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Showing 1 to 20 of 26 for “"continued fraction"”.
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Generalizations of Certain Results on Continued Fraction
… we study generalizations of the Rogers-Ramanujan continued fraction. The Rogers-Ramanujan continued fraction arises from a three-term q-difference equation. We consider (m + 1)-term q-difference equations and also a generalization of the continued fraction algorithm called a G-continued fraction. …
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Approximation of transfer functions by continued fraction expansion.
Massachusetts Institute of Technology. Dept. of Electrical Engineering. Thesis. 1972. M.S.
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The Rogers -Ramanujan Continued Fraction and a Certain Quotient of ETA Functions Found in Ramanujan's Lost Notebook
Finally, a new proof of Winquist's identity which is essential for an elementary proof for Ramanujan's famous partition identity modulo 11, p(11n + 6) ≡ 0 (mod 11) is provided in the last part of the thesis.
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Development of high-order doubly asymptotic open boundaries for wave propagation in unbounded domains by extending the scaled boundary finite element method
… in dynamic stiffness, the doubly asymptotic continued fraction solution for dynamic stiffness matrices is developed in the frequency domain using the technique of continued fraction. Factor coefficients or matrices are introduced in the continued fraction solution to improve the stability of …
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Applications of continued fractions in one and more variables
Elementary properties of continued fractions are derived from sets of three-term recurrence relations and approximation methods are developed from this simple approach. First, a well-known method for numerical inversion of Laplace transforms is modified in two different ways to obtain exponential …
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Modular equations and Ramanujan's cubic and quartic theories of theta functions
… related to the Ramanujan-G\""{o}llnitz-Gordon continued fraction that are similar to those for the famous Rogers-Ramanujan continued fraction. We give a new proof of the 8-dissection of the Ramanujan-G\""{o}llnitz-Gordon continued fraction and also show that the signs of the coefficients of …
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Hypergeometric functions, continued fractions for products of gamma functions, and q-analogues
Some of the most interesting of Ramanujan's continued fraction identities are those involving ratios of Gamma functions in Chapter 12 of his second notebook. This thesis develops a method for deriving such identities, using hypergeometric functions as the main tool. We begin by deriving a continued …
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Contributions to the Theory of Q-Series and Mock Theta Functions
… representations for the famous Rogers-Ramanujan continued fraction and the Ramanujan-Gollnitz-Gordon continued fraction. Our theorems greatly generalize the theorems of Andrews, Hirschhorn, and Ramanathan, and also have an application to another continued fraction of Ramanujan. An interesting …
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Left-orderability of Dehn surgeries on knot complements
… that the two-bridge knot that corresponds to the continued fraction [1,1,2,2,2j] for j >= 1 and the (-3,3,2j+1)-pretzel knot admit an interval of left orderable Dehn surgeries. These two families of knots gives some positive evidence for a question of Xinghua Gao.
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The Asymptotic Behavior of Birkhoff- and Lacunary Sums
… theory, specifically classical results from continued fraction theory are utilized.
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On the Convergence and Divergence of Q-Continued Fractions on and Off the Unit Circle
… implies general convergence. We show that all continued fractions in a certain class, which includes the Rogers-Ramanujan continued fractions and the three ""Ramanujan-Selberg"" continued fractions, diverge in the general sense at an uncountable set of points on the unit circle. We also show …
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Some important continued fractions of Ramanujan and Selberg
… which arise in Ramanujan and Selberg's work on q-continued fractions. From these solutions, we derive criteria for the convergence of three Ramanujan-Selberg continued fractions when q is a primitive m-th root of unity. Moreover, when the continued fractions converge, we determine their values …
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Approximations by means of continued fractions
… when extended, leads to our modern notion of continued fractions. The classical theory of continued fractions began during the Renaissance when Arabic numerals and the modern fractional notation had become common. It was studied and extended until about the end of the nineteenth century. This …
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Topics in Combinatorial Number Theory
… is irreducible; Chapter VI compares simple continued fraction convergents for SQRT.(N) with Newton approximations to SQRT.(N); and Chapter VII obtains exact formulas for a certain class of ballot problems. An introduction is included which gives preliminary discussions on various aspects of …
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Chebyshev-like polynomials, conic distribution of roots, and continued fractions
… of Chebyshev polynomials. Here we use continued fractions to give estimates for the roots that do not lie in the interval $(-1,1)$. We then show the connection between polynomials with roots on concentric circles to polynomials with roots on ellipses. In particular, we construct a …
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Examples and Applications of Infinite Iterated Function Systems
… iterated function systems derived from complex continued fraction expansions with restricted entries. Each system is obtained from an infinite number of contractions. We show that under certain conditions the limit sets of such systems possess zero Hausdorff measure and positive finite packing …
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Stability theory of differential equations
… equation is first considered in terms of a continued fraction expansion. Necessary and sufficient conditions are given for the characteristic equation to be stable. The stability of the equation is then determined by means of a determinant sequence, which was the manner originally presented …
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Quantum Metrics on Approximately Finite-Dimensional Algebras
… the Effros-Shen AF algebras associated with continued fraction expansions of irrationals, and the Cantor space, on which our construction recovers traditional ultrametrics. We also exhibit several compact classes of AF algebras for the quantum propinquity and show continuity of our family of …
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Geometric mapping theory of the Heisenberg group, sub-Riemannian manifolds, and hyperbolic spaces
… to new notions of base-$b$ expansions and continued fractions. As a metric space, $\Heis^n$ serves as an infinitesimal model (metric tangent space) of some sub-Riemannian manifolds and allows one to study derivatives of mappings between such spaces. As a subgroup of the isometry group of …
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