This paper introduces a generalized class of Weyl-type, Witt-type, and non-associative algebras constructed over an exponential–polynomial (expolynomial) framework. For fixed scalars ι1,…,ιr∈A and for fixed integers p=(p1,…,pn)∈Nn, we define the F-algebra Fe±xpeιx,eAx,xA, an expolynomial ring over a field F of characteristic zero, where A is an additive subgroup of F containing Z. This formulation extends the classical Weyl algebra through the integer power parameter p, which generates a family of non-isomorphic simple algebras. The corresponding Weyl-type algebra AF[e±xpeιx,eAx,xA], the Witt-type Lie algebra WF[e±xpeιx,eAx,xA], and their non-associative variants are examined in detail. The simplicity, grading, and automorphism structures of these algebras are established, and the dependence of these properties on the deformation parameter p is analyzed. All the constructed Weyl-type algebras, the corresponding Witt-type Lie algebras, and the non-associative algebras are shown to be simple under derivation structures. Many naturally occurring subalgebras, such as the integer-coefficient subalgebra AZ[e±xpeιx,eAx,xA], are also proven to be simple. Our analysis reveals that different choices of p result in non-isomorphic algebraic structures while retaining non-commutativity. The results obtained generalize several existing constructions of Weyl-type algebras and lay the theoretical foundation for further developments in transcendental and non-commutative algebraic frameworks.

Generic drug entry into the pharmaceutical market typically leads to a substantial decline in originator sales. Understanding the extent and trajectory of this erosion is essential for effective lifecycle management and strategic planning. This study quantified sales erosion after generic entry for originator drugs approved in the United States between 2010 and 2019 and developed a model to predict year-specific sales retention based on key product- and market-level characteristics. A total of 140 originator drugs were analyzed using FDA approval records and sales data from Evaluate Pharma. Five-year retention patterns were modeled using a three-parameter exponential decay function. Subgroup analyses were conducted by year of generic entry, therapeutic class, and product-specific features. A polynomial regression model using 700 product-year observations incorporated three binary market indicators and linear and quadratic time terms. Sales retention declined from 73.1% in the first year after generic entry to 31.7% by year five. The exponential decay model demonstrated a strong goodness-of-fit (root mean squared error [RMSE] = 0.006), capturing the initial steep decline and subsequent stabilization. Subgroup analyses showed faster erosion for blockbuster drugs and in markets with multiple first-generation generics. The regression model explained 96.4% of annual variation in retention (RMSE = 0.033), accounting for product and market heterogeneity. Sales decline after generic entry follows a predictable yet heterogeneous trajectory shaped by product and market factors. Exponential decay and polynomial regression models together offer a robust framework for forecasting sales retention and guiding strategic decisions in the pharmaceutical industry.

This study presents a multi-temporal analysis of displacement data from Sentinel-1 Synthetic Aperture Radar data at Hurst Castle with datasets sourced from the European Ground Motion Service, and temperature records from Ventnor Park and Otterbourne, with datasets sourced from the CEDA archive. To reduce speckle noise inherent in Synthetic Aperture Radar time series, three speckle filtering techniques, Boxcar, Lee, and Frost, were applied. The filtered displacement and temperature data were modelled using a range of mathematical functions, linear, quadratic, sinusoidal, step function, phenomenological, exponential, Lagrangian, and higher-order polynomial models. Model performance was evaluated using a comprehensive set of error metrics, including Root Mean Squared Error, Least Mean Squares, Sum of Squared Errors, Akaike Information Criterion, and Bayesian Information Criterion. Further regression analysis involved Median Absolute Error, Mean Absolute Percentage Error, Symmetric Mean Absolute Percentage Error, coefficient of determination, adjusted coefficient of determination, and the Durbin–Watson statistic to assess autocorrelation in residuals. The best fitting models were determined based on the coefficient of determination and Durbin– Watson values to evaluate the robustness of model selection across different filters and data sources. The study underscores the importance of integrating displacement and temperature time series modelling for environmental monitoring and structural risk assessment at heritage sites.

We introduce a new class of block methods based on a hybrid basis of Hermite probabilists’ polynomials and exponential polynomials. The proposed techniques exploit the complementary strengths of both families, offering enhanced accuracy, stability, and flexibility compared with schemes built on a single polynomial type. The methods employ interpolation and dynamic collocation and are formulated within a second-derivative framework. To strengthen their structure, additional terms are generated through the recurrence relation of Hermite probabilists’ polynomials, whose orthogonality provides further advantages over exponential functions. Since the accuracy of numerical methods depends largely on discretization constants, this hybridization, together with the clustered mesh points, help reduce discretization errors and error constants while maintaining stability. Rigorous theoretical analysis establishes A-stability and convergence of the schemes. Although their algebraic order of convergence is relatively low, numerical experiments demonstrate that the methods achieve improved accuracy and competitive precision factors compared with existing block approaches. These results suggest that hybrid polynomial bases provide a promising pathway for the development of robust and efficient block algorithms in numerical analysis.

Nonlinear eigenvalue solver for spectral element of beam structures: An exponential matrix polynomial approximation with weighted residual method

This paper deals with stability problems for an important class of differential equations of neutral type and functional difference equations with constant delays, continuous time and matrix coefficients. Based on previous work of the second author, we give an adapted proof of a known point-wise characterization of the closure set of the real parts of the characteristic roots associated with such systems for the case where all delays vary independently of each other. We also provide explicit methods to construct quasipolynomial matrices whose characteristic roots coincide with the zeros of exponential polynomials with nonzero frequencies linearly independent over the rational numbers. As a main result, sufficient conditions for stability are derived from the spectral analysis of such classes of quasipolynomial matrices. These results are illustrated by means of several examples.

Post-widder type operators through the lens of truncated exponential polynomials

The objective of this work is to investigate the global existence, general decay and blow-up results for a class of p-Biharmonic-type hyperbolic equations with delay and acoustic boundary conditions. The global existence of solutions has been obtained by potential well theory and the general decay result of energy has been established, in which the exponential decay and polynomial decay are only special cases, by using the multiplier techniques combined with a nonlinear integral inequality given by Komornik. Finally, the blow-up of solutions is established with positive initial energy. To our knowledge, the global existence, general decay, and blow-up result of solutions to p-Biharmonic-type hyperbolic equations with delay and acoustic boundary conditions has not been studied.

The choice of an approach for accurate forecasting of photovoltaic power plants and modeling of power systems with renewable energy sources depends on the availability of input data, time horizon, installation location, and weather variables. Our goal is to improve mathematical models and find new solutions to improve the performance of predicting the operation of photovoltaic power plants in energy systems using regression models. There is a problem of predicting the amount of electricity generated by photovoltaic plants in Ukraine. The data for 3 years of daily electricity production is used. The problem is solved by the application of the least squares’ method to estimate unknown parameters of the suggested dependence between the length of daylight and daily solar power generation. The main assumption is the following: daily solar power generation can be given as a linear combination of some exponential polynomials with the independent variable as the duration of the sunny day. The regression model is reduced to a system of significantly non-linear equations, which is solved numerically by the iteration method. Regression models were built using R software for big data analysis. Another novelty moment concerns grouping of the data according to the same length of daylight, and then three values were found for each such group: maximum, minimum, and average value. The proposed moving average regression models with the usage of exponential polynomials as approximating functions admit a small standard residual error between the exact values and the predicted values of solar power generation (0.3022 kWh). The forecasting horizon is one year. The significance of the created mathematical forecasting models demonstrates a possibility of using the daylight duration as a parameter in forecasting tasks, as well as evaluating the prospects to consider the parameters that affect the performance of photovoltaic power plants.

Half-cell potential (HCP) is widely acknowledged as a nondestructive method for assessing the durability of concrete, although the variability in environmental and material conditions compromises its accuracy. The reliability of traditional prediction models, which are often derived from limited data, is questionable under various conditions. This study employed a Bayesian-enhanced probabilistic model to predict corrosion reinforcement using HCP, addressing both known and unknown uncertainties. Constructed as a piecewise function, the model integrates insights from the literature with the results of an accelerated corrosion experiment conducted by the research team, thereby validating the effectiveness of the probabilistic approach. This study also examines the influence of prior knowledge on the accuracy of predictions. The findings revealed a biphasic relationship between HCP and the corroded mass reduction ratio. HCP decreased exponentially with a corroded mass reduction ratio below 15%, whereas beyond this threshold, the decline became more pronounced, modeled by a combination of exponential and cubic polynomial functions. These results underscore the critical role of employing a piecewise function to accurately define the relationship between HCP and corrosion in reinforced concrete, thereby providing a solid foundation for future durability assessments.

Application of Meyer’s theorem on quasicrystals to exponential polynomials and Dirichlet series

Abstract This study focuses on the problem of h-dichotomy for skew-evolution cocycles within Banach spaces. It outlines the necessary conditions and sufficient conditions for this framework, along with those for the notable concepts like exponential dichotomy, polynomial dichotomy and uniform h-dichotomy. These conditions are established through the use of strongly invariant families of projectors.

ABSTRACTThis paper presents a novel and efficient numerical scheme for solving multidimensional optimal control problems governed by the Caputo–Fabrizio fractional derivative (MD‐CFOCP). Unlike traditional approaches based on singular kernel derivatives, our method exploits the nonsingular and exponential nature of the Caputo–Fabrizio operator, enabling a more stable and realistic modeling of systems with short‐term memory and smooth dynamics. The core of the proposed technique lies in approximating both the state and control functions using truncated exponential polynomials (TEPs), combined with an operational matrix of fractional integration tailored to the Caputo–Fabrizio framework. This transforms the original fractional control problem into a system of algebraic equations, which is easier to analyze and solve numerically. A rigorous theoretical analysis is carried out, including error bounds and exponential convergence results. Several numerical examples are provided to demonstrate the high accuracy, computational efficiency, and practical relevance of the proposed method, including comparisons with existing schemes. This study provides a valuable computational framework for researchers and practitioners dealing with fractional dynamic systems with nonsingular memory kernels.

We investigate exponential ideals within the context of exponential polynomial rings over exponential fields. We establish two distinct notions of maximality for exponential ideals and explore their relationship to primeness. These three concepts—prime, maximal, and E-maximal—are shown to be independent, in contrast to the classical scenario. Furthermore, we demonstrate that, over an algebraically closed field K, the correspondence between points of K n and maximal exponential ideals of the ring of exponential polynomials breaks down. Finally, we introduce and characterize exponential radical ideals.

Holstein bull calves received a one-time intraperitoneal injection of Diquat to explore its effects on growth, body frame, blood oxidation indices, fecal scores, and pathogenic bacteria in weaned calves. A total of twelve 70-day-old Holstein bull calves with similar body weight (BW) and body condition were randomly assigned to one of four treatments. The treatments were as follows: Control: calves were injected with 0 mg/kg BW Diquat in 0.9% sterilized saline; treatments 6, 8, and 10 mg/kg BW Diquat, respectively. The experimental period lasted for 24 days. Measurements of BW, average daily gain (ADG), fecal scores, frame gains, fecal pathogen count, and blood samples for monitoring oxidative stress were collected on days 0, 6, 12, 18, and 24. Data were analyzed using a randomized complete block design, with days considered as a repeated measurement. In addition, exponential polynomial contrasts were used to assess the linear, quadratic, and cubic treatment responses. Growth performance (BW) and ADG showed a cubic response (p < 0.02), initially decreasing and then increasing with higher Diquat dosages. Fecal scores and fecal ratios exhibited a quadratic response (p < 0.02), rising at a diminishing rate as Diquat injection dosages increased. Frame gains for body slope, body length, hip height, and abdominal girth displayed a linear decrease (p < 0.03) with increasing Diquat injection dosages. Serum aspartate aminotransferase, glutathione, total antioxidant capacity, catalase, malondialdehyde, cortisol, and noradrenaline concentrations revealed a linear increase (p < 0.01) in response to higher Diquat injection dosages, while alanine transaminase, superoxide dismutase, and glutathione peroxidase demonstrated a quadratic response (p < 0.02), increasing at a diminishing rate. Fecal Escherichia coli concentrations demonstrated a cubic response (p < 0.01), while Staphylococcus aureus and Salmonella-Shigella demonstrated linear increases (p < 0.01) with increasing Diquat dosages. Diquat injection induced oxidative stress, leading to reduced growth performance, along with increased serum oxidative stress indices, fecal scores, and fecal pathogens, a response that may persist for up to 24 days. An optimal dosage of 8 mg/kg BW is proposed as a benchmark for elucidating oxidative stress to evaluate future technologies aimed at reducing, eliminating, or preventing oxidative stress.

In this paper, the authors introduce a new Weighted Essentially Nonoscillatory (WENO) scheme. This scheme is founded on exponential functions and utilizes arc-length smoothness indicators. The primary purpose of this WENO scheme is to provide accurate approximations for the viscosity numerical solutions of Hamilton–Jacobi equations. The arc-length smoothness indicators are derived from the derivatives of reconstructed polynomials within each sub-stencil. These smoothness indicators play a crucial role in approximating the viscosity numerical solutions of Hamilton–Jacobi equations, ensuring high-resolution results and minimizing absolute truncation errors. Numerous numerical tests have been carried out and presented to demonstrate the performance capabilities and numerical accuracy of the proposed scheme, comparing it to several traditional WENO schemes.

In this paper, we study the propagation of the second spatial-velocity moments for the kinetic Cucker–Smale model with non-compact spatial support. In contrast to compact support, non-compact support leads to a lower bound of zero for the communication weight, which makes the previous approach break down. To address this challenge, we consider two types of initial distributions: exponential decay distributions and polynomial decay distributions. Moreover, our approach uses the infinite-particle mean-field approximation as an intermediary step to analyze the kinetic Cucker–Smale model, with conservation laws of mass and momentum. When initial distributions belong to the aforementioned types of decaying classes and coupling strength exceeds a certain threshold, we show the weak flocking behavior of the kinetic Cucker–Smale model. Specifically, the second velocity moment of the solution centered around the initial average velocity converges to zero, and the second spatial moment around the position of the center of mass remains uniformly bounded in time. The emergence of weak flocking behavior illustrates that even for non-compact support, a certain degree of aggregation can be maintained for the kinetic Cucker–Smale model, as long as the initial distribution exhibits relative concentration.

In this study, we propose a novel weighted compact nonlinear scheme that enhances the performance of the known third-order scheme and addresses the issue of inconsistent accuracy between its nonlinear reconstruction step and compact weighting step. To more accurately approximate steep gradients and high oscillations, an interpolation method based on exponential polynomials with a shape parameter is introduced. Specifically, a method for selecting locally optimised parameters that achieve fourth-order accuracy at non-critical points is proposed. Additionally, we introduce a smoothness indicator based on exponential polynomials, which increases the nonlinear weights of less smooth sub-stencils and improves the resolution of the scheme. To demonstrate the optimal numerical accuracy and the shock-capturing capabilities of the proposed scheme, numerical results are presented.

Approximation processes by multidimensional Bernstein-type exponential polynomials on the hypercube

Thailand is currently grappling with a severe dengue fever outbreak, with a rising threat to public health as the rainy season and El Niño draw near. This year has witnessed a troubling surge in dengue cases, prompting the Ministry of Public Health (MoPH) to issue warnings that the numbers may hit a three-year peak. Dengue outbreaks in Thailand have historically followed a cyclical pattern, excluding COVID-19 years. This research employs data analysis and predictive modeling to forecast the forthcoming dengue case numbers in Thailand, facilitating better public health preparedness. It also incorporates data visualization for enhanced data exploration. Various forecasting models, including Exponential Smoothing, Polynomial Fitting and Random Forest, are deployed to predict dengue cases within the constraints of our data. This study offers valuable insights into the potential trajectory of dengue cases in Thailand, aiding proactive measures to combat the outbreak.