Exponential Polynomial Research Articles

Exponential Function Computation Based on DNA Strand Displacement Circuits

Due to its high programmability and storage, DNA circuits have been widely used in biological computing. In this paper, the addition, subtraction, multiplication, division, n-order and 1/n-order gates are built through DNA strand displacement reactions. The chemical reaction networks of the exponential function are established by using the six DNA analog computation gates. The integrated DNA strand displacement circuits are built through the chemical reaction networks. The exponential function polynomials can be computed through the integrated DNA strand displacement circuits. Finally, through visual DSD software verification, this design can realise the computation of exponential function polynomials, which provides a reference for solving exponential function equations and neural network computations in the future.

A Case Study in Desertified Area: Soybean Growth Responses to Soil Structure and Biochar Addition Integrating Ridge Regression Models

Desertified land covers one-fourth of the world’s total land area. Meeting the high food demands in areas affected by desertification is a major problem. This case study provided fundamental information to demonstrate the potential for utilizing the desertified land. The soybean trial was established in two sandy clay loam soils (desertified land) and one silty clay loam soil. Two types of biochar were applied as treatments. We aimed to investigate the response of soybean plants to soil structure, soil nutrient condition, and biochar amendment in the two types of soil. In addition, ridge regression was employed to model the plant growth indicators by soil structure, soil nutrients condition, soil water content, and biochar amendment. We conclude that (1) overall soil productivity in sandy clay loam soil is lower than in silty clay loam soil. The sandy clay loam soil may have high efficacy for crop production due to its higher harvest index. (2) Aggregate size 0.5–1 mm, 1–2 mm, and 2–3 mm indicated more important in plant biomass formation in silty clay loam soil. The low aggregate stability of sandy clay loam soil made the field more vulnerable to wind erosion in the semi-arid monsoon climate. (3) Cob biochar and wood biochar increased soybean shoot biomass by 48.7% and 45.0% in silty clay loam soil. (4) The higher N-fixing ability of nodules in sandy clay loam soil indicates an advantage to reduce the use of N-fertilizers in desertified areas. (5) Exponential polynomial regression ameliorated the accuracy of prediction of plant growth indicators in comparison to linear regression.

Exponential polynomialsas solutions of non-linear differential-difference equations

Exponential polynomials, an important subclass of finite order entire functions, as solutions of differential or difference or differential-difference equations are considered in [5, 10, 19, 20]. The critical domains of zeros and the quotients of exponential polynomials are considered in [6]. In this paper, we proceed to consider the exponential polynomials as solutions of some general complex differential-difference equations and extend existence results.

Proposal of a non-linear curve for reporting the performance of solar cookers

Performance parameters of solar cookers have conventionally been determined by assuming a linear trend between the cooker power and the difference between load temperature and ambient air temperature. This approach may not be convenient for some solar cooker designs. In the present work, the suitability of a non-linear regression derived from fitting the measured load temperature to a second order exponential polynomial was investigated and compared with the linear regression. Both regressions were compared with the corresponding experimental curves of a panel cooker and a box cooker. In the case of the panel cooker, the linear trend of the experimental plot was confirmed over a large period of the conducted test. Minor deviations from the experimental data were observed only at the beginning and at the end of the test. On the contrary, in the box solar cooker, significant deviations between the linear regression plot and the experimental points were observed, while smaller deviations were obtained using the non-linear regression. Thus, the proposed method can be seen as a promising approach that should be considered when updating the existing procedures for testing and reporting the performance of solar cookers.

Exponential Polynomials as Solutions of Nonlinear Differential-Difference Equations

Exponential Polynomials as Solutions of Nonlinear Differential-Difference Equations

Codimension two bifurcation in a coupled FitzHugh–Nagumo system with multiple delays

In this paper, a coupled FitzHugh–Nagumo system with multiple delays is considered. In a first step, the critical point at which a zero root of multiplicity two occurs in the characteristic equation is constructed. In a second step, in order to ensure that all the roots of the characteristic equation except for the double zero root have negative real parts, the zeros of a third and a fourth degree exponential polynomial are studied. Moreover, the critical values where the Bogdanov–Takens bifurcation occurs are derived. By using the normal form theory and reduction on the centre manifold, the truncated normal form is obtained, and throughout the bifurcation diagram, its dynamical behaviours are studied. Finally, a numerical example is given to demonstrate our results.

Analysis of Exponential–Polynomial Single Step Method for Singularly Perturbed Delay Differential Equations

This research presents Exponential – Polynomial Single Step Method (EPSSM), which uses non-linear polynomial interpolating function to solve Singularly Perturbed Delay Differential Equations (SPPDDEs). The stability polynomial of the method is obtained. The Local Grid Search Algorithm termed LGSA is applied in determining the stability region. The efficiency of these methods is compared with the other two numerical methods, namely Runge-Kutta Method (RKM) and Rational Single Step Method (RSM) through two examples of Singularly Perturbed Delay Differential Equations (SPPDDEs).

A high‐order weighted essentially nonoscillatory scheme based on exponential polynomials for nonlinear degenerate parabolic equations

AbstractIn this research the numerical solution of nonlinear degenerate parabolic equations is investigated by a new sixth‐order finite difference weighted essentially nonoscillatory (WENO) based on exponential polynomials. In smooth regions, the new scheme, named as EPWENO6, can achieve the maximal approximation order while in critical points it does not lose its accuracy. In order to better approximation near steep gradients without spurious oscillations, the EPWENO6 scheme is designed by the exponential polynomials that are incorporated into the WENO reconstruction. In design of nonlinear weights that play a very important role in WENO reconstructions, a global smoothness indicator using generalized undivided differences is introduced. To analyze the convergence order of the EPWENO6 method in full detail, the Lagrange‐type exponential functions have been used. By comparing the EPWENO6 scheme and other existing WENO schemes for solving degenerate parabolic equations, it can be seen that the new scheme offers better results while its computational cost is less. To illustrate the effectiveness of the proposed scheme, a number of numerical experiments are considered.

Stability with general decay rate of hybrid neutral stochastic pantograph differential equations driven by Lévy noise

<p style='text-indent:20px;'>This paper focuses on the <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>th moment and almost sure stability with general decay rate (including exponential decay, polynomial decay, and logarithmic decay) of highly nonlinear hybrid neutral stochastic pantograph differential equations driven by L<inline-formula><tex-math id="M3">\begin{document}$ \acute{e} $\end{document}</tex-math></inline-formula>vy noise (NSPDEs-LN). The crucial techniques used are the Lyapunov functions and the nonnegative semi-martingale convergence theorem. Simultaneously, the diffusion operators are permitted to be controlled by several additional functions with time-varying coefficients, which can be applied to a broad class of the non-autonomous hybrid NSPDEs-LN with highly nonlinear coefficients. Besides, <inline-formula><tex-math id="M4">\begin{document}$ H_\infty $\end{document}</tex-math></inline-formula> stability and the almost sure asymptotic stability are also concerned. Finally, two examples are offered to illustrate the validity of the obtained theory.</p>

Several classical identities via Mellin’s transform

We present a summation rule using Mellin’s transform to give short proofs of some important classical relations between special functions and Bernoulli and Euler polynomials. For example, the values of the Hurwitz zeta function at the negative integers are expressed in terms of Bernoulli polynomials. We also show identities involving exponential and Hermite polynomials.

Reveal the internal moisture changes of white gourd during hot air-drying process using low-field NMR

The moisture content, distribution and state changes of White gourd were studied during hot air-drying process using Low-field nuclear magnetic resonance (LF-NMR). The transverse relaxation time (T2) inversion spectrum of White gourd was measured at different temperatures during drying process, and the characteristics of internal moisture state and change were analyzed using the LF-NMR. The results showed that T2 was changed and the moisture of liquidity was reduced during drying process. In addition, the moisture with high degree of freedom changed to the moisture with low degree of freedom during drying. The amplitude of total NMR signal and the moisture content of drying base were a linear relationship during drying. The kinetics of the percentage of bound water and free water were based on exponential function and polynomial function respectively with drying time and different drying temperatures.

Characterization of exponential polynomial as solution of certain type of nonlinear delay-differential equation

In this paper, we have characterized the nature and form of solutions of the following nonlinear delay-differential equation: $$f^{n}(z)+\sum_{i=1}^{n-1}b_{i}f^{i}(z)+q(z)e^{Q(z)}L(z,f)=P(z),$$ where $b_i\in\mathbb{C}$, $L(z,f)$ are a linear delay-differential polynomial of $f$; $n$ is positive integers; $q$, $Q$ and $P$ respectively are nonzero, nonconstant and any polynomials. Different special cases of our result will accommodate all the results of [J. Math. Anal. Appl., 452(2017), 1128-1144; Mediterr. J. Math., 13(2016), 3015-3027; Open Math., 18(2020), 1292-1301]. Thus our result can be considered as an improvement of all of them. We have also illustrated a handful number of examples to show that all the cases as demonstrated in our theorem actually occur and consequently the same are automatically applicable to the previous results.

Taylor-type expansions in terms of exponential polynomials

Taylor-type expansions in terms of exponential polynomials

Shape and texture based identification of glaucoma from retinal fundus images

Among the eye disorders, glaucoma occurs due to the increase of intra-ocular pressure that causes irreversible damage of the optic nerve and leads to blindness. Therefore to avoid high cost machine usage, a novel method proposed for detecting glaucoma using shape and texture features. Initially, the retinal fundus images are decomposed using quasi bi-variate variational mode decomposition (QB-VMD) technique, the frequencies obtained from QB-VMD subjected to pyramid histogram oriented gradient (PHOG) and invariant Haralick texture features. The extracted combinational features are classified using combination of exponential polynomial support vector machines (EP-SVM) and bagged ensemble approach. The proposed method simulated on ACRIMA and Drishti-GS1 datasets using 10-fold cross validation and evaluated the performance metrics like accuracy, sensitivity, specificity and F-score. The simulation results and evaluation metrics show that the proposed approach achieves superior classification performance compared to other state-of-the-art approaches.

Definite Integral of Exponential Polynomial and Hyperbolic Function in terms of the Incomplete Gamma Function

This current paper is a table of definite integrals involving hyperbolic and logarithmic functions expressed in terms of the incomplete gamma function and fundamental constants. All the results in this work are new.

Plethystic Exponential Calculus and Characteristic Polynomials of Permutations

We prove a family of identities, expressing generating functions of powers of characteristic polynomials of permutations, as finite or infinite products. These generalize formulae first obtained in a study of the geometry/topology of symmetric products of real/algebraic tori. The proof uses formal power series expansions of plethystic exponentials, and has been motivated by some recent applications of these combinatorial tools in supersymmetric gauge and string theories. Since the methods are elementary, we tried to be self-contained, and relate to other topics such as the q-binoomial theorem, and the cycle index and Molien series for the symmetric group.

Hopf Bifurcation of an Age-Structured Epidemic Model with Quarantine and Temporary Immunity Effects

In this paper, we investigate an epidemic model with quarantine and recovery-age effects. Reformulating the model as an abstract nondensely defined Cauchy problem, we discuss the existence and uniqueness of solutions to the model and study the stability of the steady state based on the basic reproduction number. After analyzing the distribution of roots to a fourth degree exponential polynomial characteristic equation, we also derive the conditions of Hopf bifurcation. Numerical simulations are performed to illustrate the results.

Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials

Abstract In this paper, we propose a family of non-stationary combined ternary ( 2 m + 3 ) \left(2m+3) -point subdivision schemes, which possesses the property of generating/reproducing high-order exponential polynomials. This scheme is obtained by adding variable parameters on the generalized ternary subdivision scheme of order 4. For such a scheme, we investigate its support and exponential polynomial generation/reproduction and get that it can generate/reproduce certain exponential polynomials with suitable choices of the parameters and reach 2 m + 3 2m+3 approximation order. Moreover, we discuss its smoothness and show that it can produce C 2 m + 2 {C}^{2m+2} limit curves. Several numerical examples are given to show the performance of the schemes.

Moment Functions on Groups

In this paper generalized moment functions are considered. They are closely related to the well-known functions of binomial type which have been investigated on various abstract structures. The main purpose of this work is to prove characterization theorems for generalized moment functions on commutative groups. At the beginning a multivariate characterization of moment functions defined on a commutative group is given. Next the notion of generalized moment functions of higher rank is introduced and some basic properties on groups are listed. The characterization of exponential polynomials by means of complete (exponential) Bell polynomials is given. The main result is the description of generalized moment functions of higher rank defined on a commutative group as the product of an exponential and composition of multivariate Bell polynomial and an additive function. Furthermore, corollaries for generalized moment function of rank one are also stated. At the end of the paper some possible directions of further research are discussed.

Comparative Study of Bisection, Newton, Horner’s Method for Solving Nonlinear Equation

The Bisection, Newton and Horner’s method are used to compare the efficiency to solve nonlinear function such as trigonometric, exponential, logarithmic and cubic polynomial function. Although these methods are considered as alternative, the methods also possess error compared to the exact value. The efficiency is measured by the error produced at the fixed iteration. The methods are converted into C language and executed by using maple 18.