In some cases it may be difficult to describe x independently of such a limiting process involving rational numbers.

In many cases of interest there is a standard ( or " canonical ") embedding, like those of the natural numbers in the integers, the integers in the rational numbers, the rational numbers in the real numbers, and the real numbers in the complex numbers.

It can be shown that additivity implies the homogeneity in all cases where α is rational ; this is done by proving the case where α is a natural number by mathematical induction and then extending the result to arbitrary rational numbers.

In many cases, they may lack the bargaining power to act on their expectations, no matter how rational they are, or their perceptions, no matter how free of money illusion they are.

It may be inappropriate and probably inaccurate to assume that all nettles exhibit similar properties in all cases, but where an action can be attributed to principles found in the species, such as histamine, choline, formic acid and silica, a rational basis for their use is still available.

The evolving level of scrutiny applied by Federal courts to Commerce Clause cases should be considered in the context of rational basis review.

In many cases one can identify K. L as a vector space tensor product, taken over the field N that is the intersection of K and L. For example if one adjoins √ 2 to the rational field ℚ to get K, and √ 3 to get L, it is true that the field M obtained as K. L inside the complex numbers ℂ is ( up to isomorphism )

Even apart from scaling and shifting, there are infinitely many cases, e. g. by considering rational numbers of which the denominators are powers of a given prime number.

This criterion is certainly not applicable to all polynomials with integer coefficients that are irreducible over the rational numbers, but does allow in certain important particular cases to prove irreducibility with very little effort.

In April 2010, GEIPAN statistics stated that 23 % of all cases were of Type D. However, Jean-Jacques Velasco, the head of SEPRA from 1983 to 2004, wrote a book in 2004 noting that 13. 5 % of the 5, 800 cases studied by SEPRA were dismissed without any rational explanation, and stated that UFOs are extraterrestrial in origin.

This is true in the first cases mentioned, where L is a free K-module, or K contains the field of rational numbers, using the construction outlined here ( in fact, the result is a coalgebra isomorphism, and not merely a K-module isomorphism, equipping both S ( L ) and U ( L ) with their natural coalgebra structures such that for v ∈ L ).

In all cases trust is a heuristic decision rule, allowing the human to deal with complexities that would require unrealistic effort in rational reasoning.

Even in cases of racial discrimination, where the courts apply a different standard of scrutiny to government action than they do in rational basis review, evidence of disparate impact " alone is insufficient even where the Fourteenth Amendment subjects state action to strict scrutiny.

This is an unsolved problem, ; it is known that being a Hodge cycle is a necessary condition to be an algebraic cycle that is rational, and numerous particular cases of the conjecture are known.

In both cases, she claimed, the dominant ideology prevented and discouraged people from finding rational and scientifically-verifiable explanations and solutions.

In such cases the intruding provisions of the law will only be upheld if they satisfy the " rational connection " test.

In adult cases of " acute and more or less ' traumatic ' upsetting events in the life of normal persons ", Fenichel stressed that in coping, " in carrying out a ' work of learning ' or ' work of adjustment ', he must acknowledge the new and less comfortable reality and fight tendencies towards regression, towards the misinterpretation of reality ", though such rational strategies " may be mixed with relative allowances for rest and for small regressions and compensatory wish fulfillment, which are recuperative in effect ".

The question the Court resolved in these cases was not whether the death sentence imposed on each of the individual defendants was cruel, but rather whether the process by which those sentences were imposed was rational and objectively reviewable.

The drawback of having juries rather than judges fix the penalty in capital cases is the risk that they will have no frame of reference for imposing the death penalty in a rational manner.

In many cases, it is relatively easy for a director to find some rational reason for his actions and, with the courts using the business judgment rule, the case will likely be dismissed ( U. S. courts disdain getting involved in business matters ).

" The Court applied minimal scrutiny ( rational basis review ) to the economic regulation in this case, but proposed a new level of review for certain other types of cases.

While the particular limits of these groupings may seem artificially arbitrary, they do fairly express a corresponding grouping of more variable material, and they eventuate also in five classes, along a similar scale, containing approximately equal numbers of cases, namely 19, 14, 15, 11, 12 in Athabascan.

Other excavations within the Ancient Pueblo culture area produce varying numbers of unburied, and in some cases dismembered, bodies.

The uncertainty that in some cases results has made it advisable to abandon the assignation of successive numbers in the list of the popes.

In most cases, these numbers remain undisclosed to prevent abuse, but MCI maintains this widely-published, toll-free ANAC: 1-800-437-7950.

In most cases, this is the field of real numbers.

The Siege of the Alcázar in the Spanish Civil War, in which the Nationalists held out against a much larger Republican force for two months until relieved, shows that in some cases a citadel can be effective even in modern warfare ; a similar case is the Battle of Huế, where an NVA division held the citadel of Huế for 26 days against roughly their own numbers of much better-equipped US and South Vietnamese troops.

Various results in probability theory about expected values, such as the strong law of large numbers, will not work in such cases.

Check digits and parity bits are special cases of checksums, appropriate for small blocks of data ( such as Social Security numbers, bank account numbers, computer words, single bytes, etc .).

It is difficult to calculate exact numbers of cases, as many go unreported due to concerns that an outbreak may have a negative impact on the tourism of a country.

While the increased numbers of malaria victims since DDT usage collapsed document its value, many other factors contributed to the rise in cases.

According to Rummel, these were not cases of genocide because those who were killed were not selected on the basis of their race, but were killed in large numbers as a result of government policies.

The reason why there were three cases to Diophantus, while today we have only one case, is that he did not have any notion for zero and he avoided negative coefficients by considering the given numbers to all be positive in each of the three cases above.

Later ancient commentators such as Proclus ( 410 – 485 CE ) treated many questions about infinity as issues demanding proof and, e. g., Proclus claimed to prove the infinite divisibility of a line, based on a proof by contradiction in which he considered the cases of even and odd numbers of points constituting it.

Even in such cases, over-learned and rote-learned speech patterns may be retained — for instance, some patients can count from one to ten, but cannot produce the same numbers in ordinary conversation.

In such cases, economists rely on observational studies, often using data sets with many strongly associated covariates, resulting in enormous numbers of models with similar explanatory ability but different covariates and regression estimates.

Nouns and adjectives have two cases, nominative / oblique and accusative / allative, and two numbers, singular and plural ; the adjectival form of personal pronouns behaves like a genitive case.

The other important cases are the field of real numbers, finite fields, and fields of p-adic numbers.

In yet another set of related cases, in September 2003, the RIAA ( trade association of the music industry ) filed suit in civil court against several private individuals who had shared large numbers of files with Kazaa ; most of these suits were settled with monetary payments averaging $ 3, 000.

The songs usually advance the plot or develop the film's characters, though in some cases they serve merely as breaks in the storyline, often as elaborate " production numbers ".

In some cases unhealthy tissue may have a lower system Q, meaning that the system acts more like a large heavy spring as compared to higher values of system Q ( healthy tissue ) that respond to higher forcing frequencies.

An ECG may show signs of right heart strain or acute cor pulmonale in cases of large PEs — the classic signs are a large S wave in lead I, a large Q wave in lead III and an inverted T wave in lead III ( S1Q3T3 ).

The binomial distribution has a mean equal to the proportion of cases, denoted P, and a variance equal to the product of cases and noncases, PQ, wherein Q is equal to the proportion of noncases or 1-P. Accordingly, the standard deviation is simply the square root of PQ.

Consider the variance formula: e = PQ, wherein P is equal to the proportion of " 1's " or " cases " and Q is equal to ( 1-P ), the proportion of " 0's " or " noncases " in the distribution.

In both cases, an independent technique is required to provide calibration of the continuous Q analysis, as arterial PP analysis cannot account for unmeasured variables such as the changing compliance of the vascular bed.

All cases studied demonstrated some degree of mental deficiency ; I. Q.

In both cases cited by Jackson, an epistemic subject A appears to have no access to particular items of knowledge about a subject B: A cannot know that B has an experience of a particular quality Q on certain occasions.

The Q Pack was developed by the Queens Group Inc. in the mid 1990s as an alternative to regular CD jewel cases.

Since Q Pack cases are not transparent, generally cover art is applied a decal to the cover.

Recent shows have been sponsored ( with prizes going to the winner such as free Bridgestone tires or several cases of Shell Q Motor Oil ), with Series Four featuring NAPA Auto Parts as the series presenting sponsor for both the regular and All Out series, owing to the show's growing popularity.

In the U. S., demand deposits only refer to funds held in checking accounts ( or cheque offering accounts ) other than NOW accounts ; however, in a 1970s and 1980s response to the 1933 promulgation of Regulation Q in the U. S., demand deposits in some cases came to allow easier access to funds from other types of accounts ( e. g. savings accounts and money market accounts ).

Writing G for the Galois group of Q ( i )/ Q, and σ for the complex conjugation automorphism in G, there are three cases to consider.

Huebner's first work on Q Fever was a report he had done on an outbreak of 18 cases that occurred in early 1946 in an NIH laboratory, where he showed a correlation between a spike in cases and the preparation of antigens in yolk sacs, and he prepared another report on a group of 47 patients being treated for the condition at the Public Health Service Hospital in Baltimore.

Mac uses her computer hacking and cracking skills to help Veronica with her cases ; the two joke that she is a kind of Q to Veronica's James Bond.

He's very skilled in the art of disguise, and is often ordered by Dan to watch Class Q in cases by disguising as one of the suspects.

Butler argued that accepting the priority of Matthew rendered it possible to dispense with the hypothetical Q document altogether, a position he supported by arguments concerning the inadmissibility of appealing to Q as a sound explanation of the cases where Matthew appears to be more original than Mark.