continued fraction

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con·tin·ued fraction

(kən-tĭn′yo͞od)
n.
A whole number plus a fraction whose numerator is a whole number and whose denominator is a whole number plus a fraction that has a denominator consisting of a whole number plus a fraction, and so on, such as 2 + 1/(3 + 7/(1 + 2/3)).

continued fraction

n
(Mathematics) a number plus a fraction whose denominator contains a number and a fraction whose denominator contains a number and a fraction, and so on
Collins English Dictionary – Complete and Unabridged, 12th Edition 2014 © HarperCollins Publishers 1991, 1994, 1998, 2000, 2003, 2006, 2007, 2009, 2011, 2014

contin′ued frac′tion

n.
a fraction whose denominator contains a fraction whose denominator contains a fraction and so on.
[1860–65]
ThesaurusAntonymsRelated WordsSynonymsLegend:
 Noun 1 continued fraction - a fraction whose numerator is an integer and whose denominator is an integer plus a fraction whose numerator is an integer and whose denominator is an integer plus a fraction and so onfraction - the quotient of two rational numbers
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As the cardinality and number of layers of the continued fractions (1) are finite but not limited, in each point of the space-time occupied by the chain system of harmonic quantum oscillators the scalar F is defined.
The Wiener attack [16] is based on approximations using continued fractions to find the private exponent of RSA-Small-d in polynomial time if d < [N.sup.1/4], where p and q are of the same bit-length.
In [20 22], we have constructed the eigenfunctions and eigenvalues in terms of continued fractions determined by a certain three terms recurrence relation, which can be derived from the expansion of eigenfunctions relative to a basis constructed by suitably twisting the classical Hermite functions.
In section 5, we establish some explicit evaluations for the Ramanujan-Gollnitz-Gordon continued fraction, Ramanujan-Selberg continued fraction and a continued fraction of Eisenstein using the values of [h.sub.2,n].
For example, they proved that 2 and 3 are the only solutions of K(n) = [n.sup.2]; If a, b > 5, then K(a x b) > K(a) x K(b); If a > 5, then for all positive integer n, K([a.sup.n]) > n x K(a); The Fibonacci numbers and the Lucas numbers do not exist in the sequence {K(n)}; Let C be the continued fraction of the sequence {K(n)}, then C is convergent and 2 < C < 3; K([2.sup.n]-1)+1 is a triangular number; The series [[infinity].summation over (n=1)] 1/K(n) is convergent.
Ramanujan gave three more related continued fractions. A generalization (with one more parameter) has been considered by Hirschhorn [9]:
For finite continued fractions F (1), ranges of high distribution density (nodes) arise near reciprocal integers 1, 1/2, 1/3, 1/4, ...
For a good introduction to continued fractions, see http://en.wikipedia.org/wiki/Continued_fraction
The terminology has been chosen because of the resemblance between the polynomials and the (non-circular) continuant polynomials K([x.sub.1], ..., [x.sub.k]) as they arise in the study of continued fractions (see e.g.
continued fractions, Ramanujan formulas, Laplace transform
In the case of harmonic quantum oscillators, the continued fractions F (1) not only define fractal sets of natural angular frequencies wjk, angular accelerations [a.sub.jk] = c x [[omega].sub.jk], oscillation periods [[tau].sub.jk] = 1/[[omega].sub.jk] and wavelengths [[lambda].sub.jk] = c/[[omega].sub.jk] of the chain system, but also fractal sets of energies [E.sub.jk] = [??] x [[omega].sub.jk] and masses [m.sub.jk] = [E.sub.jk]/[c.sup.2] which correspond with the eigenstates of the system.
He discusses continued fractions; the geometry of complex numbers, quaternions, and spins; and Euler groups and the arithmetic of geometric progressions.

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