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 definition of basis with a continuous set of indices and consequently for a rigorous definition of position and momentum 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 instance, the following statement was rated low in compulsivity, `` She's naturally quite neat about things, but it doesn't bother her at all if her room gets messy.

For a statement of costs per kilowatt-hour would ignore the fact that many of these costs are not a function of kilowatt-hour output ( or consumption ) of energy.

For each fellowship, anyone who publicly assents to the doctrines described in the statement and is in good standing in their " home ecclesia " is generally welcome to participate in the activities of any other ecclesia.

For instance in the Central fellowship, the BASF, the standard statement of faith, has 30 doctrines to be accepted and 35 to be rejected.

For each problem, the Institute had a professional mathematician write up an official statement of the problem, which will be the main standard by which a given solution will be measured against.

For regions in R < sup > 3 </ sup > more complicated than this, the latter statement might be false ( see Poincaré lemma ).

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 clarity, this statement asserts that there is at least one person who can be fooled at all times, which is stronger than asserting that at all times at least one person exists who can be fooled.

For a statement to be questioned using observation, it needs to be at least theoretically possible that it can come in conflict with observation.

For example, the statement " All swans are white " is falsifiable, because it can come in conflict with the observation " this swan is black.

In Demonstration ( 74 ) Irenaeus reinforced his view that Jesus was at least 45 with the statement " For Herod the king of the Jews and Pontius Pilate, the governor of Claudius Caesar, came together and condemned Him to be crucified.

For the Jews slew him, although he was a most just man .” However, this statement does not appear in the extant manuscripts of Josephus.

For example, if A is some mathematical statement that an intuitionist has not yet proved or disproved, then that intuitionist will not assert the truth of " A or not A ".

For the finale, the whole city was summoned to hear a very important political statement.

President George W. Bush made note of the anniversary in a written statement, part of which echoed his remarks on the execution of Timothy McVeigh in 2001: " For the survivors of the crime and for the families of the dead the pain goes on.

For example, a statement such as " God created the universe " may be true or false, but no tests can be devised that could prove it either way ; it simply lies outside the reach of science.

For example, prime ideals in the ring of integers of quadratic number fields can be used in proving quadratic reciprocity, a statement that concerns the solvability of quadratic equations

* complete: For each statement in Presburger arithmetic, either it is possible to deduce it from the axioms or it is possible to deduce its negation.

For example, a complex statement by Richard Whately includes four puns: " Why can a man never starve in the Great Desert?

For example the statement " π is only half a pie.

John Hardy from the Legal institute of England stated " For the title to be valid, we must incorporate the company or association for the living " This statement has been used thoroughly

For example, the following statement:

For example, say we wish to reason about the following statement:

For determinantal rank, the statement is that if the row rank ( column rank ) of a matrix is p, then one can choose a p × p submatrix that is invertible: a subset of the rows and a subset of the columns simultaneously define an invertible submatrix.