Faster, simpler, and more precise calibration curves: Expanding the scope of continuous calibration.

Abstract Potential evapotranspiration (PET), defined as the evapotranspirative flux from a region under fully saturated conditions, is a critical variable in hydrologic modeling, water stress assessment, and understanding ecosystem responses to climate. The widely used Priestley–Taylor method provides a simple, low-data requirement approach for estimating PET. However, it uses a fixed coefficient (α PT = 1.26) in most applications, but this oversimplification neglects biome-specific variability, limiting its accuracy across diverse environments. Although numerous studies have attempted to derive dynamic characterizations of α PT , most estimates are developed for either obtaining reference evapotranspiration (ET 0 ) or actual evapotranspiration (ET). Consequently, these approaches do not provide α PT that accounts for ecosystem-specific aerodynamic and plant conductance constraints required to fully represent true ecosystem-level PET. In this study, we utilized 3,128 site-years of eddy covariance data from 246 FluxNet sites worldwide to optimize α PT values across a broad range of biomes. Results showed significant spatial and seasonal variability in α PT , with higher values in forests and winter months and lower values in savannas’ summer. Temperature and radiation emerged as key drivers of this variability. Using the influencing variables, we next derived functional equations to estimate α PT based on key bio-environmental variables. These equations yielded demonstrable improvements in PET estimates, and can be directly incorporated into land surface and hydrologic models, and generation of remote sensing products. Furthermore, PET derived using our simplified functional equations of α PT , when applied to obtain the Evaporative Stress Index (ESI) and ET, yielded improved estimates of both ET and plant stress. Overall, these findings offer a more ecologically representative approach to PET estimation using the Priestley-Taylor method, with broad implications for hydrologic modeling and drought assessment.

The Gardner equation is a well-known example of a nonlinear evolution equation that intersects the classical Korteweg-de Vries (KdV) and modified KdV equations, which occur in the real world in internal waves of stratified fluid, ion-acoustic waves in plasma, and in the propagation of signals in nonlinear media. In this paper, we use the Riccati-based Modified Extended Simple Equation Method (RM-ESEM) to build a comprehensive and coherent system of precise analytical solutions of the Gardner equation. The resultant solutions are of trigonometric, hyperbolic, rational, exponential, and mixed forms, which give a thorough insight into the nonlinear dynamics of the underlying waves. These solutions are known to be stable, periodical and transient, which are proven by graphical analysis and their physical relevance. The originality of the work is that it brings together several families of solutions in a single analytic framework, greatly enlarging the familiar solution space and allowing one to see interesting nonlinear phenomena, such as soliton interactions and parameter-dependent wave dynamics. These findings have found theoretical and useful applications in fluid physics, nonlinear optical systems, and plasma physics.

In anisotropic bodies in electromagnetic fields forces arise, leading to waves of stresses and deformations. The author proposed a mathematical method of matrix invariants, which made it possible to construct a characteristic form of the equations of dynamics of anisotropic bodies, which in a number of special cases breaks down into simple scalar equations, which greatly simplifies the numerical calculation. In this work, based on this method, a matrix characteristic form of Maxwell's equations is constructed in an anisotropic medium of various origins, and it is shown that for an orthotropic medium, the obtained matrix equations are reduced to scalar equations

We analysed national Tier 2 greenhouse gas (GHG) inventories for dairy and other cattle in ten East and Southern Africa (ESA) countries to evaluate the variation and predictors of enteric methane (CH 4 ) emission factors (eEFs), and highlight priorities for improvement. We compiled 313 eEFs for dairy cows (n = 26) and other cattle (n = 287) using the Intergovernmental Panel on Climate Change (IPCC) 2019 Tier 2 method and country-specific data, and explored relationships between inputs used in IPCC equations and gross energy intake (GEI, MJ head -1 day -1 ) to identify reliable predictors of eEFs. The eEFs for high (80.74 ± 2.19 kg CH 4 head -1 year -1 ) and low productivity dairy cows (62.26 ± 0.97 kg CH 4 head -1 year -1 ) were 6.5 and 4.5% lower than the IPCC Tier 1a default EFs, respectively. Milk yield solely predicted GEI in dairy cows (r 2 = 0.73, RMSEP = 11.8%). The eEF for low productivity other cattle (56.09 ± 1.0 kg CH 4 head -1 year -1 ) was 16% higher than the IPCC Tier 1a default EF. Our eEF for high productivity systems (56.16 ± 1.0 kg CH 4 head -1 year -1 ) was 6% lower than the IPCC eEF of 60 kg CH 4 head -1 year -1 . Bodyweight, average daily gain and energy digestibility explained 87 and 90% of variation in GEI in growing (RMSEP = 8.1%) and young cattle (RMSEP = 14.6%), respectively. These simplified equations are useful for predicting eEFs in cattle, and highlight the priority country-specific data for Tier 2 inventories in the ESA region.

The analytically integrable Fokas system, arising under the slowly varying envelope approximation for weakly nonlinear and weakly dispersive quasi-monochromatic waves, is used to describe pulse propagation in single-mode optical fibers and is investigated here through symbolic computational techniques. This paper establishes multiple families of exact wave solutions through the combined use of the modified simple equation strategy and the generalized exponential rational function technique. These analytical approaches enable the derivation of diverse solitary and periodic wave structures characterized by adjustable parameters that control the amplitude, shape, and propagation dynamics of the waveform. To demonstrate the physical significance of the derived solutions, comprehensive graphical visualizations are provided, highlighting symmetric propagation features and diverse parameter-dependent behaviors of the wave structures. The flexibility of the obtained solution structures allows for a detailed examination of parameter-dependent wave dynamics and waveform evolution within the considered model. Moreover, a detailed modulation instability analysis is carried out to investigate the stability characteristics of continuous-wave solutions in the context of the Fokas system. The results identify parameter regimes associated with stable and unstable wave propagation, thereby enhancing the understanding of nonlinear instability phenomena in integrable optical models. In general, the study contributes new analytical wave structures, stability interpretations, and parametric insights that extend the applicability of the Fokas system in nonlinear wave theory and optical physics.

Effect of droplet size evolution and distribution of water mist on the thermal radiation attenuation

Self-healing concrete has emerged as a sustainable and cost-effective strategy to reduce maintenance demands, with microbially induced calcium carbonate precipitation (MICP) standing out as one of its most efficient and reliable healing mechanisms. This study presents a numerical model for simulating the MICP healing process in concrete cracks. The model explicitly describes bacterial growth, decay, attachment, and encapsulation through simplified kinetic equations, while incorporating temperature effects on bacterial activity, nutrient consumption, and urea hydrolysis. The reduction in crack permeability is captured by extending the classical parallel-plate flow formulation to account for calcium carbonate deposition. To ensure numerical stability and efficiency, a semi-implicit finite difference algorithm is developed, which explicitly evaluates nonlinear reaction terms while implicitly solving diffusion-advection transport equations. The model is verified against analytical solutions and validated using experimental data. Further simulations investigate capsule- and vascular-based healing strategies. Results demonstrate that the proposed model accurately reproduces the coupled biochemical-transport processes and serves as a reliable and efficient predictive tool for assessing MICP-induced crack healing in concrete. • A novel reactive-transport model is developed to capture temperature-dependent MICP healing. • A new permeability formulation is derived to predict non-uniform crack healing in concrete. • A semi-implicit algorithm is implemented to improve numerical stability and accuracy. • The model is extended to simulate capsule- and vascular-based healing in concrete.

Abstract This paper presents a unified finite element framework based on the Special Euclidean Group SE (3) for modeling multibody systems undergoing large deformations and rotations. Rigid-body dynamics and geometrically exact beam formulations are consistently integrated within a differential geometric framework, exploiting the inherent ability of SE (3) to capture the coupling between translational and rotational motions. The generalized elastic, inertial, and external forces, together with their Jacobians, are systematically derived for both the rigid-body and flexible beam elements. Benefiting from the SE (3) framework, constant mass matrices and consistent inertial force expressions are obtained in the derivation of both rigid-body and flexible beam formulations. Moreover, compact and simplified dynamic equilibrium equations are derived. The Lie generalized-α method adapted to the SE (3) framework is employed to solve the dynamic equilibrium equations, ensuring numerical stability and geometric consistency. The multibody dynamics benchmark validates the numerical stability of the time integration algorithm, while the pure bending test of the beam element demonstrates the capability of the geometrically exact beam formulation to handle large deformations and rotations. Additionally, the dynamic response of a flexible double pendulum confirms the framework’s correctness, showing excellent agreement with the reference solution.

In this study, we investigated the optical soliton structures of the perturbed nonlinear complex Ginzburg–Landau equation (CGLE), which incorporates Kerr law nonlinearity and is influenced by multiplicative white noise, using the Extended Simple Equation Method (ESEM). The CGLE is a fundamental equation in nonlinear science, widely used to describe wave propagation and energy dissipation in systems such as Bose–Einstein condensates, fluid dynamics, plasma physics, fibre optics, and nonlinear optics. We derive various soliton solutions, including bright solitons, peakon solitons, kink wave solitons, periodic solitons, anti-kink wave solitons, dark solitons, singular solitons, mixed solitons, and other solitary wave structures. The physical characteristics of the obtained solutions are illustrated through computational simulations in contour, 2D, and 3D plots. These results highlight the potential applications of the examined optical solitons in engineering nonlinear photonic devices and developing noise-resistant optical communication systems.

Background/objectiveCarbon monoxide (CO) is a colorless, odorless toxic gas responsible for approximately 100,000 emergency department visits and more than 420 deaths annually in the United States. Although CO–hemoglobin interactions have been extensively studied, a direct relationship between CO saturation and partial pressure has not been well established. This study aimed to derive a simple equation describing this relationship using principles analogous to the oxygen–hemoglobin association model.MethodsCO saturation was defined as the fraction of hemoglobin binding sites occupied by CO. Using chemical kinetics, the concentrations of CO, O₂, and hemoglobin, together with their association constants, were incorporated into the saturation formulation. Algebraic substitution and simplification yielded a rational function with four unknown coefficients. At a fixed oxygen partial pressure of 100 mmHg, four equilibrium (PCO, CO saturation) data points were used to solve for the coefficients of a fourth-degree rational function.ResultsThe derived CO–hemoglobin association equation reproduced the four derivation data points exactly and closely approximated additional literature values. The resulting association curve was hyperbolic. Fractional analysis demonstrated that Hb, HbCO, Hb(CO)₂, Hb(CO)₃, and Hb(CO)₄ fractions peaked at CO saturations of 0%, 25%, 50%, 75%, and 100%, respectively, with the triply bound form predominating overall.ConclusionsThe CO–hemoglobin association could be described by a fourth-degree rational equation, enabling estimation of either CO saturation or CO partial pressure when one is known and providing a framework extendable to varying oxygen tensions.

In vitro fertilization is often one of the final options patients with infertility will turn to during their attempts to create a family before choosing options such as adoption or donor oocytes. Given the vast amount of information on social media around infertility, patients are often left with unrealistic expectations after undergoing an autologous oocyte retrieval. Our aim was to determine the strength of the association between baseline antral follicle count (AFC) and mature oocytes obtained at the time of autologous in vitro fertilization with intracytoplasmic sperm injection (ICSI) to assist in the appropriate counseling of how many mature oocytes patients can anticipate after a single in vitro fertilization cycle. This is a retrospective chart review of women between the ages 18-41 undergoing in vitro fertilization with ICSI at Brooke Army Medical Center between 2015 and 2022. Approval was obtained from the Institutional Review Board. A total of 362 individuals were included, and AFC at baseline and mature oocytes obtained at the time of oocyte retrieval were recorded. Data were presented as mean ± SD for numerical data or sum ± percentages for categorical data. A univariate linear regression and several multivariate regression models were performed, and a coefficient of determination squared (R2), along with its corresponding 95% CI, was calculated to determine how well the models predict the outcome of the dependent variable, mature oocytes obtained at the time of oocyte retrieval. The univariate linear regression model using AFC at baseline to predict mature oocyte obtained at time of oocyte retrieval showed an R2 value of 26.7% (95% CI, 19.0-34.8) with P < .001. This implies that one mature oocyte was obtained for every four antral follicles noted at baseline, but this was noted to have a significant right-sided skew (Figure 1). Several multivariate regression models were performed, yet R2 was only altered marginally to 32.5% (95% CI, 24.5-40.6) over the simple univariate linear regression model, and the multivariate regression model R2 value was included in the 95% CI of the univariate linear regression model. Age alone was significantly less predictive with R2 of 8.5% (95% CI, 3.7-14.8). This study demonstrated that a simplified equation or model with high accuracy to predict mature oocytes collected at the time of in vitro fertilization from a baseline AFC was unable to be obtained. The results of this retrospective study show that there is a wide prediction interval of the number of mature oocytes obtained at the time of oocyte retrieval with every numerical increase in AFC at the baseline ultrasound. Strengths of this study include the large database and broad inclusion of ages and diagnoses that enable application to a broader population. Potential limitations include the restriction to ICSI and limiting to a military facility. This further highlights the complexity of infertility and the need for appropriate counseling of patients concerning expectations and variance in response.

We study a two-stage newsvendor model where the initially uncertain mean demand is revealed midstream, allowing for a second, costlier order to be placed. The two-stage process is relevant to many retailers who have access to two supply options: a long-lead, low-cost option where orders need to be placed under much demand uncertainty, and a short-lead, high-cost option after a signal is revealed that updates the mean demand. We introduce a forecast evolution model that describes how the initial forecast for mean demand varies in response to the signal, which generalizes the popular additive and multiplicative martingale model of forecast evolution (MMFE). We show that the optimal first-stage order solves a simple ordinary differential equation (ODE), whereas the second-stage order is analytically available. We characterize asymptotics for the first-stage order and expected profit as the forecasted mean demand grows, and propose a simple, asymptotically optimal heuristic. We apply our study to data from a national retail chain, where we calibrate our model by maximum likelihood estimation (MLE). The comparison of several heuristic policies, as well as two benchmarks, shows the efficacy of our method. When the signal is more uncertain (e.g., for product lines with impulse- and trend-driven purchases), our heuristic with a simple adjusted critical fractile outperforms benchmarks; otherwise, the classic newsvendor solution performs well and is asymptotically optimal. We also extend the approach to distribution-free settings and capacitated systems. This paper was accepted by Jeannette Song, operations management. Funding: This work was supported by the Guangdong Key Lab of Mathematical Foundations for Artificial Intelligence [Grant 72192805], the National Natural Science Foundation of China [Grant 72401245], and Imperial Business School (School Research Fund). Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2024.04692 .

This work investigates the impact of a generalized quadratic-cubic form of self-phase modulation together with nonlinear chromatic dispersion on perturbed quiescent optical solitons governed by the Fokas-Lenells equation. The model integrates both generalized and linear temporal evolution to depict the whole range of nonlinear interactions affecting pulse dynamics. Soliton production, frequency shifting, spectrum broadening, and pulse compression are some of the phenomena that emerge from the interaction of self-phase modulation and nonlinear dispersion, which significantly alter the pulse phase, spectral properties, and temporal profile. These processes are essential to many photonic applications, including the creation of supercontinuum and advanced medical imaging methods like optical coherence tomography. Analytical solutions to this generalized model are obtained using the modified extended tanh method, the extended simple equation method, and the exp[Formula: see text]-expansion technique. These approaches yield a rich and diverse family of soliton solutions, including dark, periodic, singular, dark-singular, and singular-periodic structures. Among the methods used, the extended simple equation approach provides a straightforward and effective process, while the modified extended tanh method generates a richer spectrum of solutions in a compact and non-redundant form. The physical behaviour of selected soliton profiles is illustrated through real and imaginary contour plots, as well as both two-dimensional and three-dimensional visualizations. The two-dimensional plots are used to illustrate the comparisons between different values of n. Overall, this work demonstrates a novel analytical framework for regulating soliton behavior in nonlinear optical media, which has theoretical and practical implications for applications including fiber-optic communications, ultrafast lasers, and optical signal processing.

For much of the 20th century, enzyme kinetic analysis relied on deriving simplified rate equations under the steady-state approximation and later by analytical integration of differential equations for transient kinetics. This approach has since been surpassed by computational methods using numerical integration of rate equations to directly fit experimental data based on a complete user-defined model. This paradigm shift removes the constraints imposed by solving analytical equations, enabling far greater flexibility in experimental design and model complexity. Modern global fitting methods allow data from diverse experiments to be analyzed simultaneously using the minimum number of parameters supported by the information content of the data set. Global data fitting is more than just an algorithm for data analysis─it represents a fundamental change in how we design and interpret experiments, and eliminates many of the restrictions, approximations, and ambiguities inherent to equation-based analyses. In this review, we describe the principles and practice of global data fitting, compare the outcomes to conventional equation-based methods, and demonstrate its power through examples involving multiple experiments with distinct conditions and readouts. We explain why the common practice of making measurements in triplicate introduces uncertainty and we outline advanced methods for rigorously estimating errors in measurement and for establishing robust confidence limits on fitted parameters.

We describe the α–like nucleus 104Te in terms of the Multi Step Shell Model (MSM) by coupling collective two-particle states above the doubly magic nucleus 100Sn. To this purpose we first built proton-proton (pp), neutron-neutron (nn) and proton-neutron (pn) collective states describing the structure of 102Te, 102Sn and 102Sb, respectively in terms of Tamm-Dankoff Approach (TDA) with multipole-multipole residual interaction. Then, the eigenstates in 104Te are given by pp-nn and pn-pn quartets satisfying a simple equation in terms of pair energies. The experimental pair energies 2+, 4+, 6+ are obtained by extrapolating known data in the corresponding isotope chains. The agreement of theoretical predictions to these values is satisfactory. The role of the Pauli exclusion principled is analyzed.

Within the inflammatory bowel disease (IBD) research community, histological scoring of colitis is an important grading system to determine colitis severity in pre-clinical mouse models of IBD. However, there is currently no set standard for grading the severity of colitis in mouse models. There are currently 30 types of scoring systems within the IBD research community. In this paper, we propose a simple, 4-characteristic scoring equation based on one of the most popular scoring systems in the community to streamline and advance the research on pre-clinical models of colitis. Using multiple linear regression, we found that 4 of 8 canonical characteristics of colitis are particularly significant to produce a simplified model equation with a fit of R2 = 0.908. Upon testing, this model appears to fare well within both non-ulcerative and ulcerative mouse models of colitis as a severity scoring system of colitis. The Boone-Mayer Scoring System appears to be a sound proposal for standardizing scoring severity of murine models of IBD across pre-clinical IBD laboratories. Link to website can be found here: https://sites.nd.edu/lauren-mayer/ .

An advanced frequency study in thick-walled functionally graded material (FGM) spherical shells is investigated with advanced shear correction. The values of advanced shear correction can be greater than one, be a negative value, and be affected by a nonlinear term of third-order shear deformation theory (TSDT) of displacements, FGM power law index, and temperature. It is novel and interesting to consider using TSDT and advanced shear correction to derive a simple homogeneous equation with reasonable simplifications into a symmetrical sparse matrix subjected to free vibration. The zero determinant of the symmetrical sparse matrix can be expressed to calculate the natural frequency by Newton’s method. The parameter effects of advanced shear correction, a nonlinear TSDT term, temperature, and the FGM power-law index on the natural frequencies of thick-walled FGM spherical shells are presented. The natural-frequency data for the axial and circumferential mode shapes are obtained. This is a new finding, as the assumed simplification in a sparse matrix causes a numerical truncation error; the natural-frequency values of the presented sparse matrix are much greater than those in a full matrix for thick-walled FGM spherical shells.

The standard logistic map, x'=ax(1-x), serves as a paradigmatic model to demonstrate how apparently simple nonlinear equations lead to complex and chaotic dynamics. In this work, we introduce and investigate its matrix analogue defined for an arbitrary matrix X of a given order N. We show that for an arbitrary initial ensemble of Hermitian random matrices with a continuous level density supported on the interval [0,1], the asymptotic level density converges to the invariant measure of the logistic map. Depending on the parameter a, the constructed measure may be either singular, fractal or described by a continuous density. In a wider class of the map, the multiplication by a scalar logistic parameter a is replaced by transforming aX(I-X) into BX(I-X)B†, where A=BB† is a fixed positive matrix of order N. This approach generalizes the known model of coupled logistic maps and allows us to study the transition to chaos in complex networks and multidimensional systems. In particular, by associating the matrix B with a given graph, we demonstrate the gradual transfer of chaos between subsystems corresponding to vertices of a graph and coupled according to its edges.

• It is shown for the first time that the spectral signature of the glacier ice can be modelled using four parameters; it is suggested that these parameters must be reported in the output of respective glacier ice retrieval algorithms; • The spectral reflectance measurements of the glacier ice at Mitdalsbreen Glacier (Norway) has been performed and analysed using the developed theory; • The technique for the determination of the glacier properties and albedo from space using the developed theory has been applied to hyperspectral spaceborne EnMAP observations of Mitdalsbreen Glacier. We propose simple analytical equations for the modeling of clean and dusty flat glacier ice surfaces, which can be used to inversely derive the parameters of microstructure of flat bare glacier ice and snow using both ground – based and spaceborne observations of the hyperspectral solar reflectance. The retrievals are based on the asymptotic radiative transfer equations valid for the case of weak light absorption in the semi-infinite turbid medium. The light reflection at the air - ice and ice - air interfaces is fully accounted for. To demonstrate the validity of the approach, the derived equations are exemplarily applied to both ground – based and EnMAP satellite measurements over the Hardangerjokulen glacier (Norway). A number of important parameters controlling spectral signatures of the snow and glacier ice surfaces have been derived. The ground-based measurements show that the theoretical formulation presented in this work can be used to represent the solar light spectral reflectivity of glaciers. The application to satellite hyperspectral imagery shows that this novel technique allows for the determination of the glacier ice albedo (spectral, broadband) based on spaceborne glacier ice reflectance measurement. Additionally, the results show that not spectrally neutral soot but rather deposited atmospheric dust which enhances the absorption towards UV (due to the presence of iron oxides) is responsible for the light absorption by snow for the case studied. Spatial distribution maps of ice grain diameter and dust concentration are derived over the glacier. These findings show that the analytical theory presented in this work can support further research on the characterization and monitoring of glaciers based on current and upcoming hyperspectral remote sensing satellite missions.