Exponential Series Research Articles

Solutions to SIR/SEIR epidemic models with exponential series: Numerical and non numerical approaches

This study revisits the mathematical SIR/SEIR epidemic models, aiming to introduce novel exponential-type series solutions. Beyond standard non-dimensionalization, we implement a successful rescaling technique that reduces the parameter count in classical epidemiology. Consequently, solutions for the SIR model are determined solely by the basic reproduction number and initial infected fractions. Similarly, the SEIR model requires only the transmission-to-recovery ratio and initial exposed fractions. We present both numerical and non numerical solutions, alongside elucidating the limitations on the existence of exponential-type series solutions. Our analysis reveals that these solutions are valid under two key conditions: endemic situations and early epidemic stages, where the basic reproduction number is close to one. We graphically illustrate the range of physical parameters guaranteeing the existence of non numerical exponential series solutions. However, for epidemic/pandemic outbreaks with significantly higher reproduction numbers, achieving complete convergence of the exponential series across the entire physical domain becomes impossible. In such cases, we divide the exponential series solution into two zones: from initial time to peak time and from peak time to the final epidemic time. For the first zone, where convergence is slow, we successfully employ Padé approximants to accelerate the convergence of the series. This accelerated solution is then smoothly joined to the second zone solution once the peak time is identified within the first region. The presented non numerical solutions are envisioned to serve as valuable benchmarks for testing and enhancing other numerical approaches used to solve epidemic models and their variants.

Exponential series approximation of the SIR epidemiological model

IntroductionThe SIR (Susceptible-Infected-Recovered) model is one of the simplest and most widely used frameworks for understanding epidemic outbreaks.MethodsA second-order dynamical system for the R variable is formulated using an infinite exponential series expansion, and a recursion relation is established between the series coefficients. A numerical approximation scheme for the R variable is also developed.ResultsThe proposed numerical method is compared to a double exponential (DE) nonlinear approximate analytic solution, which reveals two coupled timescales: a relaxation timescale, determined by the ratio of the model’s time constants, and an excitation timescale, dictated by the population size. The DE solution is applied to estimate model parameters for a well-known epidemiological dataset—the boarding school flu outbreak.DiscussionFrom a theoretical standpoint, the primary contribution of this work is the derivation of an infinite exponential, Dirichlet, series for the model variables. Truncating the series yields a finite approximation, known as a Prony series, which can be interpreted as a sequence of coupled exponential relaxation processes, each with a distinct timescale. This apparent complexity can be approximated well by the DE solution, which appears to be of main practical interest.

Synthesis of maximally sparse conformal circular arc array with a required beam pattern by unitary matrix pencil method

This paper extends the unitary matrix pencil (UMP) method to synthesize maximally sparse conformal circular-arc array with a required beam pattern. Due to the nonlinearity between the circular-arc array pattern and its element pattern, Fourier transform preprocessing for the required beam pattern is introduced to achieve a mathematical expression, i.e., sum of a series of undamped complex exponentials, which is related to array element positions and their excitations. Then, the UMP method is used to determine the reduced number of elements and their position distributions. Moreover, the complex excitations of array elements are reconstructed by obtaining the least-square solution of an over-determined equation. A set of examples for synthesizing sparse conformal circular-arc arrays with different desired patterns and E-type patch element including the mutual coupling are conducted. Results show that the proposed UMP method can achieve a considerably lower pattern reconstruction error with a reduced number of elements than results in the literature, which demonstrates its effectiveness and robustness.

Peramalan Cadangan Devisa Menggunakan Metode Double Smoothing Exponential dan Metode Fuzzy Time Series

This study aims to forecast the amount of foreign exchange reserves using two methods, namely Double Smoothing Exponential and Fuzzy Time Series. This foreign exchange reserve data is time series data, namely monthly data starting from January 2017 to February 2024 with a total of 86 monthly data. Furthermore, to see the level of accuracy of the forecast results, the Mean Absolute Percentage Error (MAPE) and MSE values ​​are used where the smaller the error value produced, the more accurate the forecast results from the Double Smoothing Exponential Holt and Fuzzy Time Series Cheng methods. The results of forecasting the amount of foreign exchange reserves in March 2024 using the Holt method are 144434.19 and for the Cheng method are 140008.032. The double exponential smoothing method has an MSE value of 8625715.609 and a MAPE value of 1.69%. Meanwhile, with the fuzzy time series method, the MSE value is 11651319.400 and the MAPE value is 1.96%. Forecasting using these two methods has a very good level of accuracy, because it has a MAPE value below 10%. Based on the results of this analysis, it can be seen that the double exponential smoothing method from Holt has a smaller forecast error size than the Cheng fuzzy time series method. Thus, it can be concluded that the Double Exponential Smoothing method is a better method for predicting the amount of foreign exchange reserves.

Intelligent trapezoid and variable weight combination-based reconstructed GM model

The GM(1,1) model's prediction accuracy is significantly influenced by the accuracy of background value estimation. The traditional trapezoidal background value can only be applied to a specific data sequence. Therefore, this study proposes a GM(1,1) model background value reconstruction approach based on the combination of intelligent trapezoidal and variable weights in order to increase the model’s application as well as its prediction accuracy. The trapezoidal background value function with slope and point position parameters is called model I. Then, a set of point position parameter sequences, with a new background value function is constructed, called model II. A genetic algorithm is utilized to seek for the values of the parameters to be determined in both models I and II. The results showed that for the exponential growth data series, model I and II have higher prediction accuracy compared to traditional models. For data sequences, taking the traffic volume series of a road from 2014 to 2023, the prediction accuracy of this paper's model I method can be improved by 0.3643 % and 0.2725 % compared with Deng's and Wang's models. The prediction accuracy of this paper's model II method has been further improved by 0.1075 % compared with that of model I.

Exponential stochastic volatility model with Laplace returns and its variants

This paper explores the exponential stochastic volatility model generated by first-order exponential Markov sequences to model financial time series. The stationary exponential Markov sequences used to generate the volatility model are the linear exponential autoregressive model, minification model, new exponential autoregressive time series model, and an exponential mixture of autoregressive and minification processes. The statistical properties of the models are studied and the estimation of model parameters is carried out using the generalized method of moments. An approximate Kalman filtering method and a non-linear filtering approach are employed to diagnose the models. To verify the correctness of the estimations, simulation studies are conducted, and two real datasets are used to demonstrate how the models are applied. Furthermore, a comparison of the suggested models is performed using the measures root mean square error, mean absolute error, and Value at Risk via a backtesting procedure.

UHV AC/DC power grid geomagnetically induced currents harmonic characteristics analysis

The power grid’s geomagnetically induced currents (GIC) during geomagnetic storms, are associated with a broad and abrupt harmonic impact, with the potential for grid paralysis and extensive power outages in severe scenarios. Therefore, the harmonic characteristics of geomagnetically induced currents in China’s ultra-high voltage (UHV) AC/DC power grid are analyzed in this paper. Firstly, a harmonic equivalent model of transformer GIC is established, and the injection of harmonics into the UHV AC/DC power grid during GIC events is quantified.Secondly, a harmonic state-space model is established by combining the 12-pulse converter switching function theory with the exponential Fourier series. The harmonic transmission and dynamic coupling characteristics of GIC in the UHV AC/DC power grid are analyzed using the harmonic equivalent model of the main transformer GIC and the harmonic equivalent model of the converter transformer GIC. Finally, a harmonic simulation model of the UHV AC/DC power grid under GIC is constructed using PSCAD, and the distribution of harmonics in the UHV AC/DC power grid under GIC is calculated. The calculations indicate that with 30 A injected GIC in the sending end transformers, the 2nd harmonic current on the sending end network side alone is 61.45 A, and the 3rd harmonic voltage on the valve side is 8.57 kV. The characteristics of harmonic transmission in the power grid result in a 2nd harmonic current of 34.87 A on the receiving end network side. Simultaneously, the total harmonic distortion of the AC power grid at the sending end is 2.35 %, surpassing the permissible limit established by the national standard. Consequently, this issue should be prioritized by the relevant power authorities.

Prediction Of the Future Development of New Energy Vehicles in China Based on Grey Model and Times Series

Energy shortages have led countries to promote the new energy vehicle industry for sustainable development goals. Establish the grey model and exponential smoothing-based time series model to predict the future development of new energy vehicles in China. And concluded that compared to 2022, imports of new energy vehicles are projected to decrease by 77% in 2032, exports and infrastructure are projected to increase by 1.5 times, the cruising capacity of new energy vehicles is projected to triple, and the new energy vehicle industry will increase its market share by 12% in China.

Approximation Properties of Exponential Sampling Series in Logarithmic Weighted Spaces

Approximation Properties of Exponential Sampling Series in Logarithmic Weighted Spaces

A hybrid nonparametric multivariate density estimator with applications to risk management

. Multivariate density estimation is plagued by the curse of dimensionality in theory and practice. We propose a hybrid density estimator of a multivariate density f that combines the strengths of the kernel estimator and the exponential series estimator. This estimator refines a preliminary kernel estimate f ̂ 0 with a multiplicative correction that estimates the ratio r = f / f ̂ 0 with an exponential series estimator. Thanks to the consistency of the pilot estimate, the coefficients of the series expansion tend to approach zero asymptotically. Accordingly, we design a thresholding method for basis function selection. A major obstacle of multivariate exponential series estimator is the calculation of its normalization factor. We resolve this difficulty with Monte Carlo integration, using the pilot kernel estimate as the trial density for importance sampling. This approach greatly enhances the practicality of the hybrid estimator. Numerical simulations demonstrate the good finite sample performance of the hybrid estimator. We present one empirical application in financial risk management.

The Role of Smart City in Achieving Sustainable Development: Google Trends Analysis and Exponential Time Series Smoothing Models

The concept of sustainable smart urban development is an important element of the modern discourse on the future development of cities, regions, and the world as a whole, a key component of a strategy aimed at introducing innovations, improving the life quality of citizens, ensuring inclusiveness and sustainability of local development. It integrates the concept of Smart City as the application of advanced innovative technologies to solve urban problems and the concept of Sustainable Development as a balancing of socio-economic and environmental interests to achieve the Sustainable Development Goals. While discussing the problems and prospects introducing any social models or strategies, it is important to understand the level of public interest in this issue, the level of citizens’ awareness, the readiness to receive new information, and the readiness for change. The article is focused on the analysis of retrospective dynamics (from 01.01.2010 to 01.12.2023) and economic and mathematical modelling of forecast trends in changing public interest in the concepts of “Smart City” and “Sustainable Development” (identified using the Google Trends tool). The analysis was carried out for the world as a whole and for Ukraine specifically. The study identifies and describes the evolution of changes in public interest in these topics, identifies the time points of a sharp surge in interest, and conducts a comparative analysis in the content (trends for these two search queries) and spatial context (trends in Ukraine compared to global trends). The countries of the world and regions of Ukraine where user interest in the relevant topics dominates are ranked, and the countries of the world and regions of Ukraine where user interest in the relevant topics dominates are ranked, the TOP-10 are identified. The factors whose influence is relevant to the public interest dynamics are identified (consumer trends, media activity, dynamics of the introduction of relevant concepts, cultural, linguistic, and historical contexts, the general level of education, digital inclusion and consciousness of the population of different countries, etc.) Using the exponential smoothing method, models were built to predict the dynamics of time series – the number of queries from Ukrainian Internet users regarding the concepts of “Smart City” and “Sustainable Development”. Additive forecasting models, which differ in the trend component, are obtained.

Apollonian packings and Kac-Moody root systems

We study Apollonian circle packings using the properties of a certain rank 4 indefinite Kac-Moody root system Φ \Phi . We introduce the generating function Z ( s ) Z(\mathbf {s}) of a packing, an exponential series in four variables with an Apollonian symmetry group, which is a symmetric function for Φ \Phi . By exploiting the presence of affine and Lorentzian hyperbolic root subsystems of Φ \Phi , with automorphic Weyl denominators, we express Z ( s ) Z(\mathbf {s}) in terms of Jacobi theta functions and the Siegel modular form Δ 5 \Delta _5 . We also show that the domain of convergence of Z ( s ) Z(\mathbf {s}) is the Tits cone of Φ \Phi , and discover that this domain inherits the intricate geometric structure of Apollonian packings.

Simultaneous estimation of a model-derived input function for quantifying cerebral glucose metabolism with [18F]FDG PET

BackgroundQuantification of the cerebral metabolic rate of glucose (CMRGlu) by dynamic [18F]FDG PET requires invasive arterial sampling. Alternatives to using an arterial input function (AIF) include the simultaneous estimation (SIME) approach, which models the image-derived input function (IDIF) by a series of exponentials with coefficients obtained by fitting time activity curves (TACs) from multiple volumes-of-interest. A limitation of SIME is the assumption that the input function can be modelled accurately by a series of exponentials. Alternatively, we propose a SIME approach based on the two-tissue compartment model to extract a high signal-to-noise ratio (SNR) model-derived input function (MDIF) from the whole-brain TAC. The purpose of this study is to present the MDIF approach and its implementation in the analysis of animal and human data.MethodsSimulations were performed to assess the accuracy of the MDIF approach. Animal experiments were conducted to compare derived MDIFs to measured AIFs (n = 5). Using dynamic [18F]FDG PET data from neurologically healthy volunteers (n = 18), the MDIF method was compared to the original SIME-IDIF. Lastly, the feasibility of extracting parametric images was investigated by implementing a variational Bayesian parameter estimation approach.ResultsSimulations demonstrated that the MDIF can be accurately extracted from a whole-brain TAC. Good agreement between MDIFs and measured AIFs was found in the animal experiments. Similarly, the MDIF-to-IDIF area-under-the-curve ratio from the human data was 1.02 ± 0.08, resulting in good agreement in grey matter CMRGlu: 24.5 ± 3.6 and 23.9 ± 3.2 mL/100 g/min for MDIF and IDIF, respectively. The MDIF method proved superior in characterizing the first pass of [18F]FDG. Groupwise parametric images obtained with the MDIF showed the expected spatial patterns.ConclusionsA model-driven SIME method was proposed to derive high SNR input functions. Its potential was demonstrated by the good agreement between MDIFs and AIFs in animal experiments. In addition, CMRGlu estimates obtained in the human study agreed to literature values. The MDIF approach requires fewer fitting parameters than the original SIME method and has the advantage that it can model the shape of any input function. In turn, the high SNR of the MDIFs has the potential to facilitate the extraction of voxelwise parameters when combined with robust parameter estimation methods such as the variational Bayesian approach.

On some coefficients of the Artin–Hasse series modulo a prime

Let p be an odd prime, and let ∑n=0∞anXn∈Fp[[X]] be the reduction modulo p of the Artin–Hasse exponential series. We obtain a polynomial expression for akp in terms of those arp with r<k, for even k<p3−1. A conjectural analogue covering the case of odd k<p can be stated in various polynomial forms, essentially in terms of the polynomial γ(X)=∑n=1p−2(Bn/n)Xp−n, where Bn denotes the nth Bernoulli number.We prove that γ(X) satisfies the functional equation γ(X−1)−γ(X)=£1(X)+Xp−1−wp−1 in Fp[X], where £1(X) and wp are the truncated logarithm and the Wilson quotient. This is an analogue modulo p of a functional equation, in Q[[X]], established by Zagier for the power series ∑n=1∞(Bn/n)Xn. The proof of our functional equation establishes a connection with a result of Nielsen of 1915, of which we provide a fresh proof. Our polynomial framing allows us to derive congruences for certain numerical sums involving divided Bernoulli numbers.

Solution of Integral of the Fourth Power of a Finite-Length Exponential Fourier Series

For a periodic function in the form of a finite-length exponential Fourier series (i.e., a discrete finite Fourier transform), this work derives an analytical solution to the definite integral of the fourth power of the function across its periodic interval.

Ensemble forecasts in reproducing kernel Hilbert space family

A methodological framework for ensemble-based estimation and simulation of high dimensional dynamical systems such as the oceanic or atmospheric flows is proposed. To that end, the dynamical system is embedded in a family of reproducing kernel Hilbert spaces (RKHS) with kernel functions driven by the dynamics. In the RKHS family, the Koopman and Perron–Frobenius operators are unitary and uniformly continuous. This property warrants they can be expressed in exponential series of diagonalizable bounded evolution operators defined from their infinitesimal generators. Access to Lyapunov exponents and to exact ensemble based expressions of the tangent linear dynamics are directly available as well. The RKHS family enables us the devise of strikingly simple ensemble data assimilation methods for trajectory reconstructions in terms of constant-in-time linear combinations of trajectory samples. Such an embarrassingly simple strategy is made possible through a fully justified superposition principle ensuing from several fundamental theorems.

Deterministic and stochastic flexural behaviors of laminated composite thin-walled I-beams using a sinusoidal higher-order shear deformation theory

In this study, a novel sinusoidal higher-order shear deformation thin-walled beam theory is presented to examine the effects of material properties and external load uncertainty on static responses of laminated composite thin-walled beams with open sections. The solution for the deterministic flexural analysis is based on Hamilton’s principle and Ritz-type exponential shape function series. Several mechanical parameters of laminated composite materials are randomized and plugged into the beam solver to investigate the thin-walled beam’s stochastic flexural behaviors. The computational cost and accuracy of the polynomial chaos expansion (PCE) method with both projection and linear regression approaches are presented and evaluated by comparing its results with crude Monte Carlo simulation (MCS). This comparison allows for a thorough assessment of the PCE method’s performance. Additionally, a sensitivity analysis is conducted to compare the relative significance of the uncertainty in material properties and loads on the stochastic responses. The supervised training of the artificial neural network based on the MCS beam data is also conducted and compared to the PCE and MCS methods. The findings about the stochastic outputs are introduced in various statistical metrics and illustrations to demonstrate the influences of material properties’ randomness on different laminated composite thin-walled beam configurations.

Predicting future global temperature and greenhouse gas emissions via LSTM model

This work is executed to predict the variation in global temperature and greenhouse gas (GHG) emissions resulting from climate change and global warming, taking into consideration the natural climate cycle. A mathematical model was developed using a Recurrent Neural Network (RNN) with Long–Short-Term Memory (LSTM) model. Data sets of global temperature were collected from 800,000 BC to 1950 AD from the National Oceanic and Atmospheric Administration (NOAA). Furthermore, another data set was obtained from The National Aeronautics and Space Administration (NASA) climate website. This contained records from 1880 to 2019 of global temperature and carbon dioxide levels. Curve fitting techniques, employing Sin, Exponential, and Fourier Series functions, were utilized to reconstruct both NOAA and NASA data sets, unifying them on a consistent time scale and expanding data size by representing the same information over smaller periods. The fitting quality, assessed using the R-squared measure, ensured a thorough process enhancing the model's accuracy and providing a more precise representation of historical climate data. Subsequently, the time-series data were converted into a supervised format for effective use with the LSTM model for prediction purposes. Augmented by the Mean Squared Error (MSE) as the analyzed loss function, normalization techniques, and refined data representation from curve fitting the LSTM model revealed a sharp increase in global temperature, reaching a temperature rise of 4.8 °C by 2100. Moreover, carbon dioxide concentrations will continue to boom, attaining a value of 713 ppm in 2100. In addition, the findings indicated that the RNN algorithm (LSTM model) provided higher accuracy and reliable forecasting results as the prediction outputs were closer to the international climate models and were found to be in good agreement. This study contributes valuable insights into the trajectory of global temperature and GHG emissions, emphasizing the potential of LSTM models in climate prediction.

Bivariate generalized Kantorovich-type exponential sampling series

Bivariate generalized Kantorovich-type exponential sampling series

Computation of Volterra and Fredholm Integro-Differential Nonlinear Boundary Value Problems by a Modified Regula Falsi-Bisection Shooting Approach

A modified Regula Falsi shooting method approach is deployed to solve a set of Volterra and Fredholm integro- nonlinear boundary value problems. Efforts to solve this class of problems with traditional shooting methods have generally failed and the outcomes from most domain based approaches are often plagued by ill conditioning. Our modification is based on an exponential series technique embedded in a shooting bracketing method. For the purposes of validation, we initially solved problems with known closed form solutions before considering those that do not come with this property but are singular, genuinely nonlinear, and are of practical interest. Although in most of these tests, convergence was found to be super linear, the errors decreased monotonically after few iterations. This suggests that the method is robust and can be trusted to yield faithful results and by far surpasses in simplicity various other techniques that have been applied to solve similar problems. In order to buttress the effectiveness and utility of this approach, we display both the graphical and error analysis outcomes. And for each test case, the method can be seen to demonstrate the closeness of numerically generated results to the analytical solutions.