<title>MIE Scattering From Continued Fractions</title>

The purpose of this paper is to demonstrate that representations of numbers other than positional notation may lead to practical hardware realizations for digital calculation of classes of algorithms. This paper describes current research in the use of continued fractions. Although practicality has not been demonstrated, theoretical results are promising.

The Indian govt. celebrated 125𝑡ℎ anniversary of the great Mathematician of Indian soil Srinivasa Ramanujan on 22 December in the year 2012. Without any formal education and extreme poverty conditions, he emerged as one of great mathematician of India. His mathematical ideas transformed and reshaped 20th century mathematics and their ideas are inspiration for 21𝑠𝑡 century mathematicians. Srinivasa Ramanujan made substantial contributions to the analytical theory of numbers and worked on ‘elliptic functions’, ‘continued fractions’, and ‘infinite series’. He was a great Mathematician, who became world famous at the tender age of twenty-six. He was born into a family that had a humble background and that had no distinguished professional achievement, yet his mathematical ideas transformed and reshaped century mathematics and continues to inspire modern day mathematicians. Considered to be a mathematical genius, Srinivasa Ramanujan, was regarded at par with the likes of Leonhard Euler and Carl Jacobi. In spite of having almost no formal training in mathematics, Ramanujan’s knowledge of mathematics was astonishing. Even though he had no knowledge of the modern developments in the subject, he effortlessly worked out the Riemann series, the elliptic integrals, hypergeometric series, and the functional equations of the zeta function. The purpose of this paper is to introduce some of the contributions of Srinivasa Ramanujan in the field of mathematics.

In this paper we first give the value of a periodic continued fraction which was recorded incorrectly by Ramanujan on page 341 of his lost notebook. Next, we describe several pairs of equivalent continued fractions in which one is the odd part of the other. One of the results is for the Rogers-Ramanujan continued fraction which was recently proved by Berndt and Yee. Finally, using the Bauer-Muir transformation we prove the equivalence of two continued fractions. One was recorded on page 44 in Ramanujan’s lost notebook, and the other is found in the unorganized pages at the end of Ramanujan’s second notebook.

The admittances

Cohen [1] and Des Raj [2] have shown that in estimating the parameters of truncated type III populations, it is necessary to calculate for several values of $x$ the Mills ratio of the ordinate of the standardized type III curve at $x$ to the area under the curve from $x$ to $\infty$. Des Raj [3] has also noted that for large values of $x$ the existing tables of Salvosa [4] are inadequate for this purpose and he has found lower and upper bounds for the ratio. The object of this note is to improve these bounds, by obtaining monotonic sequences of lower and upper bounds through the use of continued fractions.

Let θ > 0 and α ≠ 0 be real numbers, and let θ be irrational. Khintchine has shown, by the use of continued fractions, that there is an infinite number of pairs of positive integers (p, q) which satisfy the inequalityfor any given K > 5−½; and, more recently, Jogin has shown the same is still true with K = 5−½. The condition that p and q shall be positive is, of course, essential, as otherwise there is the classical result K = ¼ due to Minkowski.

4 - Continued Fractions and Indeterminate Equations

Introduction Numerous elementary and higher transcendental functions of complex analysis possess representations in terms of continued fractions of the Stieltjes type. (An abundance of examples are given in the textbooks by WALL [1948] and PERRON [t957]; for some recent examples and applications, see MAEHLY [1960], CONTE and FRIED [t96t], MILLER, MOLMUD and MEECHAM [t962], BLANCH [1964], PATRY and KELLER [1964], W¥1~N [1964].) Similar representations also exist for the remainder terms of the asymptotic expansions of functions defined by certain Laplacian integrals (HESRICI [t963, 1965]). Such representations are of considerable computational value, both because of the ease with which a continued fraction can be evaluated recursively, and because of the fact that continued fraction representations frequently are convergent (and computationally efficient) in wider regions of the complex plane than, say, a Taylor series. In actual computation it is necessary to approximate the value of a continued fraction by some approximant of finite order. This paper makes an a t tempt to s tudy the error committed in this approximation. Both a posteriori bounds depending on the knowledge of a finite number of approximants -and a priori bounds depending only on the elements of the continued fraction or on the coefficients of its corresponding power series are presented. A solution of the problem of finding a posteriori bounds which in a sense is best possible is given in section 2. Several different a priori bounds are given in section 3. Although these bounds are not shown to be best possible, they are shown to have the correct order of magnitude in some important special cases. The tack of a generally accepted terminology and even of clear definitions in the analytic theory of continued fractions made it necessary to recollect, in a unified notation, some basic facts of the theory in section t.

Thermoluminescence glow curve deconvolution functions by continued fractions for different orders of kinetics

Continued fractions (CFs) enable straightforward representation of elementary functions and rational approximations. We improve the positional algebraic algorithm, which computes homographic functions such as y equals ax+b/cx+d, given redundant continued fractions x,y, and integers a,b,c,d. The improved algorithm for the linear fractional transformation produces exact results, given regular continued fraction input. In case the input is in redundant continued fraction form, our improved linear algorithm increases the percentage of exact results with 12-bit state registers from 78% to 98%. The maximal error of non-exact results is improved from approximately 1 to 2<SUP>-8</SUP>. Indeed, by detecting a small number of cases, we can add a final correction step to improve the guaranteed accuracy of non-exact results. We refer to the fact that a few results may not be exact as 'Semi- Exact' arithmetic. We detail the adjustments to the positional algebraic algorithm concerning register overflow, the virtual singularities that occur during the computation, and the errors due to non-regular, redundant CF inputs.

Let y = y(x) be a function defined by a continued fraction. A lower bound for |Λ| = |β 1 y 1 + β 2 y 2 + α| is given, where y 1 = y(x 1), y 2 = y(x 2), x 1 and x 2 are positive integers, α, β 1 and β 2 are algebraic irrational numbers.

O přibližných hodnotách řetězce se stálým jmenovatelem

We applied special graphical technique to investigate the influence of side/end groups of macromolecules upon long range electron transfer. We show that in case of acyclic side/end groups the Green function may be cast in the form of continued fraction, similarly to the result of electron transfer in random media. The structure of continued fraction correlates exactly to the side/end graph's structure. The rules of graphical construction of Green functions (which are solutions of Schwinger–Dyson equations) are presented in concise form.

Electronic densities of states of tight-binding s-bands for simple cubic and body-centred cubic lattices were determined using the modified-moments method. The weight functions close to the density-of-states functions for SC and BCC lattices were found. These functions define the orthogonal polynomials belonging to the class of polynomials considered by Bernstein and Szego. The analytical expressions for modified moments were derived and the densities of states in the form of continued fractions which reproduce 76 moments for the SC and 78 moments for the BCC lattice were obtained and compared with exact results. The convergence of the modified-moments method as a function of the level at which the continued fraction is terminated in an approximate way has been examined.

Starting with the Liouville equation and with use of the combined projection technique presented earlier [J. Y. Sug, C. H. Choi, and S. D. Choi, Nuovo Cimento B 109, 177 (1994)], we obtain the ensemble average of current of many-electron systems in the presence of oscillatory electric fields. For quite weak fields the dynamic conductivity is given in a continued fraction form. The present form contains the first- and second-order correction parts, while the Mori type contains only the first-order part.