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 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 a steering system that controlled the modified rocket's tail surfaces and wings equipped with ailerons, Fiedler was to land the missile on a skid especially bolted under the fuselage.

: Using * to denote a binary operation performed on a set

Using these tools, one can write a more complex text editor, and a simple compiler for a higher-level language and so on, until one can have a graphical IDE and an extremely high-level programming language.

Using pharmacogenomics, pharmaceutical companies can create drugs based on the proteins, enzymes and RNA molecules that are associated with specific genes and diseases.

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 ClearType on these display configurations will actually reduce the display quality.

Using the Cartesian coordinate system, geometric shapes ( such as curves ) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape.

Using to denote the size of the original grammar, the size blow-up in the worst case may range from to, depending on the transformation algorithm used.

Using pioneering film-looping technology, Enoch J. Rector presented the entirety of a famous 1897 prize-fight on cinema screens across the country.

* International Good Practice: Guidance on Project Appraisal Using Discounted Cash Flow, International Federation of Accountants, June 2008, ISBN 978-1-934779-39-2

Using the civil unrest in India as an excuse to move troops to the Durand Line, Afghan troops crossed the border at the western end of the Khyber Pass on 3 May 1919 and occupied the village of Bagh, the scene of an earlier uprising in April.

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 digital computers, on the other hand, both FIR and IIR filters are straightforward to implement in software.

Using humanist techniques for working on texts, he prepared important new Latin and Greek editions of the New Testament.

* Using Java, and based very closely on Weizenbaum's published description of the program: http :// www. chayden. net / eliza / Eliza. html

* Using Z80 assembly language on the TI-83 Plus: ticalc. org

Using this method, the customer does not need to authenticate his or her identity by PIN entry or signature, as on a regular machine.

Using mathematical rules and procedures based on properties of reducible configurations, Appel and Haken found an unavoidable set of reducible configurations, thus proving that a minimal counterexample to the four-color conjecture could not exist.

Using FProxy, a user can browse freesites ( web sites that use normal HTML and related tools, but whose content is stored within Freenet rather than on a traditional web server ).

Using whatever equipment the vassal could obtain by virtue of the revenues from the fief, the vassal was responsible to answer calls to military service on behalf of the lord.

Using the submarine, Akatsuki, they hope to gather information on the robot's builders, the alien simians.

Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology.

Using their data on the time occupied by travel to work, Table 2 below shows the absolute number of people who responded with travel times " at least 15 but less than 20 minutes " is higher than the numbers for the categories above and below it.

Using the Emergency-era Internal Security Act ( ISA ), the new government suspended Parliament and political parties, imposed press censorship and placed severe restrictions on political activity.