Exponential Polynomial Research Articles

Empirical path loss models at 433 MHz in Himalayan snow for health monitoring

PurposeThe purpose of this paper is to investigate the effect of snow on the radio link performance of wireless sensor nodes in Indian Himalayan conditions and to propose empirical path loss models for radio wave propagation.Design/methodology/approachAt the remote test site, one source and three listening wireless sensor nodes were deployed at frequency of 433 MHz. The path loss models are derived from experimental data collected during the period of snowfall and clear weather conditions. Linear, exponential, second and third-order polynomials path loss models have been investigated along with experimental data.FindingsWith the help of curve fitting and goodness-of-fit tests, it is found that path loss can be modelled through third-order polynomial equation during the snowfall period. However, if sensor is buried, the acceptable path loss model is exponential. Similarly, for unified modelling requirement, exponential path loss model over linear can be a preferred choice.Originality/valueResults show that path loss can be estimated priori for deciding optimum transmission energy in wireless sensor network. Presented work is usable in extending the lifetime of health monitoring devices buried in snowy environment.

Dual exponential polynomials and a problem of Ozawa

Complex linear differential equations with entire coefficients are studied in the situation where one of the coefficients is an exponential polynomial and dominates the growth of all the other coefficients. If such an equation has an exponential polynomial solution $f$, then the order of $f$ and of the dominant coefficient are equal, and the two functions possess a certain duality property. The results presented in this paper improve earlier results by some of the present authors, and the paper adjoins with two open problems.

Developing an innovative bimodal model to characterize the dynamic radar cross section of aircrafts

The distribution of aircraft dynamic radar cross section (RCS) varies at different course short-cuts and heights and can behave as a bimodal shape. Classical models, such as the Weibull distribution, the chi-square distribution, and lognormal distribution, fit poorly with bimodal RCS distributions and are unable to accurately describe the statistical characteristics of aircraft dynamic RCS. In order to overcome the shortcoming of the classical models, we propose an exponential polynomial distribution model as the probability density function for studying the RCS statistical characteristics. The model parameters were acquired through the linear least-square method. Based on the RCS statistical data, the proposed distribution model was validated through simulation experiments. We calculated the fitting error between the new model and the RCS statistical distribution and determined the optimal exponential polynomial distribution model. We then computed the coefficients of the determination to evaluate the proposed model. The Kolmogorov–Smirnov test showed that the proposed approach yielded smaller values compared to conventional distribution models. For all the analyzed cases, the exponential polynomial model can fit very well against the RCS distribution data with bimodal shape, and the fitting effect is significantly better than the results of the three classical models. The advantages of the proposed model are further demonstrated by comparing it with the Gaussian mixture distribution model from fitting error, determination coefficient, goodness-of-fit, mean error and variance error. The results suggest that the exponential polynomial distribution model provides a more effective alternative in describing the fluctuation characteristics of aircraft dynamic RCS at the different course short-cuts and heights. Given the advantages exhibited by the exponential polynomial distribution model in this study, the methodology and findings in this study can be used to improve in early warning detection of radar targets and as a theoretical basis to enhance the overall performance of radars. 2009 Elsevier Ltd. All rights reserved.

Analytic Functions in Local Shift-Invariant Spaces and Analytic Limits of Level Dependent Subdivision

In this paper we characterize all subspaces of analytic functions in finitely generated shift-invariant spaces with compactly supported generators and provide explicit descriptions of their elements. We illustrate the differences between our characterizations and Strang-Fix or zero conditions on several examples. Consequently, we depict the analytic functions generated by scalar or vector subdivision with masks of bounded and unbounded support. In particular, we prove that exponential polynomials are indeed the only analytic limits of level dependent scalar subdivision schemes with finitely supported masks.

Navigating in Trees with Permanently Noisy Advice

We consider a search problem on trees in which an agent starts at the root of a tree and aims to locate an adversarially placed treasure, by moving along the edges, while relying on local, partial information. Specifically, each node in the tree holds a pointer to one of its neighbors, termed advice . A node is faulty with probability q . The advice at a non-faulty node points to the neighbor that is closer to the treasure, and the advice at a faulty node points to a uniformly random neighbor. Crucially, the advice is permanent , in the sense that querying the same node again would yield the same answer. Let Δ denote the maximum degree. For the expected number of moves (edge traversals) until finding the treasure, we show that a phase transition occurs when the noise parameter q is roughly 1 √Δ. Below the threshold, there exists an algorithm with expected number of moves O ( D √Δ), where D is the depth of the treasure, whereas above the threshold, every search algorithm has an expected number of moves, which is both exponential in D and polynomial in the number of nodes n . In contrast, if we require to find the treasure with probability at least 1 − δ, then for every fixed ɛ > 0, if q < 1/Δ ɛ , then there exists a search strategy that with probability 1 − δ finds the treasure using (Δ −1 D ) O (1/ε) moves. Moreover, we show that (Δ −1 D ) Ω(1/ε) moves are necessary.

Essential bounds of Dirichlet polynomials

In this paper we have given conditions on exponential polynomials \(P_{n}(s)\) of Dirichlet type to be attained the equality between each of two pairs of bounds, called essential bounds, \(a_{P_{n}(s)}\), \(\rho _{N}\) and \(b_{P_{n}(s)}\), \(\rho _{0}\) associated with \(P_{n}(s)\). The reciprocal question has been also treated. The bounds \(a_{P_{n}(s)}\), \(b_{P_{n}(s)}\) are defined as the end-points of the minimal closed and bounded real interval \(I= [ a_{P_{n}(s)},b_{P_{n}(s)} ] \) such that all the zeros of \(P_{n}(s)\) are contained in the strip \(I\times {\mathbb {R}}\) of the complex plane \({\mathbb {C}}\). The bounds \(\rho _{N}\), \(\rho _{0}\) are defined as the unique real solutions of Henry equations of \(P_{n}(s)\). Some applications to the partial sums of the Riemann zeta function have been also showed.

Analytical Solution of Hyperchaotic Zhou Equations by Multistage Homotopy Analysis Method

The hyperchaotic system is a condition of phenomenal uncertainty in which at least two Lyapunov exponents are present in the system. For the analytical solution of the hyperchaotic Zhou system, the multistage homotopy analysis approach is used in this article. The solution constructs a convergent series in terms of the exponential combination and polynomial functions that use a few terms of this series to obtain higher accuracy. To demonstrate the efficiency of the proposed method, the residual error is presented.

Datko criteria for uniform instability in Banach spaces

The main objective of this paper is to present some necessary and sufficient conditions of Datko type for the uniform exponential and uniform polynomial instability concepts for evolution operators in Banach spaces.

Stochastic response of MDOF system to non-stationary random excitation

An approximate scheme is developed for efficiently determining non-stationary response PDF of multi-degree-of-freedom (MDOF) dynamical system to time-modulated random excitation. It further generalizes state-space-split exponential polynomial closure (SSS-EPC) technique and enhances its application for determining non-stationary response probability density function (PDF). To this aim, time variable is introduced into the approximate solution by setting polynomial coefficients as time dependent ones. With the proposed procedure, non-stationary solution to high-dimensional FPK equation can be obtained. Meanwhile, Monte Carlo simulation (MCS) method is conducted to assess the reliability of the proposed method. Three typical examples of linear and nonlinear MDOF dynamical systems to different modulated Gaussian white noise excitations are considered. Comparisons with the simulated results and computational time of the MCS method demonstrate the relatively high efficiency and accuracy of the proposed procedure.

On amalgamation of truncated exponential and Gould-Hopper polynomials

In this article, the truncated exponential polynomials are taken as base with the Gould-Hopper polynomials to introduce a hybrid family of truncated exponential-Gould-Hopper polynomials. These polynomials are framed within the context of monomiality principle and their properties are established. Certain properties including operational representations, expansion formula, integral representation and summation formula are also derived for this family. Further, the graphs of these polynomials are drawn for different values of indices using MATLAB.

Ameliorating Sandy Soil Properties: Application of Mathematical Model to Explore Spinach (Amaranthus tricolor L.) Plant Response

Amelioration is an effort to increase soil fertility by manipulating chemical and physical properties of the soil. In this study, biochar and manure were adopted for ameliorating sandy soil. The purposes of this study were to identify the response of spinach plants, to apply mathematical model for exploring spinach plant growth, and to determine the appropriate ameliorant dosage. The ameliorant were rice husk biochar and poultry manure. The study was conducted in a greenhouse with 4 different dose comparisons (in kg; sandy soil: rice husk biochar: poultry manure), i.e. 1: 0: 0 (C), 1: 0.007: 0 (B), 1: 0.0035: 0, 0035 (MB1), and 1: 0.007: 0.007 (MB2). Each treatment was repeated 5 times in a Complete Block Design. Growth parameters observed were plant height, number of leaves, and plant weight. Observation was carried out for 70 days. The analysis used was one-way ANOVA test, Turkey test, logistic equation and exponential polynomials models, and linear regression. The study revealed that the best spinach growth response by MB1. The growth rate from plant height side of C, B, MB1, and MB2 treatments were -0.0509, -0.0406, -0.0556, and -0.0489 respectively. In terms of number of leaves, the optimum of growing period of C, B, MB1, MB2 treatments were 48, 65, 96, and 66 days after planting respectively. One-way ANOVA test showed that rice husk biochar and poultry manure ameliorant had a significant effect on plant growth both height and number of leaves (sign. <0.05). While, the wet and dry weight were not significantly effect (sign. > 0.05). Mathematical model with logistic and exponential equations were acceptable to describe spinach plant growth (R2> 80%). Ameliorant rice husk and poultry naure 5 tons ha/ha (MB1 treatment) was recommended for optimum spinach growth in sandy soil.

POLYNOMIAL TERM STRUCTURE MODELS

In this paper, we explore a class of tractable interest rate models that have the property that the price of a zero-coupon bond can be expressed as a polynomial of a state diffusion process. Our results include a classification of all such time-homogeneous single-factor models in the spirit of Filipović’s maximal degree theorem for exponential polynomial models, as well as an explicit characterization of the set of feasible parameters in the case when the factor process is bounded. Extensions to time-inhomogeneous and multi-factor polynomial models are also considered.

A new geometrical perspective on Bohr-equivalence of exponential polynomials

Based on Bohr’s equivalence relation for general Dirichlet series, in this paper we connect the families of equivalent exponential polynomials with a geometrical point of view related to lines in crystal-like structures. In particular we characterize this equivalence relation, and give an alternative proof of Bochner’s property referring to these functions, through this new geometrical perspective.

Transient probabilistic solutions of stochastic oscillator with even nonlinearities by exponential polynomial closure method

The current work is devoted to analyze the transient probability density function solutions of stochastic oscillator with even nonlinearities under external excitation of Gaussian white noise by applying the extended exponential polynomial closure method. Specifically, the Fokker–Planck–Kolmogorov equation which governs the probability density function solutions of the nonlinear system is presented first. The residual error of the Fokker–Planck–Kolmogorov equation is then derived by assuming the probability density function solution as the type of exponential polynomial with time-dependent variables. Finally, by making the projection of the residual error vanish, a set of nonlinear ordinary differential equations is established and solved numerically. Numerical analysis show that the extended exponential polynomial closure method with polynomial order being six is both effective and efficient for solving the transient analysis of the stochastic oscillator with even nonlinearities by comparing the numerical results obtained by the proposed method with those obtained by Monte Carlo simulation method. Numerical results also show that the transient probability density function solutions of the system responses are not symmetric about their nonzero means due to the existence of even nonlinearities.

A third-order WENO scheme based on exponential polynomials for Hamilton-Jacobi equations

In this study, we provide a novel third-order weighted essentially non-oscillatory (WENO) method to solve Hamilton-Jacobi equations. The key idea is to incorporate exponential polynomials to construct numerical fluxes and smoothness indicators. First, the new smoothness indicators are designed by using the finite difference operator annihilating exponential polynomials such that singular regions can be distinguished from smooth regions more efficiently. Moreover, to construct numerical flux, we employ an interpolation method based on exponential polynomials which yields improved results around steep gradients. The proposed scheme retains the optimal order of accuracy (i.e., three) in smooth areas, even near the critical points. To illustrate the ability of the new scheme, some numerical results are provided along with comparisons with other WENO schemes.

A non-uniform corner-cutting subdivision scheme with an improved accuracy

The aim of this paper is to construct a new non-uniform corner-cutting (NUCC) subdivision scheme that improves the accuracy of the classical (stationary and non-stationary) methods. The refinement rules are formulated via the reproducing property of exponential polynomials. An exponential polynomial has a shape parameter so that it may be adapted to the characteristic of the given data. In this study, we propose a method of selecting the shape parameter, so that it enables the associated scheme to achieve an improved approximation order (that is, three), in case that either the initial data or its derivative is bounded away from zero. In contrast, the classical methods attain the second-order accuracy. An analysis of convergence and smoothness of the proposed scheme is conducted. The proposed scheme is shown to have the same smoothness as the classical Chaikin’s corner-cutting algorithm, that is, C1. Finally, some numerical examples are presented to demonstrate the advantages of the new corner-cutting algorithm.

An annihilator-based strategy for the automatic detection of exponential polynomial spaces in subdivision

Exponential polynomials are essential in subdivision for the reconstruction of specific families of curves and surfaces, such as conic sections and quadric surfaces. It is well known that if a linear subdivision scheme is able to reproduce a certain space of exponential polynomials, then it must be level-dependent, with rules depending on the frequencies (and eventual multiplicities) defining the considered space. This work discusses a general strategy that exploits annihilating operators to locally detect those frequencies directly from the given data and therefore to choose the correct subdivision rule to be applied. This is intended as a first step towards the construction of self-adapting subdivision schemes able to locally reproduce exponential polynomials belonging to different spaces. An application of the proposed strategy is shown explicitly on an example involving the classical butterfly interpolatory scheme. This particular example is the generalization of what has been done for the univariate case in Donat and López-Ureña (2019), which inspired this work.

Arched beams of Bresse type: observability and application in thermoelasticity

This is the first paper of a trilogy intended by the authors in what concerns a unified approach to the stability of thermoelastic arched beams of Bresse type under Fourier’s law. Our main goal in this starting work is to develop an original observability inequality for conservative Bresse systems with non-constant coefficients. Then, as a powerful application, we prove mathematically that the stability of a partially damped model in thermoelastic Bresse beams is invariant under the boundary conditions. The exponential and optimal polynomial decay rates are addressed. This approach gives a new view on the stability of Bresse systems subject to different boundary conditions as well as it provides an accurate answer for the related issue raised by Liu and Rao (Z. Angew. Math. Phys. 60(1): 54–69, 2009) from both the physical and mathematical points of view.

A Comprehensive Analysis of Quantum Clustering : Finding All the Potential Minima

Quantum clustering (QC), is a data clustering algorithm based on quantum mechanics which is accomplished by substituting each point in a given dataset with a Gaussian. The width of the Gaussian is a σ value, a hyper-parameter which can be manually defined and manipulated to suit the application. Numerical methods are used to find all the minima of the quantum potential as they correspond to cluster centers. Herein, we investigate the mathematical task of expressing and finding all the roots of the exponential polynomial corresponding to the minima of a two-dimensional quantum potential. This is an outstanding task because normally such expressions are impossible to solve analytically. However, we prove that if the points are all included in a square region of size σ, there is only one minimum. This bound is not only useful in the number of solutions to look for, by numerical means, it allows to to propose a new numerical approach “per block”. This technique decreases the number of particles by approximating some groups of particles to weighted particles. These findings are not only useful to the quantum clustering problem but also for the exponential polynomials encountered in quantum chemistry, Solid-state Physics and other applications.

Fourier Quasicrystals with Unit Masses

The sum of δ-measures sitting at the points of a discrete set Λ⊂ℝ forms a Fourier quasicrystal if and only if Λ is the zero set of an exponential polynomial with imaginary frequencies.