Using the chi-squared distribution to interpret Pearson's chi-squared statistic requires one to assume that the discrete probability of observed binomial frequencies in the table can be approximated by the continuous chi-squared distribution.

Using binomial coefficients, the formula can be expressed as

Using the above theorem it is easy to see that the original Borsuk Ulam statement is correct since if we take a map f: S < sup > n </ sup > → ℝ < sup > n </ sup > that does not equalize on any antipodes then we can construct a map g: S < sup > n </ sup > → S < sup > n-1 </ sup > by the formula

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 a uniqueness theorem and showing that a potential satisfies Laplace's equation ( second derivative of V should be zero i. e. in free space ) and the potential has the correct values at the boundaries, the potential is then uniquely defined.

Using the parallel axis theorem, the radius of gyration L of a rigid pendulum can be shown to be

Using the formula for Bayes ' theorem, we have:

Using the disintegration theorem we can define a restriction of Lebesgue measure to the-dimensional affine subspace of where the Gaussian distribution is supported, i. e..

Using Fibonacci numbers, he proved that when finding the greatest common divisor of integers a and b, the algorithm runs in no more than 5k steps, where k is the number of ( decimal ) digits of b. He also proved a special case of Fermat's last theorem.

Using Bayes ' theorem, we write

Using the above rule, we can complete the proof of Fermat's little theorem quite easily, as follows.

Using Noether's theorem, the types of Lagrangians that conserve X through a continuous symmetry may be determined, and their fitness judged by further criteria.

Using the fundamental theorem of calculus often requires finding an antiderivative.

Using the Pythagorean theorem twice, b < sup > 2 </ sup > =

Using Dirichlet's theorem on primes in arithmetic progressions, the law of quadratic reciprocity, and the Chinese remainder theorem ( CRT ) it is easy to see that for any M > 0 there are primes p such that the numbers 1, 2, …, M are all residues modulo p.

Using Gauss's theorem and Poisson's equation gives

Using the close correspondence between divisors and holomorphic line bundles on a Riemann surface, the theorem can also be stated in a different, yet equivalent way: let L be a holomorphic line bundle on X.

Using topological cobordism and K-homology one may provide a full statement of an index theorem on quasiconformal manifolds ( see page 678 of ).

Using the compactness theorem, the resulting theory is easily seen to be consistent.

This theorem was never published by Desargues ( 1591 – 1661 ), but appeared in an appendix entitled Universal Method of M. Desargues for Using Perspective ( Maniére universelle de M. Desargues pour practiquer la perspective ) of a practical book on the use of perspective published in 1648 by his friend and pupil Abraham Bosse ( 1602 – 1676 ).

Using the theorem we can treat every abelian category as if it is the category of R-modules concerning theorems about existence of morphisms in a diagram and commutativity and exactness of diagrams.

Using Bloch's theorem, we only need to find a solution for a single period, make sure it is continuous and smooth, and to make sure the function u ( x ) is also continuous and smooth.

Using case 3 of the master theorem ( with ε = 1 and c = 2 / 3 ), we know that T ( n ) = Θ ( n ).

Using a theorem of Schwartz on compact operators in Frechet spaces, Cartan and Serre proved that compact complex manifolds have the property that their sheaf cohomology for any coherent sheaf consists of vector spaces of finite dimension.

Using Taylor's theorem,

Using the Riesz representation theorem, one can show that there exists a unique continuous linear operator

Using his excellent knowledge of Greek, which was then rare in the West, to his advantage, he studied the Hebrew Bible and Greek authors like Philo, Origen, Athanasius, and Basil of Caesarea, with whom he was also exchanging letters.

Using a small Department of Defense International Military Education and Training ( IMET ) budget, the U. S. Embassy has established English-language courses at an LAT military base, and has brought LAT officers to attended officer basic training courses in the U. S. The government of Burkina Faso has also accepted additional U. S. training assistance in counter-terrorism tactics and humanitarian assistance.

Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space.

Using more than one codec or encoding scheme successively can also degrade quality significantly.

Using his favorite model in his last painting of her, a fellow painter, Victorine Meurent, also the model for Olympia and the Luncheon on the Grass, sits before an iron fence holding a sleeping puppy and an open book in her lap.

Using needles of varying shape and thickness as well as different varieties of yarn can also change the effect.

Using the thoroughbred racing convention of one length equaling one-fifth of a second to calculate Sham ’ s time, he also finished in under two minutes.

Using this format can also lead to accusations that the reporter has created a misleading appearance that viewpoints have equal validity ( sometimes called " false balance ").

Using the corresponding strict relation "<", one can also define the interval ( a, b ) as the set of points x satisfying a < x and x < b, also written a < x < b. An open interval may be empty even if a < b.

Using statistics collected from tens of thousands of coincidence events, a set of simultaneous equations for the total activity of each parcel of tissue along many LORs can be solved by a number of techniques, and, thus, a map of radioactivities as a function of location for parcels or bits of tissue ( also called voxels ) may be constructed and plotted.

Using λ instead of φ ( n ) allows more choices for d. λ can also be defined using the Carmichael function, λ ( n ).

Using the view factor reciprocity, A < sub > i </ sub > F < sub > ij </ sub > = A < sub > j </ sub > F < sub > ji </ sub >, the update equation can also be re-written in terms of the view factor F < sub > ji </ sub > seen by each sending patch A < sub > j </ sub >:

Using stock sets for other shows results in odd subcategories like the science fiction western ; some established shows also have the occasional episode.

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 purple as Mr. Glass ' color to David Dunn's green was also Jackson's idea.

Using a higher or lower pitch than normal will also increase stress in the speech organs.

Using a smaller wakeboard will make the wakeboard feel lighter, spin faster and seem more aggressive but also make landing neatly more difficult.

Using nigger as a self-referential identity term also was a way of avoiding white suspicion, of encountering an intelligent slave, and so put whites at their ease.

Using a less strict definition of the electro-optic effect allowing also electric fields oscillating at optical frequencies, one could also include nonlinear absorption ( absorption depends on the light intensity ) to category a ) and the optical Kerr effect ( refractive index depends on the light intensity ) to category b ).

Using coils with parallel capacitors will form a filter with the topology of an m-derived filter and a band of frequencies above the cut-off will also be passed.

" Using a 50 power refracting telescope that he designed himself, Huygens also discovered the first of Saturn's moons, Titan.

Using greases specifically designed for this purpose is also a good idea, as these are often better at sealing under vacuum, thicker and so less likely to flow out of the taper, become fluidic at higher temperatures than Vaseline ( a common substitute ) and are more chemically inert than other substitutes.

Using the eggbeater, swimmers can also perform " boosts ", where they use their legs to momentarily propel themselves out of the water to their hips or higher.