying this principle leads to taking the more difficult ( unharmonized ) reading as being more likely to be the original.

Applying heat to the inner cable for soldering may weaken the steel, and although soft soldering is less strong than silver solder, a lower temperature is required to form the joint, and there is less likelihood of the inner cable being damaged as a result.

Applying this concept to human society, Kropotkin presented mutual aid as one of the dominant factors of evolution, the other being self assertion, and concluded that In the practice of mutual aid, which we can retrace to the earliest beginnings of evolution, we thus find the positive and undoubted origin of our ethical conceptions ; and we can affirm that in the ethical progress of man, mutual support not mutual struggle – has had the leading part.

Applying a low thrust over a longer period of time is referred to as a non-impulsive maneuver ( where ' non-impulsive ' refers to the maneuver not being of a short time period rather than not involving impulse-change in momentum, which clearly must take place ).

Applying these principles, the Payne child was allowed to be taken to New Zealand by the mother since otherwise the effect on her of being forced to stay in England would be " devastating ".