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Any rational function of a real variable can be written as the sum of a polynomial function and a finite number of algebraic fractions.
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Any and rational
Any rational number with a denominator whose only prime factors are 2 and / or 5 may be precisely expressed as a decimal fraction and has a finite decimal expansion.
* Any field, including fields of rational numbers, real numbers, and complex numbers.
Any process of evaluation or analysis, that may be called rational, is expected to be highly objective, logical and " mechanical ".
Roger Ebert's review for the Chicago Sun-Times was :" Any attempt to defend this movie on rational grounds is futile.
Any conic section defined over F with a rational point in F is a rational curve.
Any one of these models must be explicitly constructed, and most of these models are built using the basic properties of the rational number system as an ordered field.
Any discrete subset of Euclidean space is countable, since the isolation of each of its points ( together with the fact the rationals are dense in the reals ) means that it may be mapped 1-1 to a set of points with rational co-ordinates, of which there are only countably many.
Any other cost price would yield an arbitrage opportunity and riskless profit ( see rational pricing for the arbitrage mechanics ).
Any positive rational number can be written as the sum of unit fractions, in multiple ways.
Any Heronian triangle can be split into two right-angled triangles whose side lengths form Pythagorean triples with rational values.
Any such field is either equal to K or is also rational, i. e. L = K ( F ) for some rational function F. In geometrical terms this states that a non-constant rational map from the projective line to a curve C can only occur when C also has genus 0.
Any pairing function can be used in set theory to prove that integers and rational numbers have the same cardinality as natural numbers.
Any polynomial, and more generally any rational function p ( x )/ q ( x ) where p and q are polynomials, transforms the Riemann sphere by mapping it to itself.
For example, can be written as Any positive rational number can be written as a sum of unit fractions in infinitely many ways.
Any and function
Any function itself defines an equivalence relation on according to which if and only if.
Any cryptographic hash function, such as MD5 or SHA-1, may be used in the calculation of an HMAC ; the resulting MAC algorithm is termed HMAC-MD5 or HMAC-SHA1 accordingly.
Any compression algorithm can be viewed as a function that maps sequences of units ( normally octets ) into other sequences of the same units.
The blitter's block copying mode takes zero to three data sources in memory, called A, B and C, performs a programmable boolean function on the data sources and writes the result to a destination area, D. Any of these four areas can overlap.
Any real-valued differentiable function, H, on a symplectic manifold can serve as an energy function or Hamiltonian.
Any non-constant algebraic function of a single variable yields a transcendental value when applied to a transcendental argument.
Any finite number of initial terms of the Taylor series of a function is called a Taylor polynomial.
Any decrease in the Gibbs function of a system is the upper limit for any isothermal, isobaric work that can be captured in the surroundings, or it may simply be dissipated, appearing as T times a corresponding increase in the entropy of the system and / or its surrounding.
Any poles on the imaginary axis have real strictly positive residues, and similarly at any zeros on the imaginary axis, the function has a real strictly positive derivative.
Any building requires a certain amount of internal infrastructure to function, which includes such elements like heating / cooling, power and telecommunications, water and wastewater etc.
Any such function is called a Urysohn function for A and B.
Any function can be restricted to a subset of its domain.
Any such K is referred to as a Lipschitz constant for the function ƒ.
Any impairment which limits the physical function of limbs or fine or gross motor ability is a physical disability.
* Any polynomial ( real or complex ) is an analytic function.
Any Taylor series for this function converges not only for x close enough to x < sub > 0 </ sub > ( as in the definition ) but for all values of x ( real or complex ).
* Any analytic function is smooth, that is, infinitely differentiable.
Any real analytic function on some open set on the real line can be extended to a complex analytic function on some open set of the complex plane.
" Any sequence of numbers that may be modeled by a mathematical function can be considered a pattern.
Any such satisficing problem can be formulated as an ( equivalent ) optimization problem using the Indicator function of the satisficing requirements as an objective function.
Any particular Boolean function can be represented by one and only one full disjunctive normal form, one of the two canonical forms.
Any and real
# Any real number can be added to both sides.
# Any real number can be subtracted from both sides.
# Any real number can be multiplied to both sides.
# Any non-zero real number can divide both sides.
Any given individual or organization attempting to implement this model in a real situation would be unable to comply with the three requirements.
* Any Lie group G defines an associated real Lie algebra.
* Any closed interval b of real numbers is Lebesgue measurable, and its Lebesgue measure is the length b − a.
Any real mode operating system ( such as Xenix ) could also be made to run using OS / 2's virtual machine capabilities, subject to certain direct hardware access limitations.
Any Dedekind-complete ordered field is isomorphic to the real numbers.
Any non-negative real number can be represented as a base-φ numeral using only the digits 0 and 1, and avoiding the digit sequence " 11 "-this is called a standard form.
Any destruction by the Occupying Power of real or personal property belonging individually or collectively to private persons, or to the State, or to other public authorities, or to social or cooperative organizations, is prohibited, except where such destruction is rendered absolutely necessary by military operations.
Furthermore, in the opening credits there is a mock disclaimer which reads ( in translation ): “ Any resemblance to real events, to persons dead or living, is not accidental.
Any real advantages to the system were far outweighed by the additional cost of manufacture associated with the device.
Any hope of real progress by the Poles was then lost.
Any entity from the real or virtual world that lends itself to dramatic interpretation may be taken up as a subject.
* Any open subset of a finite-dimensional real, and therefore complex, vector space is a diffeological space.
Any public employee who fell under his power would be replaced by someone in whom he saw real ability instead of a mere useful recipient of a favor.
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