Exponential Series Research Articles

The Artin-Hasse series and Laguerre polynomials modulo a prime

For an odd prime p, let Ep(X)=∑n=0∞anXn∈Fp[[X]]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{E}\\,}}_{p}(X)=\\sum _{n=0}^{\\infty } a_{n}X^{n}\\in {\\mathbb {F}}_p[[X]]$$\\end{document} denote the reduction modulo p of the Artin-Hasse exponential series. It is known that there exists a series G(Xp)∈Fp[[X]]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G(X^p)\\in {\\mathbb {F}}_{p}[[X]]$$\\end{document}, such that Lp-1(-T(X))(X)=Ep(X)·G(Xp)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_{p-1}^{(-T(X))}(X)={{\\,\ extrm{E}\\,}}_{p}(X)\\cdot G(X^p)$$\\end{document}, where T(X)=∑i=1∞Xpi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T(X)=\\sum _{i=1}^{\\infty }X^{p^{i}}$$\\end{document} and Lp-1(α)(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_{p-1}^{(\\alpha )}(X)$$\\end{document} denotes the (generalized) Laguerre polynomial of degree p-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p-1$$\\end{document}. We prove that G(Xp)=∑n=0∞(-1)nanpXnp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G(X^p)=\\sum _{n=0}^{\\infty }(-1)^n a_{np}X^{np}$$\\end{document}, and show that it satisfies G(Xp)G(-Xp)T(X)=Xp.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ G(X^p)\\,G(-X^p)\\,T(X)=X^p. $$\\end{document}

Prediction and Analysis of the Price of Carbon Emission Rights in Shanghai: Under the Background of COVID-19 and the Russia–Ukraine Conflict

In the spring of 2022, a new round of epidemic broke out in Shanghai, causing a shock to the Shanghai carbon trading market. Against this background, this paper studied the impact of the new epidemic on the price of Shanghai carbon emission rights and tried to explore the prediction model under the unexpected event. First, because a model based on point value data cannot capture the information hidden in inter-day price fluctuation, based on the interval price of Shanghai carbon emission rights (SHEA) and its influencing factors, an autoregressive conditional interval model with jumping and exogenous variables (ACIXJ) was established to explore the influence of the Russian–Ukrainian conflict and COVID-19 on the interval price of SHEA, respectively. The empirical results show that the conflict between Russia and Ukraine has no obvious influence on the price of SHEA, but COVID-19 led to a decline in the price trend of SHEA over four days before the city was closed, and the volatility changed significantly on the day before the city was closed. The price fluctuation was the strongest within 3 days after the city was closed; In addition, in order to accurately predict the interval data of SHEA against the background of COVID-19, based on the interval data decomposition algorithm (BEMD), a hybrid forecasting model of NDGM-ACIXJ/CNN-LSTM was constructed, in which the discrete gray model of approximate nonhomogeneous exponential series (NDGM) combined with the ACIXJ model is used to predict the high-frequency sub-interval, and the convolution neural network long-term and short-term memory model (CNN-LSTM) is used to predict the low-frequency sub-interval. The empirical results show that the prediction model proposed in this article has higher prediction precision than the reference models (ACIX, ACIXJ, NDGM-ACIXJ, BEMD-ACIX/CNN-LSTM, BEMD-ACIXJ/CNN-LSTM).

Approximation Results for Hadamard-Type Exponential Sampling Kantorovich Series

Approximation Results for Hadamard-Type Exponential Sampling Kantorovich Series

Prony Analysis of Left Ventricle Pressure and Volume.

Direct measurement of cardiac pressure-volume (PV) relationships is the gold standard for assessment of ventricular hemodynamics, but few innovations have been made to "multi-beat" PV analysis beyond traditional signal processing. The Prony method solves the signal recovery problem with a series of dampened exponentials or sinusoids. It achieves this by extracting the amplitude, frequency, dampening, and phase of each component. Since its inception, application of the Prony method to biologic and medical signal has demonstrated a relative degree of success, as a series of dampened complex sinusoids easily generalizes to multifaceted physiological processes. In cardiovascular physiology, the Prony analysis has been used to determine fatal arrythmia from electrocardiogram signals. However, application of the Prony method to simple left ventricular function based on pressure and volume analysis is absent. We have developed a new pipeline for analysis of pressure volume signals recorded from the left ventricle. We propose fitting pressure-volume data from cardiac catheterization to the Prony method for pole extraction and quantification of the transfer function. We implemented the Prony algorithm using open-source Python packages and analyzed the pressure and volume signals before and after severe hemorrhagic shock, and after resuscitation with stored blood. Each animal (n=6 per group) underwent a 50% hemorrhage to induce hypovolemic shock, which was maintained for 30min, and resuscitated with 3-week-old stored RBCs until 90% baseline blood pressure was achieved. Pressure-volume catheterization data used for Prony analysis were 1s in length, sampled at 1000Hz, and acquired at the time of hypovolemic shock, 15 and 30min after induction of hypovolemic shock, and 10, 30, and 60min after volume resuscitation. We next assessed the complex poles from both pressure and volume waveforms. To quantify deviation from the unit circle, which represents deviation from a Fourier series, we counted the number of poles at least 0.2 radial units away from it. We found a significant decrease in the number of poles after shock (p=0.0072vs. baseline) and after resuscitation (p=0.0091vs. baseline). No differences were observed in this metric pre and post volume resuscitation (p=0.2956). We next found a composite transfer function using the Prony fits between the pressure and volume waveforms and found differences in both the magnitude and phase Bode plots at baseline, during shock, and after resuscitation. In summary, our implementation of the Prony analysis shows meaningful physiologic differences after shock and resuscitation and allows for future applications to broader physiological and pathophysiological conditions.

Wave Attenuation in Additively Manufactured Polymer Acoustic Black Hole Structures Considering the Viscoelastic Effect.

The acoustic black hole (ABH) is a feature commonly found in thin-walled structures that is characterized by a diminishing thickness and damping layer with an efficient wave energy dissipation effect, which has been extensively studied. The additive manufacture of polymer ABH structures has shown promise as a low-cost method to manufacture ABHs with complex geometries, exhibiting even more effective dissipation. However, the commonly used elastic model with viscous damping for both the damping layer and polymer ignores the viscoelastic changes that occur due to variations in frequency. To address this, we used Prony exponential series expansion to describe the viscoelastic behavior of the material, where the modulus is represented by a summation of decaying exponential functions. The parameters of the Prony model were obtained through experimental dynamic mechanical analysis and applied to finite element models to simulate wave attenuation characteristics in polymer ABH structures. The numerical results were validated by experiments, where the out-of-plane displacement response under a tone burst excitation was measured by a scanning laser doppler vibrometer system. The experimental results illustrated good consistency with the simulations, demonstrating the effectiveness of the Prony series model in predicting wave attenuation in polymer ABH structures. Finally, the effect of loading frequency on wave attenuation was studied. The findings of this study have implications for the design of ABH structures with improved wave attenuation characteristics.

Applications of Fourier series and Fourier Transform in Electrical and Electronics Devices

Abstract: Fourier series are of great importance in both theoretical and ap­ plied mathematics. For orthonormal families of complex­valued functions Fourier Series are sums of the φn that can approximate periodic, complex­valued functions with arbitrary precision. This paper will focus on the Fourier Series of the complex exponentials. Of the many possi­ ble methods of estimating complex­valued functions, Fourier series are especially attractive because uniform convergence of the Fourier series (as more terms are added) is guaranteed for continuous, bounded functions. Furthermore, the Fourier coefficients are designed to minimize the square of the error from the actual function. Finally, complex exponentials are relatively simple to deal with and ubiquitous in physical phenomena. This paper first defines generalized Fourier series, with an emphasis on the se­ ries with complex exponentials. Then, important properties of Fourier series are described and proved, and their relevance is explained. A com­plete example is then given, and the paper concludes by briefly mentioning some of the applications of Fourier series and the generalization of Fourier series, Fourier transforms. In the mid­eighteenth century, physical problems such as the conduction pat­ terns of heat and the study of vibrations and oscillations led to the study of Fourier series. Of central interest was the problem of how arbitrary real­valued functions could be represented by sums of simpler functions. As we shall see later, a Fourier series is an infinite sum of trigonometric functions that can be used to model real­valued, periodic functions. We shall begin by giving a brief description of the trigonometric polynomials,and especially of their relation to the complex exponentials. The Fourier series, the founding principle behind the field of Fourier analysis, is an infinite expansion of function in terms sine’s and cosines. Fourier transform provides a continuous complex frequency of a function. It is useful in the study of frequency response of a filter, solution PDE, Discrete Fourier transform and FFT in the signals analysis. The advent of Fourier transformation method has greatly extended our ability to implement Fourier methods on digital multimedia visualization system. It covers the mathematical foundations of Digital signal processing (DSP), classical sound, synthesis algorithms and multi-time frequency domain analysis associated musical sound. In this paper, we are analysis of square wave in terms of Fourier component, may occur in electric circuits designed to handle sharply rising pulses and how to convert analog to digital system by using Fourier transform and its applications in electronics and digital multimedia visualization signal process of communication system.

Universal families of Eulerian multiple zeta values in positive characteristic

We study positive characteristic multiple zeta values associated to general curves over Fq together with an Fq-rational point ∞ as introduced by Thakur. For the case of the projective line these values were defined as analogues of classical multiple zeta values. In the present paper we first establish a general non-commutative factorization of exponential series associated to certain lattices of rank one. Next we introduce universal families of multiple zeta values of Thakur and show that they are Eulerian in full generality. In particular, we prove a conjecture of Lara Rodríguez and Thakur in [28]. One of the main ingredients of the proofs is the notion of L-series in Tate algebras introduced by the third author [31] in 2012.

Stieltjes analytic functions and higher order linear differential equations

In this work we develop a theory of Stieltjes-analytic functions. We first define the Stieltjes monomials and polynomials and we study them exhaustively. Then, we introduce the g-analytic functions locally, as an infinite series of these Stieltjes monomials and we study their properties in depth and how they relate to higher order Stieltjes differentiation. We define the exponential series and prove that it solves the first order linear problem. Finally, we apply the theory to solve higher order linear homogeneous Stieltjes differential equations with constant coefficients.

On bivariate Kantorovich exponential sampling series

In this paper, we introduce and analyze the approximation properties of bivariate generalization for the family of Kantorovich type exponential sampling series. We derive the basic convergence result and Voronovskaya type theorem for the proposed sampling series. Using logarithmic modulus of smoothness, we establish the quantitative estimate of order of convergence for the Kantorovich type exponential sampling series. Furthermore, we study the convergence results for the generalized Boolean sum (GBS) operator associated with bivariate Kantorovich exponential sampling series. At the end, we provide a few examples of kernels to which the presented theory can be applied along with the graphical representation and error estimates.

Analysis of Milk Production and Failure Data: Using Unit Exponentiated Half Logistic Power Series Class of Distributions

The unit exponentiated half logistic power series (UEHLPS), a family of compound distributions with bounded support, is introduced in this study. This family is produced by compounding the unit exponentiated half logistic and power series distributions. In the UEHLPS class, some interesting compound distributions can be found. We find formulas for the moments, density and distribution functions, limiting behavior, and other UEHLPS properties. Five well-known estimating approaches are used to estimate the parameters of one sub-model, and a simulation study is created. The simulated results show that the maximum product of spacing estimates had lower accuracy measure values than the other estimates. Ultimately, three real data sets from various scientific areas are used to analyze the performance of the new class.

Long-Term Scenario Analysis of Electricity Supply and Demand in Iran: Time Series Analysis, Renewable Electricity Development, Energy Efficiency and Conservation

Electricity plays a vital role in the economic development and welfare of countries. Examining the electricity situation and defining scenarios for developing power plant infrastructure will help countries avoid misguided policies that incur high costs and reduce people’s welfare. In the present research, three scenarios from 2021–2040 have been defined for Iran’s electricity status. The first scenario continues the current trend and forecasts population, electricity consumption, and carbon dioxide emissions from power plants with ARIMA and single and triple exponential smoothing time series algorithms. As part of the second scenario, only non-hydro renewable resources will be used to increase the electricity supply. By ensuring the existence of potential, annual growth patterns have been defined, taking into account the renewable electricity generation achieved by successful nations. The third scenario involves integrating operating gas turbines into combined cycles in exchange for buyback contracts. Economically, this scenario calculates return on investment through an arrangement of various contracts for the seller company and fuel savings for the buyer.

Exponential sampling with a multiplier

The exponential sampling formula has some limitations. By incorporating a Mellin bandlimited multiplier, we extend it to a wider class of functions with a series that converges faster. This series is a generalized exponential sampling series with some interesting properties. Moreover, under a side condition, any generalized exponential sampling series that is interpolating can be generated by a Mellin bandlimited multiplier. For an error analysis, we consider a truncated series with 2N+1 terms and look for a highest speed of convergence as Nrightarrow infty . We show by using a certain non-bandlimited multiplier, which introduces in addition an aliasing error, that we can achieve a higher rate of convergence to the function, namely mathcal {O}(e^{-alpha N}) with alpha >0, than with the truncated series of an exact formula. The results are illustrated by three examples.

Singular Nonlinear Problems for Phase Trajectories of Some Self-Similar Solutions of Boundary Layer Equations: Correct Formulation, Analysis, and Calculations

We study a singular initial value problem for a nonlinear non-autonomous ordinary differential equation of the second order, defined on a semi-infinite interval and degenerating in the initial data for the phase variable. The problem arises in the dynamics of a viscous incompressible fluid as an auxiliary problem in the study of self-similar solutions of the boundary layer equations for a stream function with a zero pressure gradient (plane-parallel laminar flow in a mixing layer). It is also of independent mathematical interest. Using the previously obtained results on singular nonlinear Cauchy problems and parametric exponential Lyapunov series, a correct formulation and a complete mathematical analysis of this singular initial value problem are given. Restrictions on the “self-similarity parameter” for the global existence of solutions are formulated, two-sided estimates of solutions, and results of calculations of the phase trajectories of solutions for different values of this parameter are given.

Modified version of the cross-correlation function to measure drought occurrence time-delay correlation

Abstract According to the importance of assessing the presence of time delay between the occurrence of various hydrological and meteorological phenomena, the study aim is to introduce a new method (with high ability and non-sensitivity to the abnormality of datasets and the existence of outliers) for determining the time delay in mentioned data series. In this research, a new measure to detect the time delay between two stationary time series (Non-Parametric Cross-Correlation Function or NCCF, called Spearman's CCF or SCCF) is introduced, which has very low sensitivity to abnormality of data series and also the existence of outliers in the data series. The numerical studies verify the ability of the proposed measure. In standard uniform and exponential (with mean 1) time series, at 100% of numerical analyses and in standard Gaussian time series at more than 60% of numerical studies, the ability of SCCF was more than the CCF. The applicability of the proposed measure in practice was also studied using the Reconnaissance Drought Index (RDI) data series of 20 stations over Iran during 1967–2019 in 1, 3, and 12-month time scales. The results of the practical study also proved the appropriate performance of the proposed model in all time scales.

Исследование решений линейной однородной системы дифференциальных уравнений

The paper shows that the fundamental zero-normalized solution of a linear homogeneous differ-ential equations system can be represented as an exponential matrices products formal series. If the system satisfies the equations system triangulation Perron theorem conditions, then the system solution can be represented as an exponential matrices finite product. In addition, an exponential matrix function differentiating formula is derived. Also, the transformation constructing problem is considered. Such, a homogeneous differential equations system allows to reduce to a triangular form.

The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching

Abstract It is more difficult to give Laplace transform directly in a defined form or derive it by Fourier transform in mathematics teaching. The article gives a solution for solving high exponential series sum by using Laplace transform. With the help of Laplace transform, calculus operations can be transformed into complex plane algebra operations. The application of the algorithm to the option hedging strategy verifies the applicability of the algorithm proposed in this article.

Propagation of skew-symmetric unsteady shear waves from thick-walled shell in elastic space

Problems of propagation and diffraction of nonstationary waves in elastic bodies are of great theoretical and practical importance in such fields of science and technology as aircraft construction, shipbuilding, seismic exploration of minerals, seismic resistance of structures, and many others. The paper considers the problem of the propagation of skew-symmetric unsteady shear waves from a thick-walled spherical shell in elastic space. To solve the problem, the integral Laplace transforms in dimensionless time, and the method of incomplete separation of variables was used. In the image space, the problem is reduced to an infinite system of linear algebraic equations, the solution of which is sought in the form of an infinite exponential series. Formulas for the displacement vector components and the stress tensor are obtained. The transition to the originals is carried out using the theory of residues. Numerical experiments have been carried out, the results of which are presented in graphs. The obtained results of the work can be used in geophysics, seismology, and design organizations in the construction of structures and in the design of underground reservoirs.

Effect of calcium leaching on the fracture properties of concrete

In this work, acceleration leaching and three-point bending (TPB) tests were performed on single-notched TPB beam specimens to explore the evolution of the fracture properties of concrete subjected to calcium leaching. The load-crack mouth opening displacement (P-CMOD) curves obtained for concrete at various leaching times were adopted to analytically calculate parameters that can reflect the fracture properties of concrete, such as the crack extension resistance (KR), fracture toughness (KIcini and KIcun), elastic modulus (E), tensile strength (ft), and fracture energy (Gf). Furthermore, a phase-field fracture model for concrete that considers calcium leaching effects is established by incorporating the time-dependent damage variables E, ft, and Gf. The results reveal that the KR curves of the leached concrete increase with crack propagation but lower and flatten with more leaching time. The fracture parametersKIcini, KIcunE, ft, and Gf rapidly decrease with leaching time in the first 60 days. However, with more leaching time, the impact of calcium leaching on the concrete fracture properties gradually diminishes. The variation in fracture parameters of leached concrete can be described by a series of exponential functions. The residual value fracture parameters of the fully leached concrete are estimated to be approximately 60–80 % of those of sound concrete. The established fracture model reproduces the P-CMOD curves of concrete at each leaching time, which verifies the rationality of the model. The numerical simulation reveals that crack propagate faster in leached concrete than in sound concrete, and leached concrete is more prone to cracking.

Factorization of coefficients for exponential and logarithm in function fields

Let X be an elliptic curve or a ramifying hyperelliptic curve over Fq. We will discuss how to factorize the coefficients of the exponential and logarithm series for a Hayes module over such a curve. This allows us to obtain v-adic convergence results for such exponential and logarithm series, for v a “finite” prime. As an application, we can show that the v-adic Goss L-value Lv(1,Ψ) is log-algebraic for suitable characters Ψ.

Consolidation settlement of vertically loaded pile groups in multilayered poroelastic soils

Pile groups are commonly used as the foundations of many structures including those used in transportation infrastructures. Consolidation settlement of a pile foundation is an important design parameter. A theoretical model is developed in this study to estimate the consolidation settlement and axial load transfer of vertically loaded pile groups in multilayered poroelastic soils. The multilayered saturated soil is modeled according to Biot’s poroelasticity theory. In order to determine quasi-static response of pile groups, the interaction problem is first formulated in the Laplace transform domain. Vertical displacement compatibility is enforced at the pile-soil interface to simulate the pile group-soil interaction. Axial deformation of each pile is represented by an exponential series with undetermined coefficients, which are obtained from a variational approach. Vertical displacement influence functions due to a buried uniform vertical load applied to the layered soil are required in the formulation. The application of an exact stiffness matrix method yields the required influence functions. Time-domain solutions are obtained by employing a numerical Laplace inversion method. Numerical results for time-dependent vertical stiffness and consolidation settlement are presented for different pile group configurations, layer profiles, pile elastic moduli and pile lengths.