For some forms of philosophy, this very division between the empirical and the rational becomes a sign of the metaphysical superiority of the latter.

For what concerns all scientific disciplines is precisely that which can be captured for the rational, i.e., for the scientific determination of what in past ages was considered ultimate and irrational.

For example, the field extension R / Q, that is the field of real numbers as an extension of the field of rational numbers, is transcendental, while the field extensions C / R and Q (√ 2 )/ Q are algebraic, where C is the field of complex numbers.

For instance, the field of all algebraic numbers is an infinite algebraic extension of the rational numbers.

For Friedrich Schiller aesthetic appreciation of beauty is the most perfect reconciliation of the sensual and rational parts of human nature.

" For Steiner, the human capacity for rational thought would allow individuals to comprehend spiritual research on their own and bypass the danger of dependency on an authority.

For example, the species, man, may be defined as an animal ( genus ) that is rational ( difference ).

For example, the division example above is surjective ( or onto ) because every rational number may be expressed as a quotient of an integer and a natural number.

For instance, the set of rational numbers is not complete, because e. g. is " missing " from it, even though one can construct a Cauchy sequence of rational numbers that converges to it.

For a prime p, the p-adic numbers arise by completing the rational numbers with respect to a different metric.

For others, psychological inconsistency is not a stumbling block to good drama: " Euripides is in pursuit of a larger insight: he aims to set forth the two modes, emotional and rational, with which human beings confront their own mortality.

For the various sectarian gnostics, gnosis was obtained as speculative gnosis, instigated by the contemplation of their religio-philosophical ( Cosmological, Metaphysical, salvational and rational ) systems.

In fact, for many fields K one does not know in general precisely which finite groups occur as Galois groups over K. This is the inverse Galois problem for a field K. ( For some fields K the inverse Galois problem is settled, such as the field of rational functions in one variable over the complex numbers.

For a simple example of the first kind, consider the proposition, ¬ p: " there is no smallest rational number greater than 0 ".

For other examples, see proof that the square root of 2 is not rational and Cantor's diagonal argument.

For example, every rational solution of the equation

For example, a surprise decrease in the money supply may shock rational economic factors and suddenly inhibit aggregate demand.

For example, the set of rational numbers — those numbers which can be written as a quotient of integers — contains the natural numbers as a subset, but is no bigger than the set of natural numbers since the rationals are countable: There is a bijection from the naturals to the rationals.

For the rest of their lives, every appearance of the word subconsciously generates a feeling of uneasiness and confusion, and prevents rational consideration of the subject.

For example, it is the means by which rational beings understand themselves to think about cause and effect, truth and falsehood, and what is good or bad.

For example in modern economics, rational choice is assumed to equate to logically consistent choice.

For this use of the term see Padé approximation and Chebyshev rational functions.

For example, the set of rational numbers, as a subset of R has the property that the interior has an empty closure, but it is not nowhere dense ; in fact it is dense in R.

For the interpretation of these significant investigations, it should be remembered that reciprocal relations exist in the hypothalamus with respect to autonomic and somatic functions which are closely associated with the emotions.

For many years he had provided music and entertainment for functions throughout the Northwest.

For the Greeks, Apollo was all the Gods in one and through the centuries he acquired different functions which could originate from different gods.

For liturgical functions other than the Mass the bishop typically wears the cope.

For example, one cannot define all the trigonometric functions in a Banach algebra without identity.

For non-integer α, the functions J < sub > α </ sub >( x ) and J < sub >− α </ sub >( x ) are linearly independent, and are therefore the two solutions of the differential equation.

For the spherical Bessel functions the orthogonality relation is:

For instance, NASA has extremely rigorous software testing procedures for many operating systems and communication functions.

For a certain class of Green functions coming from solutions of integral equations, Schmidt had shown that a property analogous to the Arzelà – Ascoli theorem held in the sense of mean convergence — or convergence in what would later be dubbed a Hilbert space.

For those priests over 80 who became cardinal-deacons and were not ordained to the episcopacy, this is the highest position they can normally attain in the Church hierarchy ( though all cardinals rank above bishops in rank and order of precedence, those cardinals who are not bishops do not have the right to perform the functions reserved solely to bishops, such as ordination ).

For some distributed parameter systems the vectors may be infinite-dimensional ( typically functions ).

For instance, instead of a hardware multiplier, a calculator might implement floating point mathematics with code in ROM, and compute trigonometric functions with the CORDIC algorithm because CORDIC does not require hardware floating-point.

For example: clairsentience is one of the six human special functions mentioned or recorded in Buddhism.

For the same reason, filter functions whose critical response is at lower frequencies ( compared to the sampling frequency 1 / T ) require a higher order, more computationally intensive FIR filter.

For polynomials of degree four and smaller there are explicit formulae for the roots ( the formulae are elementary functions ).

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.

For instance, by assigning to every open set U the associative algebra of real-valued continuous functions on U, one obtains a presheaf of algebras on X.

For periodic functions, both the Fourier transform and the DTFT comprise only a discrete set of frequency components ( Fourier series ), and the transforms diverge at those frequencies.

Direction of causal flow is upward. For a multivariate function, functional decomposition generally refers to a process of identifying a set of functions such that

For crimes that are not committed in the exercise of his functions, the President of the Republic answers before the common courts, once his term of office has ended.

For example, operation-preserving functions exist between the set of real numbers with addition and the set of positive real numbers with multiplication.

For most types of hashing functions the choice of the function depends strongly on the nature of the input data, and their probability distribution in the intended application.

For instance, the inverse of the sine function is typically called the arcsine function, written as arcsin, which is, like sin, conventionally denoted in roman type and not in italics ( note that software libraries of mathematical functions often use the name < tt > asin </ tt >):

For example, scientists developing a new viral drug to treat an infection with a pathogenic virus ( e. g. HIV-1 ) may find that a candidate drug functions to prevent viral replication in an in vitro setting ( typically cell culture ).

For this purpose the Internet Protocol defines an addressing system that has two functions: identifying hosts and providing a logical location service.