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 the same formula in Toronto, Jacobs rallied local groups to oppose development of the Spadina Expressway project, while twice being arrested in the process.

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 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 the techniques developed at Harvard for generating a paradigm from a representative form and its classification, we can add all forms of a word to the dictionary at once.

Applying source separation techniques on the measured signals can help remove undesired artifacts from the signal.

* Greening The Desert: Applying natural farming techniques in Africa, interview with Masanobu Fukuoka ( retrieved 30 November 2010 )

Applying Six Sigma techniques to ITIL brings the engineering approach to ITIL's framework.

Applying Lean techniques promotes continuous improvement of the ITIL's best practices.

Applying proper motivational techniques can be much harder than it seems.

Applying these techniques, Arthur Cayley has found a general criterion for determining whether any given quintic is solvable.

# Applying protective / supportive techniques that allow the athlete to regain physically active lifestyle

Applying linearization techniques to the equation of motions for both species, to the equation of continuity, and Poisson's equation, and introducing the spatial and temporal harmonic operators, we can get the following expression:

Applying a conception of such illocutionary acts according to which they are ( roughly ) acts of saying something with the intention of communicating with an audience, he describes indirect speech acts as follows: " In indirect speech acts the speaker communicates to the hearer more than he actually says by way of relying on their mutually shared background information, both linguistic and nonlinguistic, together with the general powers of rationality and inference on the part of the hearer.

Applying his mathematical prowess to the problem, Piet Hein proposed to use the superellipse which became the hallmark of modern Scandinavian architecture.

Applying the Phase correlation method to a pair of images produces a third image which contains a single peak.

Applying the cosmological principle, this suggests that heavier elements were not created in the Big Bang but were produced by nucleosynthesis in giant stars and expelled across a series of supernovae explosions and new star formation from the supernovae remnants, which means heavier elements would accumulate over time.

Applying to the study of the French Revolution the rules of historical criticism which had produced such rich results in the study of ancient and medieval history, he devoted himself to profound research in the archives, and to the publication of numerous important contributions to the political, administrative and moral history of that period.

Applying this to a space of dimension 0, which contains exactly one vector ( the zero vector ), the only candidate for a basis is the empty set ( both because 0 elements are required, and because the zero vector cannot be in any basis ).

Applying this model to the experiences of Mexican Americans forces one to see Mexican American immigrants as positive additions to the " American melting pot ," in which as more additions are made to the pot, the more equal and accepting society will become.

Applying for a commercial pilot's licence on 2 June 1921 ( in which he gave his name as ' Charles Edward Kingsford-Smith '), he became one of Australia's first airline pilots when he was chosen by Norman Brearley to fly for the newly formed West Australian Airways.

He also wrote An Essay on Electricity: Explaining the Theory and Practice of that Useful Science, and the Mode of Applying it to Medical Purposes ( London, 1784 ), which ran to four editions in his lifetime.

Applying a Newton or Picard iteration produces a system of linear equations which is nonsymmetric in the presence of advection and indefinite in the presence of incompressibility.

Applying the operator L to each side of this equation results in the completeness relation, which was assumed true.

) Applying this to the geometric progression 31, 62, 124, 248, 496 ( which results from 1, 2, 4, 8, 16 by multiplying all terms by 31 ), we see that 62 minus 31 is to 31 as 496 minus 31 is to the sum of 31, 62, 124, 248.

Applying the Tannakian formalism, one concludes that M < sub > num </ sub > is equivalent to the category of representations of an algebraic group G, which is called motivic Galois group.

Applying the above bundle map ( pointwise ) to this section yields the pullback of α by φ, which is the 1-form φ < sup >*</ sup > α on M defined by

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 flap technique ( first used by Sir Harold Gillies in 1951 ), the spongiform erectile tissue of the penis is removed, and the skin, with its nerves and vascular system ( blood supply ) still attached, is used to create a vestibule area and labia minora, which then are inverted into the neovaginal cavity created in the pelvic tissue.

Applying more power ( increasing collective pitch ) serves to further accelerate the downwash through which the main-rotor is descending, exacerbating the condition.

Applying and drying the paste added days to the production of each vehicle, which was unacceptable as there was a shortage of tanks.

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 Wang's views on reading, journalism, and news reports, readers should focus their attention on news of particular interest to them, ones which they feel comfortable to read and criticize.

Applying the e < nowiki >'</ nowiki > th iterate of F to a ring which contains a field K of p < sup > e </ sup > elements gives us a fixed point set equal to K, similar to the example above.

Applying appropriate lowpass filter to get rid of the high frequency component on can obtain the value of the equivalent control, which contains more information about the state of the estimated system.