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.

An alternative and equivalent definition of Carmichael numbers is given by Korselt's criterion.

The authors ' caveats and repeated efforts at alternative definition have been echoed in subsequent scholarship: in the more than five decades since, there have been innumerable further attempts at definition, yet in the words of cinema historian Mark Bould, film noir remains an " elusive phenomenon ... always just out of reach ".

Keynes asked his friend Piero Sraffa to respond publicly to Hayek's challenge ; instead of formulating an alternative theory, Sraffa elaborated on the logical inconsistencies of Hayek's argument, especially concerning the effect of inflation-induced " forced savings " on the capital sector and about the definition of a " natural " interest rate in a growing economy.

Another alternative is to give the definition of the word.

Radon measures have an alternative definition in terms of linear functionals on the locally convex space of continuous functions with compact support.

From these three assumptions, neoclassical economists have built a structure to understand the allocation of scarce resources among alternative ends — in fact understanding such allocation is often considered the definition of economics to neoclassical theorists.

An alternative definition for " weakly typed " refers to languages, such as Perl and JavaScript, which permit a large number of implicit type conversions.

A popular alternative definition of chronic pain, involving no arbitrarily fixed durations is " pain that extends beyond the expected period of healing.

* Choice Corollary: People choose for themselves the particular alternative in a dichotomized construct through which they anticipate the greater possibility for extension and definition of their system.

An alternative and disputable definition of QoS, used especially in application layer services such as telephony and streaming video, is requirements on a metric that reflects or predicts the subjectively experienced quality.

In atmosphere science, an alternative definition is usually given, from 3 Hz to 3 kHz.

By this broader definition the two-round system is not the only form of run-off voting, and others include the exhaustive ballot and instant run-off voting ( also known as the alternative vote ).

Like the " dimension of image " characterization this can be generalized to a definition of the rank of a linear map: the rank of a linear map f from V → W is the minimal dimension k of an intermediate space X such that f can be written as the composition of a map V → X and a map X → W. While this definition does not suggest an efficient manner to compute the rank ( for which it is better to use one of the alternative definitions ), it does allow to easily understand many of the properties of the rank, for instance that the rank of the transpose of A is the same as that of A.

The US edition, once again by A & E Home Video, includes the first North American release of an alternative edit of " Arrival " ( in high definition ), as well as " The Alternate Version of ' The Chimes of Big Ben '" from the earlier DVD / VHS releases ( in standard definition due to the degraded source material ) and assorted documentaries and behind-the-scenes footage.

An alternative spelling in English is ' vocal chords ', possibly due to the musical connotations or to confusion with the geometrical definition of the word " chord ".

Swiss psychoanalyst Carl Jung, who studied archetypes, proposed an alternative definition of symbol, distinguishing it from the term sign.

An alternative definition of SNR is as the reciprocal of the coefficient of variation, i. e., the ratio of mean to standard deviation of a signal or measurement:

If the signal data are not constant, then can be calculated from data where the signal is zero or relatively constant .</ ref > Notice that such an alternative definition is only useful for variables that are always non-negative ( such as photon counts and luminance ).

Sometimes SNR is defined as the square of the alternative definition above.

Yet another alternative, very specific and distinct definition of SNR is employed to characterize sensitivity of imaging systems ; see signal to noise ratio ( imaging ).

The alternative to this is that if a conservative candidate is nominated the national committee will have to appeal to the trusts for their campaign funds, and in doing this will incur obligations which would make a Democratic victory absolutely fruitless.

There is a workable alternative to this potentially dangerous and harmful C. & O. - B. & O. merger scheme '' --

He is already beginning to box the compass of alternative lines of action, including doing nothing.

Accordingly, if it is not repealed by the Congress at its present session, I shall have no alternative thereafter but to direct the Secretary of Defense to disregard the section unless a court of competent jurisdiction determines otherwise.

On review the Supreme Court, via Mr. Justice Frankfurter, found southern racial problems `` a sensitive area of social policy on which the federal courts ought not to enter unless no alternative to adjudication is open ''.

A sense of self-certainty and the freedom to experiment with different roles, or confidence in one's own unique behavior as an alternative to peer-group conformity, is more easily developed during adolescence if, during early childhood, the individual was permitted to exercise initiative and encouraged to develop some autonomy.

The largest hurdle the Republicans would have to face is a state law which says that before making a first race, one of two alternative courses must be taken: 1

It is one that most try to avoid, as long as they can see an alternative approach to the problem.

`` My feeling at the moment '', he said, `` is that we have no alternative, irrespective of some of the arguments about him.

Expressed differently: if the price for becoming a faithful follower of Jesus Christ is some form of self-destruction, whether of the body or of the mind -- sacrificium corporis, sacrificium intellectus -- then there is no alternative but that the price remain unpaid.

In any event, it is an irreversible step, and if we are at all honest with ourselves, we will know we have no other alternative than to live in the world in which God has seen fit to place us.

For we have said, in effect, that of the two alternatives to his position variously represented by the other participants in the demythologizing discussion, only one is really an alternative.

It follows, then, provided the possibilities have been exhausted, that the only real alternative is the general viewpoint of the `` left '', which has been represented on the Continent by Fritz Buri and, to some extent at least, is found in much that is significant in American and English theology.

It will be recalled from the discussion in Section 7 that the position of the `` right '', as represented by Barth, rests on the following thesis: The only tenable alternative to Bultmann's position is a theology that ( 1 ) rejects or at least qualifies his unconditioned demand for demythologization and existential interpretation ; ;

Certainly, in analyzing an action which truly faced such alternatives, `` it is never possible that no world would be preferable to some worlds, and there are in truth no circumstances in which the destruction of human life presents itself as a reasonable alternative ''.

`` Your alternative is breathtaking '', he said, `` and, I'm frank in saying, a bit mad.

As with most mythology, there is a tale which offers an alternative version of these events: in Argonautica ( iv. 760 ) Zeus ' sister and wife Hera alludes to Thetis ' chaste resistance to the advances of Zeus, that Thetis was so loyal to Hera's marriage bond that she coolly rejected him.

Rudra could bring diseases with his arrows, but he was able to free people of them, and his alternative Shiba, is a healer physician god.

The ANOVA F – test ( of the null-hypothesis that all treatments have exactly the same effect ) is recommended as a practical test, because of its robustness against many alternative distributions.

Power analysis can assist in study design by determining what sample size would be required in order to have a reasonable chance of rejecting the null hypothesis when the alternative hypothesis is true.

Augmentative and alternative communication ( AAC ) is an umbrella term that encompasses methods of communication for those with impairments or restrictions on the production or comprehension of spoken or written language.