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

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 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 the treatment of bacterial scours, feed 100-200 grams.

For prevention or treatment of bacterial scours, feed 0.1 to 5 milligrams per pound of body weight daily.

For the prevention or treatment of the early stages of shipping fever complex, increase feeding level to 0.5 to 2 grams per head per day.

For treatment of shipping fever, this level should be fed at the onset of the disease symptoms until symptoms disappear.

For treatment of bloat, drug is fed at a higher level.

For the prevention or treatment of acetonemia ( ketosis ) in dairy cows.

For the treatment of ketosis feed 1/4 to 1/2 pound per day for 10 days.

For treatment of ketosis, feed 1/2 pound daily until symptoms disappear.

For example, the level of improvement noted in a recent experiment with a short course of immediate treatment for parent-child relationship problems compared favorably with the results reported by typical child guidance clinics where the hours spent in purely diagnostic study may equal or exceed the number of hours devoted to actual treatment interviews in the experimental project.

For several days, she was ill as a result of Lee's treatment.

For example, the model for a simplified ANOVA with one type of treatment at different levels.

For tonal languages, further classification can be based on their treatment of tone, though names do not yet exist to distinguish the various types.

For example, " aloin " is derived from Aloe vera and Aloe ferox and has important medical uses ( e. g. as a laxative and in the treatment of burns ) as well as cosmetic uses ( e. g. in skin and hair products ).

For the most part, treatment of the eye conditions consists of visual rehabilitation.

For many individuals with bipolar disorder a good prognosis results from good treatment, which, in turn, results from an accurate diagnosis.

For a fully relativistic treatment, the equation of state used will interpolate between the equations for small ρ and for large ρ.

For example, a court might certify a case for class treatment where a number of individual bond-holders sue to determine whether they may convert their bonds to common stock.

For example, a Bahá ' í who refuses to follow guidance on treatment of Covenant-breakers is at risk of being named one.

For a physician in the acute treatment setting, acute alcohol intoxication can mimic other acute neurological disorders, or is frequently combined with other recreational drugs that complicate diagnosis and treatment.

For example in a paper reporting on a study involving human subjects, there typically appears a table giving the overall sample size, sample sizes in important subgroups ( e. g., for each treatment or exposure group ), and demographic or clinical characteristics such as the average age, the proportion of subjects of each sex, and the proportion of subjects with related comorbidities.

For example, 330 for economics +. 9 for geographic treatment +. 04 for Europe = 330. 94 European economy ; 973 for United States +. 05 form division for periodicals

For example, in his writings about a young man afflicted with the plague, he concentrated on the treatment of internal and external ulcerations.