Multidimensional regular C-fraction with independent variables corresponding to formal multiple power series

2019 ◽  
Vol 150 (4) ◽  
pp. 1853-1870 ◽  
Author(s):  
R. I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Continued Fractions ◽  
Numerical Experiments ◽  
The Other ◽  
Independent Variables ◽  
Classical Algorithm ◽  
Multiple Power ◽  
The One ◽  
The Given ◽  
Multiple Power Series

AbstractIn the paper the correspondence between a formal multiple power series and a special type of branched continued fractions, the so-called ‘multidimensional regular C-fractions with independent variables’ is analysed providing with an algorithm based upon the classical algorithm and that enables us to compute from the coefficients of the given formal multiple power series, the coefficients of the corresponding multidimensional regular C-fraction with independent variables. A few numerical experiments show, on the one hand, the efficiency of the proposed algorithm and, on the other, the power and feasibility of the method in order to numerically approximate certain multivariable functions from their formal multiple power series.

scholarly journals Approximation of functions of several variables by multidimensional $S$-fractions with independent variables

2021 ◽  
Vol 13 (3) ◽  
pp. 592-607
Author(s):  
R.I. Dmytryshyn ◽  
S.V. Sharyn
Keyword(s):  
Power Series ◽  
Continued Fractions ◽  
Numerical Experiments ◽  
Sufficient Conditions ◽  
Approximation Of Functions ◽  
Independent Variables ◽  
Several Variables ◽  
Multiple Power ◽  
Necessary And Sufficient ◽  
Multiple Power Series

The paper deals with the problem of approximation of functions of several variables by branched continued fractions. We study the correspondence between formal multiple power series and the so-called "multidimensional $S$-fraction with independent variables". As a result, the necessary and sufficient conditions for the expansion of the formal multiple power series into the corresponding multidimensional $S$-fraction with independent variables have been established. Several numerical experiments show the efficiency, power and feasibility of using the branched continued fractions in order to numerically approximate certain functions of several variables from their formal multiple power series.

scholarly journals On the convergence of multidimensional regular C-fractions with independent variables

2018 ◽  
Vol 26 (1) ◽  
pp. 18 ◽  
Author(s):  
R.I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Open Disk ◽  
Efficient Tool ◽  
Independent Variables ◽  
Multidimensional Generalization ◽  
Multiple Power ◽  
Multiple Power Series ◽  
Branched Continued Fraction

In this paper, we investigate the convergence of multidimensional regular С-fractions with independent variables, which are a multidimensional generalization of regular С-fractions. These branched continued fractions are an efficient tool for the approximation of multivariable functions, which are represented by formal multiple power series. We have shown that the intersection of the interior of the parabola and the open disk is the domain of convergence of a multidimensional regular С-fraction with independent variables. And, in addition, we have shown that the interior of the parabola is the domain of convergence of a branched continued fraction, which is reciprocal to the multidimensional regular С-fraction with independent variables.

scholarly journals On the convergence criterion for branched continued fractions with independent variables

2018 ◽  
Vol 9 (2) ◽  
pp. 120-127 ◽  
Author(s):  
R.I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Continued Fractions ◽  
Special Form ◽  
Absolute Convergence ◽  
Convergence Criterion ◽  
Complex Numbers ◽  
Independent Variables ◽  
Nonnegative Coefficients ◽  
Multiple Power ◽  
Multiple Power Series

In this paper, we consider the problem of convergence of an important type of multidimensional generalization of continued fractions, the branched continued fractions with independent variables. These fractions are an efficient apparatus for the approximation of multivariable functions, which are represented by multiple power series. We have established the effective criterion of absolute convergence of branched continued fractions of the special form in the case when the partial numerators are complex numbers and partial denominators are equal to one. This result is a multidimensional analog of the Worpitzky's criterion for continued fractions. We have investigated the polycircular domain of uniform convergence for multidimensional C-fractions with independent variables in the case of nonnegative coefficients of this fraction.

scholarly journals On the convergence of multidimensional S-fractions with independent variables

2020 ◽  
Vol 12 (2) ◽  
pp. 353-359
Author(s):  
O.S. Bodnar ◽  
R.I. Dmytryshyn ◽  
S.V. Sharyn
Keyword(s):  
Power Series ◽  
Analytic Functions ◽  
Continued Fractions ◽  
Special Class ◽  
Convergence Criterion ◽  
Convergence Problem ◽  
Independent Variables ◽  
Several Variables ◽  
Multiple Power ◽  
Multiple Power Series

The paper investigates the convergence problem of a special class of branched continued fractions, i.e. the multidimensional S-fractions with independent variables, consisting of \[\sum_{i_1=1}^N\frac{c_{i(1)}z_{i_1}}{1}{\atop+}\sum_{i_2=1}^{i_1}\frac{c_{i(2)}z_{i_2}}{1}{\atop+} \sum_{i_3=1}^{i_2}\frac{c_{i(3)}z_{i_3}}{1}{\atop+}\cdots,\] which are multidimensional generalizations of S-fractions (Stieltjes fractions). These branched continued fractions are used, in particular, for approximation of the analytic functions of several variables given by multiple power series. For multidimensional S-fractions with independent variables we have established a convergence criterion in the domain \[H=\left\{{\bf{z}}=(z_1,z_2,\ldots,z_N)\in\mathbb{C}^N:\;|\arg(z_k+1)|<\pi,\; 1\le k\le N\right\}\] as well as the estimates of the rate of convergence in the open polydisc \[Q=\left\{{\bf{z}}=(z_1,z_2,\ldots,z_N)\in\mathbb{C}^N:\;|z_k|<1,\;1\le k\le N\right\}\] and in a closure of the domain $Q.$

scholarly journals On the convergence of multidimensional S-fractions with independent variables

2018 ◽  
Vol 10 (1) ◽  
pp. 58-64
Author(s):  
O.S. Bodnar ◽  
R.I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Continued Fraction ◽  
Small Region ◽  
Open Disk ◽  
Convergence Criteria ◽  
Efficient Tool ◽  
Independent Variables ◽  
Multiple Power ◽  
Multiple Power Series ◽  
Branched Continued Fraction

In this paper, we investigate the convergence of multidimensional S-fractions with independent variables, which are a multidimensional generalization of S-fractions. These branched continued fractions are an efficient tool for the approximation of multivariable functions, which are represented by formal multiple power series. For establishing the convergence criteria, we use the convergence continuation theorem to extend the convergence, already known for a small region, to a larger region. As a result, we have shown that the intersection of the interior of the parabola and the open disk is the domain of convergence of a multidimensional S-fraction with independent variables. And, also, we have shown that the interior of the parabola is the domain of convergence of a branched continued fraction, which is reciprocal to the multidimensional S-fraction with independent variables. In addition, we have obtained two new convergence criteria for S-fractions as a consequences from the above mentioned results.

scholarly journals Convergence criterion for branched contіnued fractions of the special form with positive elements

2017 ◽  
Vol 9 (1) ◽  
pp. 13-21 ◽  
Author(s):  
D.I. Bodnar ◽  
I.B. Bilanyk
Keyword(s):  
Continued Fractions ◽  
Special Form ◽  
Sufficient Conditions ◽  
Convergence Criterion ◽  
Sufficient Condition ◽  
Independent Variables ◽  
Positive Elements ◽  
Multiple Power ◽  
Necessary And Sufficient ◽  
Multiple Power Series

In this paper the problem of convergence of the important type of a multidimensional generalization of continued fractions, the branched continued fractions with independent variables, is considered. This fractions are an efficient apparatus for the approximation of multivariable functions, which are represented by multiple power series. When variables are fixed these fractions are called the branched continued fractions of the special form. Their structure is much simpler then the structure of general branched continued fractions. It has given a possibility to establish the necessary and sufficient conditions of convergence of branched continued fractions of the special form with the positive elements. The received result is the multidimensional analog of Seidel's criterion for the continued fractions. The condition of convergence of investigated fractions is the divergence of series, whose elements are continued fractions. Therefore, the sufficient condition of the convergence of this fraction which has been formulated by the divergence of series composed of partial denominators of this fraction, is established. Using the established criterion and Stieltjes-Vitali Theorem the parabolic theorems of branched continued fractions of the special form with complex elements convergence, is investigated. The sufficient conditions gave a possibility to make the condition of convergence of the branched continued fractions of the special form, whose elements lie in parabolic domains.

scholarly journals A multidimensional generalization of the Rutishauser $qd$-algorithm

2016 ◽  
Vol 8 (2) ◽  
pp. 230-238 ◽  
Author(s):  
R.I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Independent Variables ◽  
Multidimensional Generalization ◽  
Multiple Power ◽  
Necessary And Sufficient ◽  
Multiple Power Series

In this paper the regular multidimensional $C$-fraction with independent variables, which is a generalization of regular $C$-fraction, is considered. An algorithm of calculation of the coefficients of the regular multidimensional $C$-fraction with independent variables correspondence to a given formal multiple power series is constructed. Necessary and sufficient conditions of the existence of this algorithm are established. The above mentioned algorithm is a multidimensional generalization of the Rutishauser $qd$-algorithm.

Critical Value-Based Power Options Pricing Problems in Uncertain Financial Markets

2021 ◽  
pp. 2150002
Author(s):  
Guimin Yang ◽  
Yuanguo Zhu
Keyword(s):  
Financial Markets ◽  
Financial Market ◽  
Numerical Experiments ◽  
Critical Value ◽  
The Other ◽  
Options Pricing ◽  
Underlying Asset ◽  
Pricing Problems ◽  
Fractional Differential ◽  
The One

Compared with investing an ordinary options, investing the power options may possibly yield greater returns. On the one hand, the power option is the best choice for those who want to maximize the leverage of the underlying market movements. On the other hand, power options can also prevent the financial market changes caused by the sharp fluctuations of the underlying assets. In this paper, we investigate the power option pricing problem in which the price of the underlying asset follows the Ornstein–Uhlenbeck type of model involving an uncertain fractional differential equation. Based on critical value criterion, the pricing formulas of European power options are derived. Finally, some numerical experiments are performed to illustrate the results.

Influence of Food and Larval Age on the Defensive Chemistry of Satumia pyri

2001 ◽  
Vol 56 (1-2) ◽  
pp. 89-94 ◽  
Author(s):  
Reinhold Deml
Keyword(s):  
Body Fluids ◽  
The Other ◽  
Larval Diet ◽  
Larval Instars ◽  
Defensive Chemistry ◽  
Crataegus Monogyna ◽  
Secondary Compound ◽  
Influence Of Food ◽  
The One ◽  
The Given

Abstract Scolus secretions and hemolymph of caterpillars of Satumia pyri fed with two different foodplants (Crataegus monogyna, Prunus spinosa) were chemically analyzed and their chemical similarities determined. The secondary-compound patterns obtained for the two body fluids showed no significant differences when compared between the two groups of alterna­ tively fed last-instar larvae. Thus, the composition of these fluids of full-grown caterpillars is not influenced by the larval diet. However, younger larvae on P. spinosa revealed a diversity of compounds differing significantly from that of larger caterpillars fed with either C. mono­gyna (both body fluids) or P. spinosa (hemolymph only). This indicates that, on the one hand, the hemolymph composition is adapted to the changing physiological requirements of the given instars whereas, on the other hand, the defensive mixtures remain unaltered in the late larval instars due to a constant spectrum of potential enemies.

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