Transcendental Function Research Articles

Rotation Domains and Stable Baker Omitted Value

A Baker omitted value, in short bov of a transcendental meromorphic function f is an omitted value such that there is a disk D centered at the bov for which each component of the boundary of \(f^{-1}(D)\) is bounded. Assuming all the iterates \(f^n\) to be analytic in a neighborhood of its bov, this article proves a number of results on the rotation domains of the function. The number of p-periodic Herman rings is shown to be finite for each \(p \ge 1\). It is also proved that every Julia component intersects the boundaries of at most finitely many Herman rings. Further, if the bov is the only limit point of the critical values then it is shown that f has infinitely many repelling fixed points. If a repelling periodic point of period p is on the boundary of a p-periodic rotation domain then the periodic point is shown to be on the boundary of infinitely many Fatou components. As a corollary we have shown that if D is a p-periodic rotation domain of f and \(f^p\) is univalent in a neighbourhood of the boundary \( \partial D \) of D then f has no repelling p-periodic point on \( \partial D \). Under additional assumptions on the critical points, a sufficient condition is found for a Julia component to be a singleton. As a consequence, it is proved that if the boundary of a wandering domain W accumulates at some point of the plane under the iteration of f then each limit of \(f^n\) on W is either a parabolic periodic point or in the \(\omega \)-limit set of recurrent critical points. Using the same ideas, the boundaries of rotation domains are shown to be in the \(\omega \)-limit set of recurrent critical points.

FDI inflow, ICT and economic performance of India: an empirical investigation

PurposeThe aim of this study is to examine the impact of foreign direct investment (FDI) inflow and information and communication technology (ICT) on the economic performance of India by analysing annual data from 1991 to 2019.Design/methodology/approachThis study has used data collected from secondary sources. The variables considered for the analysis are based on the review of theoretical and empirical literature. Moreover, apart from the quantitative variables, two qualitative variables have also been considered through the use of dummy variables. The Cobb–Douglas, Transcendental logarithmic and Simultaneous equations models have been used for the study.FindingsThe result reveals that the partial elasticities of the per-capita gross domestic product (PCGDP) of India with respect to FDI, mobile density (MD) and internet density (ID) are 0.074, 0.024 and 0.036, respectively. The positive and significant coefficient of the interaction among FDI, MD and ID in the estimation of the transcendental logarithmic function indicates the importance of ICT infrastructure in extracting the best out of FDI (the coefficient is 0.011 for the model without any control variables and it is 0.005 with control variables).Originality/valueThe findings of this study are more reliable as the latest available data have been analysed through the appropriate econometric models. This study will be useful for the policymakers in the formulation of policies with regard to foreign capital and digitalisation.

Low‐frequency variable charge solitary waves in a collisionless dusty plasma with a Cairns‐Gurevich ion velocity distribution

AbstractIn this paper, the problem of large amplitude dust acoustic (DA) solitons has been addressed in a charge varying dusty plasma with ions following a Cairns‐Gurevich distribution. Based on the orbit motion limited approach, the correct Cairns‐Gurevich ion charging current is presented for the first time. The expression relating the variable dust charge to the plasma potential is given in terms of the Lambert function and we take advantage of this transcendental function to, carefully, analyse DA solitons in a charge varying dusty plasma with trapped nonthermal ions. Our results show that the spatial patterns of the variable charge solitary wave are significantly changed due to the presence of ion population modelled by the Cairns‐Gurevich distribution. An addition of a small concentration of trapped nonthermal ions makes the solitary structure less spiky, grows the net negative charge residing on the dust grain surface, and contributes to the electron depletion. Finally, our investigation is extended to highlight the effect of the grain dust charge variation. We have shown that under certain conditions, the impact of dust charge fluctuation may furnish an alternate physical mechanism rasing anomalous dissipation, which becomes more strong and may predominate over the dispersion as the nonthermal character of ions following the Cairns‐Gurevich distribution increases.

On Fatou Sets Containing Baker Omitted Value

An omitted value of a transcendental meromorphic function f is called a Baker omitted value, in short bov if there is a disk D centered at the bov such that each component of the boundary of \(f^{-1}(D)\) is bounded. Assuming that the bov is in the Fatou set of f, this article investigates the dynamics of the function. Firstly, the connectivity of all the Fatou components are determined. If U is the Fatou component containing the bov then it is proved that a Fatou component \(U'\) is infinitely connected if and only if it lands on U, i.e. \(f^{k}(U') \subset U\) for some \(k \ge 1\). Every other Fatou component is either simply connected or lands on a Herman ring. Further, assuming that the number of critical points in the Fatou set whose forward orbits do not intersect U is finite, we have shown that the connectivity of each Fatou component belongs to a finite set. This set is independent of the Fatou components. It is proved that the Fatou component containing the bov is completely invariant whenever it is forward invariant. Further, if the invariant Fatou component is an attracting domain and compactly contains all the critical values of the function then the Julia set is totally disconnected. Baker domains are shown to be non-existent whenever the bov is in the Fatou set. It is also proved that, if there is a 2-periodic Baker domain (these are not ruled out when the bov is in the Julia set), or a 2-periodic attracting or parabolic domain containing the bov then the function has no Herman ring. Some examples exhibiting different possibilities for the Fatou set are discussed. This includes the first example of a meromorphic function with an omitted value which has two infinitely connected Fatou components.

The Periodicity on a Transcendental Entire Function with its Differential Polynomials

The Periodicity on a Transcendental Entire Function with its Differential Polynomials

Iterating the Minimum Modulus: Functions of Order Half, Minimal Type

For a transcendental entire function f, the property that there exists r>0 such that m^n(r)rightarrow infty as nrightarrow infty , where m(r)=min {|f(z)|:|z|=r}, is related to conjectures of Eremenko and of Baker, for both of which order 1/2 minimal type is a significant rate of growth. We show that this property holds for functions of order 1/2 minimal type if the maximum modulus of f has sufficiently regular growth and we give examples to show the sharpness of our results by using a recent generalisation of Kjellberg’s method of constructing entire functions of small growth, which allows rather precise control of m(r).

Output and substitution elasticity estimates between renewable and non-renewable energy: implications for economic growth and sustainability in India.

The energy profile of India is dominated by fossil fuels, which create concerns over resource and environmental sustainability as fossil fuels are non-renewable and high carbon emitting. This scenario has necessitated the call for more renewables to replace fossil fuels to address resource and environmental sustainability concerns. This study, therefore, investigates the possibility of switching the fossil fuels of oil, coal, and natural gas for renewable energy in India. Using annual Indian data spanning more than four decades, a transcendental logarithmic production function based on a second-order Taylor Series approximation is estimated with the ridge regression technique. To achieve robustness, two equations with gross domestic product and adjusted net savings as regressands are estimated to proxy economic growth and sustainable development, respectively. The empirical results show substantial substitution possibilities between the fuels for both gross domestic product and adjusted net savings equations. The empirical findings show that India has the capacity to satisfy its energy needs through renewables to pursue not only economic growth but sustainable development. To actualize this potential, the Indian government should promote investment in renewables as this also promotes economic growth and development.

Unicity on Entire Function Concerning Its Differential-Difference Operators

In this paper, we study the uniqueness of entire function and its differential-difference operators. We prove the following result: let f be a transcendental entire function of finite order, let \(\eta \) be a non-zero complex number, \(n\ge 1, k\ge 0\) two integers and let a and b be two distinct finite complex numbers. If f and \((\Delta _{\eta }^{n}f)^{(k)}\) share a CM and share b IM, then \(f\equiv (\Delta _{\eta }^{n}f)^{(k)}\).

Combinatorics of criniferous entire maps with escaping critical values

A transcendental entire function is called criniferous if every point in its escaping set can eventually be connected to infinity by a curve of escaping points. Many transcendental entire functions with bounded singular set have this property, and this class has recently attracted much attention in complex dynamics. In the presence of escaping critical values, these curves break or split at critical points. In this paper, we develop combinatorial tools that allow us to provide a complete description of the escaping set of any criniferous function without asymptotic values on its Julia set. In particular, our description precisely reflects the splitting phenomenon. This combinatorial structure provides the foundation for further study of this class of functions. For example, we use these results in another paper to give the first full description of the topological dynamics of a class of transcendental entire maps with unbounded postsingular set.

Analysis of the properties of a derived one-step numerical method of a transcendental function

The paper presents a new numerical method for the solution of Initial Value Problems (IVPs) of first order Ordinary Differential Equations (ODEs) via a transcendental basis function. FORTRAN 95 programing language was used for the implementation of the derived numerical scheme. The analysis of the properties of the newly derived method was investigated which includes the stability, accuracy and consistency through which the suitability of it was established. This scheme avoids the need for higher derivative of differential equation. This scheme was tested on some illustrative examples and the results compared favourably with exact solutions. We conclude that the derived numerical method is consistent, convergent, quite stable and more accurate for application to numerical problems and can be widely used in solving first order IVPs in ODEs.

On relative deficiencies of difference polynomials from the view point of integrated moduli of logarithmic derivative

Let be a transcendental entire function defined in the open complex plane ℂ. A difference-monomial generated by . Now for the sake of definiteness let us take a difference-polynomial generated by . In this paper we compare the Valiron defect with the relative Nevanlinna defect of a particular type of differential-difference polynomial generated by a transcendental entire function with respect to integrated moduli of logarithmic derivative. Some examples are provided in order to justify the results obtained.

The in-plane mechanical properties of highly compressible and stretchable 2D lattices

Highly compressible and stretchable lattice materials are perfectly suitable to be exploited in a range of cutting edge engineering applications such as low band-gap acoustic metamaterials, vibration absorbers, soft robotics, stretchable electronics and stent devices. Physics-based understanding and efficient computational methods are of paramount importance for the analysis and design of such cellular metamaterials. This paper develops the analytical framework to understand the nonlinear mechanics of hexagonal lattices subject to in-plane compressive and tensile stresses. Nonlinear equivalent elastic moduli and Poisson’s ratios of the stressed lattice are expressed through the coefficients of the stiffness matrices of the constitutive beam elements. The stiffness coefficients, in turn, are derived from the transcendental displacement function which is the exact solution of the corresponding governing ordinary differential equation with appropriate boundary conditions. The closed-form analytical expressions of the equivalent elastic properties of the lattice are expressed in terms of trigonometric functions for the case of compressive stress and hyperbolic functions for the case of tensile stress. The general expressions are then used to investigate three special cases of wide interest, namely, auxetic hexagonal lattices, rhombus-shaped lattices and rectangular lattices. Analytical expressions are validated using independent nonlinear finite element simulation results. Numerical results are displayed for applied compressions and tensions in both directions separately and together. The equivalent elastic moduli show a softening effect under compression and a stiffening effect under tension. The Poisson’s ratios are not significantly affected by the applied stresses. The proposed analytical approach and the new closed-form expressions provide a computationally efficient and physically intuitive framework for the analysis and parametric design of lattice materials under external stresses.

The Periodicity of Entire Functions with Finite Order

This paper is concerned with the periodicity of entire functions with finite growth order, and some sufficient conditions are given. Let f is a transcendental entire function with finite growth order, zero is a Picard exceptional value of f , and a given differential monomial Q f of f is periodic, then f is also periodic. We are also interested in finding the following: let f is a transcendental entire function with finite growth order, d is a Picard exceptional value of f and f z Δ c n f z is a periodic function, then f is also a periodic function. These results extend Yang’s conjecture.

A Computational Approach to Solve a System of Transcendental Equations with Multi-Functions and Multi-Variables

A system of transcendental equations (SoTE) is a set of simultaneous equations containing at least a transcendental function. Solutions involving transcendental equations are often problematic, particularly in the form of a system of equations. This challenge has limited the number of equations, with inter-related multi-functions and multi-variables, often included in the mathematical modelling of physical systems during problem formulation. Here, we presented detailed steps for using a code-based modelling approach for solving SoTEs that may be encountered in science and engineering problems. A SoTE comprising six functions, including Sine-Gordon wave functions, was used to illustrate the steps. Parametric studies were performed to visualize how a change in the variables affected the superposition of the waves as the independent variable varies from x1 = 1:0.0005:100 to x1 = 1:5:100. The application of the proposed approach in modelling and simulation of photovoltaic and thermophotovoltaic systems were also highlighted. Overall, solutions to SoTEs present new opportunities for including more functions and variables in numerical models of systems, which will ultimately lead to a more robust representation of physical systems.

Глибинні основи щастя. Засадничі основи різних типів розуміння щастя

The choice of the article’s topic is conditioned by the necessity to develop happiness management (such as knowledge management, time management, etc.). With considerable attention to the definition of “happiness”, the analysis of recent publications reveals that it is crucial to understand the deep foundations of happiness, create a typology, reveal the basic principles of different types of understanding of happiness, which was chosen as the topic research. Theoretical approaches to the definition of “happiness” are chosen sociology of imagination of G. Durand, the theory of archetypes of C. Jung, and the theory of images and dreams of G. Bachelard. In the context of this system of views, the idea of happiness is the result of a free play of the imagination, which, while being on the path from past to future, is transformed, revealed, comes accurate as a result of previous collective and individual intermediate ideas, and is enriched and concretised by individual people, social groups, individuals. The mythos of happiness across nations and people differs in the way, method, and tools of individuation, the discovery of the Self. The anthropological tract of happiness has an end to its existence. It is determined by the cessation of existence, the life of nations and individuals. The desire to experience pleasure (according to Freud), the desire to rise (according to Durand) are reflexive, which determines the physiological basis of happiness. Representations of happiness determine priorities, coordinate the direction of thinking, actions, reactions to external circumstances, and choose ways to achieve happiness. This is done through the transcendental function (according to Jung) – a psychological function that arises from the connection of the content of the unconscious with the content of consciousness. Achieving happiness allows one to strengthen the subjectivity and reveal their uniqueness, which allows them to identify typological features (archetypes) of behavioural reactions of people based on individual and group ideas about happiness. The authors reveal the basic foundations of such archetypes of happiness as hedonism, eudemonia, “rat racing”, nihilism, subjective well-being.

Use of production functions in assessing the profitability of shares of insurance companies

In this study the production functions (Cobb-Douglas, Zener-Rivanker, and the transcendental production function) have been used to assess the profitability of insurance companies, by reformulating these nonlinear functions based on the introduction of a set of variables that contribute to increase the explanatory capacity of the model. Then the best production function commensurate with the nature of the variable representing the profitability of insurance companies was chosen, to use it to assess the efficiency of their profitability versus the use of different factors of production and thus the possibility of using it in forecasting. It was found that the proposed model of the production function "Zener-Rivanker" is the best production functions representing the profitability of the Tawuniya and Bupa Insurance Companies. The proposed model of the Cobb-Douglas production function is suitable for the results of both Enaya and Sanad Cooperative Insurance Companies. The explanatory capacity of the production functions was also increased when the proposed variables were added (net subscribed premiums-net claims incurred).

Relative Growth of Series in Systems of Functions and Laplace—Stieltjes-Type Integrals

For a regularly converging-in-C series A(z)=∑n=1∞anf(λnz), where f is an entire transcendental function, the asymptotic behavior of the function Mf−1(MA(r)), where Mf(r)=max{|f(z)|:|z|=r}, is investigated. It is proven that, under certain conditions on the functions f, α, and the coefficients an, the equality limr→+∞α(Mf−1(MA(r)))α(r)=1 is correct. A similar result is obtained for the Laplace–Stiltjes-type integral I(r)=∫0∞a(x)f(rx)dF(x). Unresolved problems are formulated.

Escaping sets are not sigma-compact

Let $f$ be a transcendental entire function. The escaping set $I(f)$ consists of those points that tend to infinity under iteration of $f$. We show that $I(f)$ is not $\sigma$-compact, resolving a question of Rippon from 2009.

A note on the value distribution of $\phi f^2 f^{(k)}-1$

In this paper, we study the value distribution of the differential polynomial $\varphi f^2f^{(k)}-1$, where $f(z)$ is a transcendental meromorphic function, $\varphi (z)\;(\not\equiv 0)$ is a small function of $f(z)$ and $k\;(\geq 2)$ is a positive integer. We obtain an inequality concerning the Nevanlinna Characteristic function $T(r,f)$ estimated by reduced counting function only. Our result extends the result due to J.F. Xu and H.X. Yi [J. Math. Inequal., 10 (2016), 971-976].

Robust stochastic frontier using Cauchy distribution for noise component to measure efficiency of rice farming in East Java

Several studies have been conducted to estimate efficiency of rice farming in Indonesia. Those studies generally apply standard stochastic frontier (SF) model which assume that noise component has normal distribution. However, the normality assumption of noise is inadequate if the outliers are present. This paper employs Cauchy distribution for noise component to produce a more robust SF model. There are two variants of models that are employed: Cauchy-Half Normal and Cauchy-Exponential models. These models are applied to measure technical efficiency of rice farming in East Java. The findings presented in this study are based primarily on data from Cost Structure of Paddy Cultivation Household Survey 2017 (SOUT2017-SPD) that is compiled by Statistics Indonesia. Output and input variables are linked using transcendental logarithmic production function, while the parameters are estimated using maximum simulated likelihood. The results showed that Cauchy-Half Normal and Cauchy-Exponential models effectively reduce the effect of outliers. The performance of model is improved after the outliers are handled. Cauchy-Half Normal and Cauchy-Exponential models revise the technical efficiency scores of several rice farming units.