On the Padé table for e x and the simple continued fractions for e and e L/M

A recently observed connection between some Padé approximants for the exponential series and the convergents of the simple continued fraction for e is established, leading to an alternative proof of the latter. Similar results for the simple continued fraction e ²,e ^1/M and e ^2/M , when M is a natural number greater than one, are derived.      

Simultaneous hyperaccumulation of arsenic and antimony in Cretan brake fern: Evidence of plant uptake and subcellular distributions

Arsenic (As) and antimony (Sb) show similar chemical properties and often present together in sulfide ores. Currently, phenomenon of co-contamination of As and Sb at some sites of the world has been increasingly emerged. The present study was conducted to explore the potential of Pteris cretica L. (Cretan brake fern), an arsenic (As) hyperaccumulator, to simultaneously accumulate As and Sb under hydroponic conditions. Arsenic was imposed at medium and high levels of 5 mg L^− 1 and 20 mg L^− 1, while Sb was imposed either single or co-presence with As at medium and high levels of 10 mg L^− 1 and 20 mg L^− 1, with no As and Sb addition as the control. The single and interactive effects of As and Sb on their uptake and subcellular distributions were analyzed. Cretan brake fern could accumulate high concentrations of As and Sb, with the highest concentrations of As and Sb been recorded as 1677.2 mg kg^− 1 and 1516.5 mg kg^− 1 in the fronds, respectively. Arsenic and Sb were found mainly in cytosol, while less in cell wall and cytoplasmic organelles. Sb uptake by Cretan brake fern was enhanced with increasing As levels, which was accompanied with an increase of Sb but a decrease of As in cytosol fractions. Arsenic uptake was slightly enhanced whereas suppressed when Sb was co-present in a medium and high level, respectively; however, in both conditions, As was found to be decreased in cytosol of the above ground parts as fronds and stems of Cretan brake fern. The results demonstrate Cretan brake fern can simultaneously hyperaccumulate As and Sb, thus is valued in phytoremediation of As and Sb co-contamination.     

Flat Power Series over a Finite Field

We define and describe a class of algebraic continued fractions for power series over a finite field. These continued fraction expansions, for which all the partial quotients are polynomials of degree one, have a regular pattern induced by the Frobenius homomorphism.This is an extension, in the case of positive characteristic, of purely periodic expansions corresponding to quadratic power series.     

An M/M/1 Driven Fluid Queue – Continued Fraction Approach

This paper presents complete solutions of the stationary distributions of buffer occupancy and buffer content of a fluid queue driven by an M/M/1 queue. We assume a general boundary condition when compared to the model discussed in Virtamo and Norros [Queueing Systems 16 (1994) 373–386] and Adan and Resing [Queueing Systems 22 (1996) 171–174]. We achieve the required solutions by transforming the underlying system of differential equations using Laplace transforms to a system of difference equations leading to a continued fraction. This continued fraction helps us to find complete solutions. We also obtain the buffer content distribution for this fluid model using the method of Sericola and Tuffin [Queueing Systems 31 (1999) 253–264].     

Explicit continued fractions with expected partial quotient growth

For let be the continued fraction expansion of . Write

We construct some numbers 's with

     

Solution of the Semi-Infinite Toda Lattice for Unbounded Sequences

The semi-infinite Toda lattice is the system of differential equations d _n (t)/dt = _n (t)(b _n+1(t) – b _n (t)), db _n (t)/dt = 2( _n ²(t) – _n–1 ²(t)), n = 1, 2, ..., t > 0. The solution of this system (if it exists) is a pair of real sequences _n (t), b _n (t) which satisfy the conditions _n (0) = _n ,, b _n (0) = b _n , where _n > 0 and b _n are given sequences of real numbers. It is well known that the system has a unique solution provided that both sequences _n and b _n are bounded. When at least one of the known sequences _n and b _n is unbounded, many difficulties arise and, to the best of our knowledge, there are few results concerning the solution of the system. In this letter we find a class of unbounded sequences _n and b _n such that the system has a unique solution. The results are illustrated with a typical example where the sequences _i (t), b _i (t), i = 1, 2, ... can be exactly determined. The connection of the Toda lattice with the semi-infinite differential-difference equation d²/dt ² log h _n = h _n+1 + h _n–1 – 2h _n , n = 1, 2, ... is also discussed and the above results are translated to analogous results for the last equation.     

Partial fractions expansion: a review of computational methodology and efficiency

Nine methods for expressing a proper rational function in terms of partial fractions are presented for the case where the denominator polynomial has been reduced to linear factors. Only those methods which are amenable to computation algorithms are considered. To the extent possible, Newton's divided difference formula is used to provide a uniform derivational tool. Each method is illustrated numerically. The efficiency of the methods are compared on the basis of the number of multiplications and divisions required in the computation.     

On Tori Triangulations Associated with Two-Dimensional Continued Fractions of Cubic Irrationalities

The notion of equivalence of multidimensional continued fractions is introduced. We consider some properties and state some conjectures related to the structure of the family of equivalence classes of two-dimensional periodic continued fractions. Our approach to the study of the family of equivalence classes of two-dimensional periodic continued fractions leads to revealing special subfamilies of continued fractions for which the triangulations of the torus (i.e., the combinatorics of their fundamental domains) are subjected to clear rules. Some of these subfamilies are studied in detail; the way to construct other similar subfamilies is indicated.     

The distribution of special subsets of the Farey sequence

We will examine the subset of Farey fractions of order Q consisting of those fractions whose denominators are not divisible by a fixed prime p. In particular, we will provide an asymptotic result on the distribution of H-tuples of consecutive fractions in , as Q→∞.