We can be virtuous only if we control our lower natures, the passions in this case, and strengthen our rational side ; ;

If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.

A common example of a vertical asymptote is the case of a rational function at a point x such that the denominator is zero and the numerator is non-zero.

In the case of the rational numbers this means that any number has two irreducible fractions, related by a change of sign of both numerator and denominator ; this ambiguity can be removed by requiring the denominator to be positive.

This might come in the form of a proof that the number in question is in fact irrational ( or rational, as the case may be ); or a finite algorithm that could determine whether the number is rational or not.

In the 18th century Alexander Baumgarten considered the special case of sensitive knowledge and called it ' gnoseologia inferior ' ( as opposed to a superior rational one ) and with Kant it became aesthetics.

He argued that bureaucratic political and economic systems emerging in the Middle Ages were essential in the rise of modern capitalism ( including rational book-keeping and organization of formally free labour ), while they were a hindrance in the case of ancient capitalism, which had a different social and political structure based on conquest, slavery, and the coastal city-state.

These algorithms are not practicable for hand written computation, but are available in any Computer algebra system ( see Berlekamp's algorithm for the case in which the coefficients belong to a finite field or the Berlekamp – Zassenhaus algorithm when working over the rational numbers ).

In algebraic geometry, the parabola is generalized by the rational normal curves, which have coordinates the standard parabola is the case and the case is known as the twisted cubic.

The rational root theorem is a special case ( for a single linear factor ) of Gauss's lemma on the factorization of polynomials.

The integral root theorem is a special case of the rational root theorem if the leading coefficient a < sub > n </ sub > = 1.

In this particular case there is exactly one rational root.

It may also be the case that none of the candidates is a solution ; in this case the equation has no rational solution.

He argues that it is possible to distinguish the moral impulse of utilitarianism, which is “ to define the right as good consequences and to motivate people to achieve these ” from our ability to correctly apply rational principles which will among other things “ depend on the perceived facts of the case and on the particular moral actor ’ s mental equipment .” The fact that the latter is limited and can change doesn ’ t mean that the former has to be rejected.

Other languages, such as MATLAB and every computer algebra system return a rational number as the answer, as in case 3 above.

A very popular complex dynamical system is given by the family of quadratic polynomials, a special case of rational maps.

Alarms have the capability of causing a fight-or-flight response in humans ; a person under this mindset will panic and either flee the perceived danger or attempt to eliminate it, often ignoring rational thought in either case.

According to the definition above, some exotic functions, for example the Dirichlet function, are also periodic ; in the case of Dirichlet function, any nonzero rational number is a period.

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 the case of rational function of a complex variable, all denominators will have a polynomial of degree 1, or some positive integer power of such a polynomial.

In that case, the pressures and the velocity are assumed to be time harmonic functions of the form

In Ancient Greek, there was no ablative case ; its functions were taken by the genitive, so that the genitive had functions belonging to the Proto-Indo-European genitive and ablative cases.

As in Ancient Greek, the functions of the ablative case in Serbian are performed by the genitive case.

ASCII included the distinction of upper and lower case alphabets and a set of control characters to facilitate the transmission of written language as well as printing device functions, such as page advance and line feed, and the physical or logical control of data flow over the transmission media.

When α is an integer, moreover, as was similarly the case for the functions of the first kind, the following relationship is valid:

Faà di Bruno's formula for higher-order derivatives of single-variable functions generalizes to the multivariable case.

The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers.

In addition, this article discusses the definition for the more general case of functions between two metric spaces.

One of the functions ( in this case g ) is first reflected about and then offset by t, making it.

In some languages, the dative case has assimilated the functions of other now-extinct cases.

The pronoun whom is a remnant of the dative case in English, descending from the Old English dative pronoun " hwām " ( as opposed to the nominative " who ", which descends from Old English " hwā ") — though " whom " also absorbed the functions of the Old English accusative pronoun " hwone ".

Clenshaw – Curtis quadrature is essentially a change of variables to cast an arbitrary integral in terms of integrals of periodic functions where the Euler – Maclaurin approach is very accurate ( in that particular case the Euler – Maclaurin formula takes the form of a discrete cosine transform ).

With most IIR transfer functions there are related transfer functions having a frequency response with the same magnitude but a different phase ; in most case the so-called minimum phase transfer function is preferred.

For the case of a carrier modulated by a single sine wave, the resulting frequency spectrum can be calculated using Bessel functions of the first kind, as a function of the sideband number and the modulation index.

The major change in nominal morphology was the loss of the dative case ( its functions being largely taken over by the genitive ); in the verb, the major change was the loss of the infinitive, with a concomitant rise in new periphrastic forms.

Shotshell presses are a special case, and are generally a single unit of the " H " configuration that handles all functions, dedicated to reloading just one gauge of shotshell.

It was cleanly programmed and relatively bug-free, as well as speed gained from being written completely in x86 assembly language ( this remained the case for all versions until 4. 0 when Lotus switched to C ) and wrote directly to video memory rather than use the slow DOS and / or BIOS text output functions.

The is not strictly necessary at the end of a function like this, but is good programming practice in case other functions are added below later.

The firms may face differing cost functions in which case the reaction functions would not be identical nor would the equilibrium quantities.

In the simple case of periodic functions, this result is based on the theorem that any continuous periodic function can be represented as the sum of a series of sine waves and cosine waves:

This is the use of the factor n-1 in the denominator of the formula, rather than just n. This occurs when the sample mean rather than the population mean is used to centre the data and since the sample mean is a linear combination of the data the residual to the sample mean overcounts the number of degrees of freedom by the number of constraint equations — in this case one.

In case a denominator and numerator remain that are too large to ensure they are coprime by inspection, a greatest common divisor computation is needed anyway to ensure the fraction is actually irreducible.

In each case, however, if the limits of the numerator and denominator are evaluated and plugged into the division operation, the resulting expression is 0 / 0.

When programming a computer, one should use the function rather than the ordinary arctangent function (), in order to simplify handling of the case where the denominator is zero, and to compute unambiguously in all quadrants.

The appearance of in the denominator suggests that the required threshold gain would be decreased by lengthening the gain medium, but this is not generally the case.

The denominator, m-n, is the statistical degrees of freedom ; see effective degrees of freedom for generalizations for the case of correlated observations.

" Proportional to " in this case means that the denominator is not a function of and thus is the same for all values of ; it forms part of the normalization constant for the distribution over.

In the case a word has never been met during the learning phase, both the numerator and the denominator are equal to zero, both in the general formula and in the spamicity formula.

Journals may also attempt to limit the number of " citable items ", ie the denominator of the IF equation, either by declining to publish articles ( such as case reports in medical journals ) which are unlikely to be cited or by altering articles ( by not allowing an abstract or bibliography ) in hopes that Thomson Scientific will not deem it a " citable item ".