Using Wick theorem on the terms of the Dyson series, all the terms of the S-matrix for quantum electrodynamics can be computed through the technique of Feynman diagrams.

Using quantum mechanics it can be shown that for an electron in an external electric field E, the acceleration a < sub > ℓ </ sub > along coordinate direction ℓ is:

Using this approach he has argued that near a black hole, quantum fields could be described by a theory in a lower dimension.

Using mathematical formulations of quantum mechanics, such as the Schrödinger equation, the wave function can be solved.

Using some semiclassical electrodynamics and non-relativistic quantum mechanics, in this section we present a relatively simple and quantitative description of the spin-orbit interaction for an electron bound to an atom, up to first order in perturbation theory.

Using methods similar to those developed for silicon, the substrate can be built free of the defects called dislocations, and the atoms laid down so the distance between the ones making up the ground and those of the quantum wells are the same.

Using Born's rule, one can derive the simplest form of quantum tomography.

Using the once popular vortex theory of gravity, the possibility of matter with negative gravity was discussed by William Hicks in the 1880s.

Using terms from formal language theory, the precise mathematical definition of this concept is as follows: Let S and T be two finite sets, called the source and target alphabets, respectively.

Using the language of category theory, many areas of mathematical study can be cast into appropriate categories, such as the categories of all sets, groups, topologies, and so on.

Using Huygens ' theory and the principle of superposition of waves, the complex amplitude at a further point P is found by summing the contributions from each point on the sphere of radius r < sub > 0 </ sub >.

Using a statistical description for data, information theory quantifies the number of bits needed to describe the data, which is the information entropy of the source.

Using an early theory of inertia, Galileo could explain why rocks dropped from a tower fall straight down even if the earth rotates.

Using his theory, he discovered homologies showing that slightly changed body parts served different functions to meet new conditions, and he found an intermediate stage in the evolution of distinct sexes.

Using Bayesian reasoning, a simple theory is preferred to a complicated one because of a higher prior probability.

Using these simulations in a training program allows for the application of theory into a something akin to real life.

Using Mikhail Bakhtin's theory of the carnivalesque, Jungian and historical writings on the images of the fool in myth and history, and ruminations on the mingling of ecstasy and dread in the Information Age, Dery asserts the evil clown is an icon of our times.

Using the Coupon collector's problem in probability theory, it can be shown that it takes, on average, 72 questions of the Magic 8 Ball for all 20 of its answers to appear at least once.

Using Galois theory, certain problems in field theory can be reduced to group theory, which is in some sense simpler and better understood.

Using Kaldor – Hicks efficiency, an outcome is more efficient if those that are made better off could in theory compensate those that are made worse off, so that a Pareto improving outcome results.

Using Kuratowski's representation for an ordered pair, the second definition above can be reformulated in terms of pure set theory:

* Algorithm design – Using ideas from algorithm theory to creatively design solutions to real tasks.

* Reynolds, John C. Using category theory to design implicit conversions and generic operators.

Using theory and methods drawn from such behavioral sciences as industrial / organizational psychology, industrial sociology, communication, cultural anthropology, administrative theory, organizational behavior, economics, and political science, the change agent's main function is to help the organization define and solve its own problems.

Using game theory, they added the concept of complementors ( also called " the 6th force "), helping to explain the reasoning behind strategic alliances.

** Protective Coating: Bearing the Weight of Pynchon Using the Spectrum of Freud's Insight An analysis of Pynchon's use of parody in relation to Freudian theory

Using perturbation theory, we can use the known solutions of these simple Hamiltonians to generate solutions for a range of more complicated systems.

Using this theory one would expect that there would be considerably more high-energy light than low-energy, due to the presence of stars and other hot bodies.

Using the Dirac notation, let the system initially be in the state where

Using all of the above information to plug into the Dirac equation results in

Using the Dirac comb distribution and its Fourier series:

Using the Dirac adjoint, the conserved probability four-current density for a spin-1 / 2 particle field

Using meconium, a Canadian research group at the Hospital for Sick Children, University of Toronto, showed that by measuring a by-product of alcohol ( FAEE ) they could objectively detect babies exposed to excessive maternal drinking of alcohol in pregnancy.

Using similar measurements he showed that comets were also not atmospheric phenomena, as previously thought, and must pass through the supposed " immutable " celestial spheres.

Using Vinogradov's method, Chudakov, van der Corput, and Estermann showed that almost all even numbers can be written as the sum of two primes ( in the sense that the fraction of even numbers which can be so written tends towards 1 ).

Using event-related potentials, Blanke and colleagues showed the selective activation of the TPJ 330 – 400 ms after stimulus onset when healthy volunteers imagined themselves in the position and visual perspective that generally are reported by people experiencing spontaneous OBEs.

Using this method, he showed how to solve the Hamiltonian cycle problem in arbitrary n-vertex graphs by a Monte Carlo algorithm in time O ( 1. 657 < sup > n </ sup >); for bipartite graphs this algorithm can be further improved to time O ( 1. 414 < sup > n </ sup >).

Using these definitions, Noether showed that the N quantities

Using a carefully bundled group of these DNA threads and keeping them hydrated, Wilkins and a graduate student Raymond Gosling obtained x-ray photographs of DNA that showed that the long, thin DNA molecule in the sample from Signer had a regular, crystal-like structure in these threads.

Using his background in advertising to market the team, Modell showed a flair for promotions, with one popular innovation coming in 1962 by scheduling pro football preseason doubleheaders at Cleveland Stadium.

Using Abelian gauge models, they showed that, if such a Goldstone boson is formed, it is " eaten " by the Higgs mechanism, becoming the longitudinal component of the now massive gauge boson.

Using MBBK's alternates, the results WRR returned showed equivalent or better support for the existence of the codes, and so challenged the " wiggle room " assertion of MBBK.

Using fMRI, neuroscientist Tania Singer showed that empathy-related neural responses are significantly lower in males when observing an " unfair " person experiencing pain.

Using footage recorded by HotDog Media she showed where the flight was scored to.

Using mice as test animals, one experiment showed that heating BSE positive brain tissue at for 18 minutes resulted in only a 2. 5 log decrease in prion infectivity.

Using MEG, they also showed that differences between synesthetes and non-synesthetes begin very quickly after the grapheme is presented.

Using the principle of locality, in a famous paper he and his co-authors articulated the Einstein-Podolsky-Rosen Paradox which showed that position and momentum were simultaneous " real " physical properties of a particle.

Their last two studio albums, Using My Gills As a Roadmap ( 1998 ) and Everydaydream ( 2000 ) showed the band moving away from basic guitar rock and making greater use of electronics, citing Kraftwerk, David Bowie's Low and obscure Euro dance records as influences.

Using a mouse model that allowed the visual tracking and purification of melanocytes using a green fluorescent dye, data showed that UVB-induced, macrophage-enhanced interferon-γ release results in melanoma growth, proliferation and immunoevasion.

Using a concealed camera installed by the CIA, Seal took pictures during the Nicaragua sting operation that clearly showed Pablo Escobar, Jorge Luis Ochoa Vásquez, and other members the medellin Cartel loading kilos of cocaine on to a C-123 transport plane.

Using spectroscopy data from both the CFHT and the Hubble Space Telescope, the group showed that the speed of revolution of the stars within the center of the galaxy could not be maintained unless a mass 1 billion times the mass of the Sun, or, is present in the center.

where e is the identity of G. Though the f ″ defined a couple of paragraphs back is only defined for G-irreducible representations, and though A ⊗ B is not a G-irreducible representation in general, we claim this argument could be made correct since A ⊗ B is simply the direct sum of copies of Bs, and we have shown that each copy all maps to the same G × G-irreducible subrepresentation of C, we have just showed that as an irreducible G × G-subrepresentation of C is contained in A ⊗ B as another irreducible G × G-subrepresentation of C. Using Schur's lemma again, this means both irreducible representations are the same.

If the order of a group G is relatively small and the subgroup H is known to be uncomplicated ( for example, a cyclic group ), then the algorithm can be carried out by hand and gives a reasonable description of the group G. Using their algorithm, Coxeter and Todd showed that certain systems of relations between generators of known groups are complete, i. e. constitute systems of defining relations.

Using clinical neuropsychological and MRI evaluations at baseline and one year later showed that the cognitive impairment was still there one year after the stroke.

Using β-galactosidase as a marker for the melanopsin gene, X-gal labeling of these ipRGCs showed that their axons directly target the SCN, providing further evidence that melanopsin is important in entrainment through the retinohypothalamic tract ( RHT ).

Using bell-jars over water Mayow showed that the active substance that we today call oxygen constitutes about a fifth part of the air.