For more rigorous and precise analysis, the numerical approximations must be used.

For a rigorous treatment of the Dirac inner product of non-normalizable states see the definition given by D. Carfì in and.

For a rigorous statement of the expansion of an S-diagonalizable operator-observable-in its eigenbasis or in another basis see.

For public corporations, these responsibilities are typically much more rigorous and complex than for those of other types.

For instance, NASA has extremely rigorous software testing procedures for many operating systems and communication functions.

For instance, a 2006 meta analysis on 14 studies concluded that there is " no discernable effect " while a 2007 systemic review of intercessory prayer reported inconclusive results, noting that 7 of 17 studies had " small, but significant, effect sizes " but the review noted that the most methodologically rigorous studies failed to produce significant findings.

For children living with CF, preliminary studies show pediatric massage therapy may improve patients and their families quality of life, though more rigorous studies must be done.

For example, one can specify the density operators describing microcanonical, canonical, and grand canonical ensembles of quantum mechanical systems, in a mathematically rigorous fashion.

For a more rigorous mathematical derivation of Schrödinger's equation, see also.

For a rigorous mathematical treatment, see electromagnetic induction and Maxwell's equations.

For example, Robert K. Merton asserts that all ideas must be tested and are subject to rigorous, structured community scrutiny ( see Mertonian norms ).

For decades, it was generally thought that this proof was flawed and that the first rigorous proof was carried out by Oswald Veblen.

For conditions with a large enough potential market, more rigorous trials were sponsored by pharmaceutical companies that were making growth hormone to achieve approval to market for those specific indications.

For example, judges routinely assign numbers to properties such as the perceived beauty of an exercise ( e. g. 1-10 ) without necessarily establishing quantitative structure in any sort of rigorous fashion.

For a basic introduction to sets see the article on sets, for a fuller account see naive set theory, and for a full rigorous axiomatic treatment see axiomatic set theory.

It is a one-year, rigorous programme with a heavy focus on the manufacturing sector, and is drawn on the lines of MIT's Leaders for Global Operations ( LGO ) ( formerly called Leaders For Manufacturing ( LFM )) programme, run jointly by MIT Sloan School of Management and MIT's School of Engineering.

For specimens known only from fossils, the rigorous analysis of characters to determine evolutionary relationships between different groups of animals ( clades ) proved incredibly useful.

For around 1300 years, from 605 to 1905, mandarins were selected by merit through the extremely rigorous imperial examination.

For example, aerospace contractors are particularly rigorous in the analysis and implementation of Safety factors in the manufacture of manned space flight equipment or even launch equipment for unmanned satellites.

For this he was sentenced to one year rigorous imprisonment and sent to Hissar jail on 8 October 1930.

For want of provisions, the Sikhs were reduced to rigorous straits.

: For a detailed presentation of the various points of view around the definition of " algorithm " see Algorithm characterizations.

For Lie algebras the definition is slightly different.

For planets, comets and asteroids a different definition of absolute magnitude is used which is more meaningful for nonstellar objects.

For example, one definition of bandwidth could be the range of frequencies beyond which the frequency function is zero.

For example, they might interpret a tomato as a vegetable — according to the English definition of tomato — even though the ideal Blissymbol of vegetable was restricted by Bliss to just vegetables growing underground.

For example, in Virginia, the definition of the conduct that constitutes the crime of robbery exists only in the common law, and the robbery statute only sets the punishment.

For example, jīngjì ( 经济 ／ 經濟, keizai ), which in the original Chinese meant " the workings of the state ", was narrowed to " economy " in Japanese ; this narrowed definition was then re-imported into Chinese.

For alternative ( equivalent ) formulations of the definition in terms of a bijective function or a surjective function, see the section Formal definition and properties below.

In his preface to Jürgen Osterhammel's Colonialism: A Theoretical Overview, Roger Tignor says, " For Osterhammel, the essence of colonialism is the existence of colonies, which are by definition governed differently from other territories such as protectorates or informal spheres of influence.

For example, in formal languages like mathematics, a ' stipulative ' definition guides a specific discussion.

For this reason, Locking adds that a definition of a term must not comprise of terms which are synonymous with it.

For terms such as these it is not possible and indeed not necessary to state a definition ; rather, one simply comes to understand the use of the term.

For the mathematical definition, see Decimal representation.

For square matrices with entries in a non-commutative ring, for instance the quaternions, there is no unique definition for the determinant, and no definition that has all the usual properties of determinants over commutative rings.

For example, should animal experimentation become illegal in a society, it will no longer be an ethical issue on Hoy's definition.

For example, the statement " A society is free if and only if liberty is maximized and people are required to take responsibility for their actions " is true or paradoxical, depending on the individual's definition of liberty.

For a long time this definition was used with a derogatory intention.

For all intents and purposes, the 1947 ICC definition confirmed the 1895 MCC definition and gave it international recognition and usage.

For a planar graph, the crossing number is zero by definition.

For the making of selections on the basis of excellence requires that any foundation making the selections shall have available the judgments of a corps of advisors whose judgments are known to be good: such judgments can be known to be good only by the records of those selected, by records made subsequent to their selection over considerable periods of time.

For example, on the basis of the regression equation, a child with an I.Q. of 120 in this sample would be expected to earn an achievement test score of 4.8 ( grade equivalent ).

For the first time, the tactic of using two express bowlers in tandem paid off as Jack Gregory and Ted McDonald crippled the English batting on a regular basis.

For example, automated DNA sequencing machines were the basis to complete human genome projects leading to the birth of genomics.

For a finite-dimensional vector space, using a fixed orthonormal basis, the inner product can be written as a matrix multiplication of a row vector with a column vector:

For this reason, casuistry is widely considered to be the basis for the English common law and its derivatives.

For diatomic molecules, a systematic study using a minimum basis set and the first calculation with a larger basis set were published by Ransil and Nesbet respectively in 1960.

For instance, it defined chemical engineering to be a " science of itself, the basis of which is ... unit operations " in a 1922 report ; and with which principle, it had published a list of academic institutions which offered " satisfactory " chemical engineering courses.

For example, Italy allows citizenship almost entirely on the basis of jus sanguinis ( having an Italian Ancestor ).

For a useful interpretation of human reality, to share faith experience he said that we must each make certain " over-beliefs " in things which, while they cannot be proven on the basis of experience, help us to live fuller and better lives.

For the most part, however, the powers of the Crown are exercised on a day-to-day basis by elected and appointed individuals, leaving the governor to perform the various ceremonial duties the sovereign otherwise carries out when in the country ; at such a moment, the governor removes him or herself from public, though the presence of the monarch does not affect the governor's ability to perform governmental roles.

For quadratic forms of any signature, an orthogonal basis

For example, producer Mark " Hot Rod " Trollan copied bass-line sections from the 1983 Italo disco song " Feels Good ( Carrots & Beets )" ( by Electra featuring Tara Butler ) to form the basis of his 1986 production of " Your Love " by Jamie Principle.

" For Ourselves Council for Generalized Self-Management " discusses Stirner and speaks of a " communist egoism ," which is said to be a " synthesis of individualism and collectivism ," and says that " greed in its fullest sense is the only possible basis of communist society.

For basis vector in the context of crystals, see crystal structure.

For instance, the Godbeites broke from the LDS Church in the late 19th century on the basis of both political and religious liberalism, and in 1985 the Restoration Church of Jesus Christ broke from the LDS Church as an LGBT-friendly denomination.

For whole companies, outcome-based evaluations are the basis of stock exchange prices: Companies which produce higher profit growth are more valuable than companies which perform poorly.

For example, the state ( a, b, c, d, e, f, g, h ) in the computational basis can be written as:

For example, the notion of gauge invariance forms the basis of the well-known Mattis spin glasses, which are systems with the usual spin degrees of freedom for i = 1 ,..., N, with the special fixed " random " couplings Here the ε < sub > i </ sub > and ε < sub > k </ sub > quantities can independently and " randomly " take the values ± 1, which corresponds to a most-simple gauge transformation This means that thermodynamic expectation values of measurable quantities, e. g. of the energy are invariant.

For example, given two basis vectors of and two basis vectors of, the following is an entangled state: