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Continued fractions
Published in John Bird, Mathematics Pocket Book for Engineers and Scientists, 2019
A continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. These approximations to fractions are used to obtain practical ratios for gearwheels or for a dividing head (used to give a required angular displacement).
Non-rigid rank-one infinite measures on the circle
Published in Dynamical Systems, 2023
Hindy Drillick, Alonso Espinosa-Dominguez, Jennifer N. Jones-Baro, James Leng, Yelena Mandelshtam, Cesar E. Silva
Now we address the question of the measure of the set of all irrational numbers such that converges. We use a corollary of the famous theorem due to Gauss and Kuzmin to answer this question. We will first want to compute the probability that for the kth continued fraction coefficient of α equals n. Without loss of generality we will assume for the remainder of this section that . This is because the continued fraction expansion of α is identical the continued fraction expansion of with the exception of the first coefficient. We can consider the standard Lebesgue measure on , then the kth continued fraction coefficient is a well defined random variable.
Critical Depth of Buried Isothermal Circular Pipes
Published in Heat Transfer Engineering, 2018
The heat conduction problem for a single pipe is addressed using Fourier series analysis in bicylindrical coordinates. Using a term-by-term approach, we obtain a tri-diagonal matrix for the Fourier coefficients. An explicit expression for the heat transfer rate was obtained in the form of a continued fraction. Most important, the results reveal that, for any Biot number, there exists a critical depth for a buried pipe where maximum heat transfer rate is achieved. For a periodic array of pipes, the problem is addressed using a finite element analysis to reveal that, besides a critical depth, there also exists a critical Biot number beyond which the critical depth is zero. Furthermore, insulating the pipe reduces the critical depth, and the heat transfer rate does not vary significantly with respect to the depth.
Order Reduction Techniques via Routh Approximation: A Critical Survey
Published in IETE Journal of Research, 2019
Amit Kumar Choudhary, Shyam Krishna Nagar
While blending this survey paper, authors encountered few popular reduction techniques that consider RA, as a tool to compute the required parameters for stable reduced denominator polynomials. Once the parameters are acquired, their individual algorithm procedure proceeds with the derivation of the reduced models. The techniques that take help of RA both in frequency and time domain for both continuous-time and discrete-time includes (a) continued fraction or Cauer forms [78,80,129–144], (b) factor-division [145–149], and (c) differentiation method [150].