This will be more or less our general method for the second problem.

If the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.

Combinatorics is an example of a field of mathematics which does not, in general, follow the axiomatic method.

A general method for analysis of concentration involves the creation of a calibration curve.

Justice Holmes cautioned that “ the proper derivation of general principals in both common and constitutional law ... arise gradually, in the emergence of a consensus from a multitude of particularized prior decisions .” Justice Cardozo noted the “ common law does not work from pre-established truths of universal and inflexible validity to conclusions derived from them deductively ,” but “ ts method is inductive, and it draws its generalizations from particulars .”

As a general rule of thumb, if the condition number, then you may lose up to digits of accuracy on top of what would be lost to the numerical method due to loss of precision from arithmetic methods.

The book describes the general pattern of the past official dynastic historiography on structure, method, order of arrangement, sequence, caption and commentary back to the pre-Qin era.

Some of the suspicions about hidden weaknesses in the S-boxes were allayed in 1990, with the independent discovery and open publication by Eli Biham and Adi Shamir of differential cryptanalysis, a general method for breaking block ciphers.

Processes ( in general ) are treated as a product of design, not the method of design.

“ Our author ( Diophantos ) not the slightest trace of a general, comprehensive method is discernible ; each problem calls for some special method which refuses to work even for the most closely related problems.

Exponentiating by squaring is a general method for fast computation of large integer powers of a number.

* Fixed-point iteration, a general method to compute a function's fixed point

This method ( and the general idea of an FFT ) was popularized by a publication of J. W. Cooley and J. W. Tukey in 1965, but it was later discovered ( Heideman & Burrus, 1984 ) that those two authors had independently re-invented an algorithm known to Carl Friedrich Gauss around 1805 ( and subsequently rediscovered several times in limited forms ).

Though this method can backfire, as the general opinion of feet can depend on the culture and society of the area.

Paul Erdős was the first to prove the general result, using the probabilistic method.

For he, following the example of no previous general, with teachers summoned from the gladiatorial training school of C. Aurelus Scaurus, implanted in the legions a more sophisticated method of avoiding and dealing a blow and mixed bravery with skill and skill back again with virtue so that skill became stronger by bravery's passion and passion became more wary with the knowledge of this art.

In practice, this method is only feasible for small numbers ; computing prime factorizations in general takes far too long.

According to many Marxists influenced by Soviet Marxism, historical materialism is a specifically sociological method, while dialectical materialism refers to a more general, abstract, philosophy.

The extension ψ is in general not uniquely specified by φ, and the proof gives no explicit method as to how to find ψ: in the case of an infinite dimensional space V, it depends on Zorn's lemma, one formulation of the axiom of choice.

For the benefit of users wishing to produce statistics and indexes oriented towards medical care, the Ninth Revision included an optional alternative method of classifying diagnostic statements, including information about both an underlying general disease and a manifestation in a particular organ or site.

Wallis generalized Cavalieri's method, computing integrals of x to a general power, including negative powers and fractional powers.

He was best known to the general public for inventing a method to treat milk and wine in order to prevent it from causing sickness, a process that came to be called pasteurization.

The general method of matrix addressing consists of sequentially addressing one side of the matrix, for example by selecting the rows one-by-one and applying the picture information on the other side at the columns row-by-row.

The Greek mathematician Menaechmus solved problems and proved theorems by using a method that had a strong resemblance to the use of coordinates and it has sometimes been maintained that he had introduced analytic geometry.

They can be solved by the following approach, known as an integrating factor method.

Āryabhaṭa ( 476 – 550 CE ) showed that pairs of simultaneous congruences, could be solved by a method he called kuṭṭaka, or pulveriser ; this is a procedure close to ( a generalisation of ) the Euclidean algorithm, which was probably discovered independently in India.

Numerical modeling techniques such as the finite element method, the boundary element method and the transmission-line matrix method can be used to model the propagation of light in systems which cannot be solved analytically.

Another element of philosophical method is to formulate questions to be answered or problems to be solved.

Brahmagupta solved many Pell equations with this method ; in particular he showed how to obtain solutions starting from an integer solution of for k =± 1, ± 2, or ± 4.

Quadratic equations can be solved by factoring, completing the square, graphing, Newton's method, and using the quadratic formula ( given below ).

The structure of graphite was solved in 1916 by the related method of powder diffraction, which was developed by Peter Debye and Paul Scherrer and, independently, by Albert Hull in 1917.

Although this is the original method by which protein crystal structures were solved, it has largely been superseded by MAD phasing with selenomethionine.

He than solved the high order equation by Southern Song dynasty mathematician Qin Jiushao's " Ling long kai fang " method published in Shùshū Jiǔzhāng (“ Mathematical Treatise in Nine Sections ”) in 1247 ( more than 570 years before English mathematician William Horner's method using synthetic division ).

* Find out how boundary value problems governed by Laplace's equation may be solved numerically by boundary element method

The equations are then discretized onto a grid using the finite difference method and solved subject to the constraints of Gauss's law and Linear elasticity.

In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved ; this is known as the method of generating functions.

The cubic was first partly solved by the 15th / 16th century Italian mathematician Scipione del Ferro, who did not however publish his results ; this method only solved one of three classes, as the others involved taking square roots of negative numbers, and complex numbers were not known at the time.

A common method is Platt's Sequential Minimal Optimization ( SMO ) algorithm, which breaks the problem down into 2-dimensional sub-problems that may be solved analytically, eliminating the need for a numerical optimization algorithm.

Fuchs and co-authors showed that constructing a BSP tree solved both of these problems by providing a rapid method of sorting polygons with respect to a given viewpoint ( linear in the number of polygons in the scene ) and by subdividing overlapping polygons to avoid errors that can occur with the painter's algorithm.

LLSQ solutions can be computed using direct methods, although problems with large numbers of parameters are typically solved with iterative methods, such as the Gauss – Seidel method.

The Legendre differential equation may be solved using the standard power series method.

Above we used the method of arbitrage-free pricing (“ delta-hedging ”) to derive the Black – Scholes PDE, and then solved the PDE to get the valuation formula.

Lima's Solar Refinery General Manager John Van Dyke and Herman Frasch, Standard's chemist, solved the distillation problem for sour crude by devising a method for removing the sulfur.