Estimation-free spatial-domain image reconstruction of structured illumination microscopy

Structured illumination microscopy(SIM)achieves super-resolution(SR)by modulating the high-frequency information of the sample into the passband of the optical system and subsequent image reconstruction.The traditional Wiener-filtering-based reconstruction algorithm operates in the Fourier domain,it requires prior knowledge of the sinusoidal illumination patterns which makes the time-consuming procedure of parameter estimation to raw datasets necessary,besides,the parameter estimation is sensitive to noise or aberration-induced pattern distortion which leads to reconstruction artifacts.Here,we propose a spatial-domain image reconstruction method that does not require parameter estimation but calculates patterns from raw datasets,and a reconstructed image can be obtained just by calculating the spatial covariance of differential calculated patterns and differential filtered datasets(the notch filtering operation is performed to the raw datasets for attenuating and compensating the optical transfer function(OTF)).Experiments on reconstructing raw datasets including nonbiological,biological,and simulated samples demonstrate that our method has SR capability,high reconstruction speed,and high robustness to aberration and noise.

Generation of structured light beams with polarization variation along arbitrary spatial trajectories using tri-layer metasurfaces

Conventionally,the spatially structured light beams produced by metasurfaces primarily highlight the polarization modulation of the beams propagating along the optical axis or the beams'spatial transmission trajectory.In particular,along the optical axis,the polarization state is either constant or varies continuously in each output plane.Here,we develop innovative spatially structured light beams with continually changing polarization along any arbitrary spatial transmission trajectories.With tri-layer metallic metasurfaces,the geometric characteristics of each layer structure can be adjusted to modulate the phase and polarization state of the incident terahertz(THz)wave.The beam will converge to the predefined trajectory along several paths to generate a Bessel-like beam with longitudinal polarization changes.We demonstrate the versatility of the approach by designing two THz-band structured light beams with varying polarization states along the spatial helical transmission trajectory.Continuous linear polarization changes and linear polarization to right circular polarization(RCP)and back to linear polarization changes are realized respectively.The experimental results are basically consistent with the simulated results.Our proposal for arbitrary trajectory structured light beams with longitudinally varying polarization offers a practical method for continuously regulating the characteristics of spatial structured light beams with non-axial transmission.This technique has potential uses in optical encryption,particle manipulation,and biomedical imaging.

Algorithms and statistical analysis for linear structured weighted total least squares problem

Weighted total least squares(WTLS)have been regarded as the standard tool for the errors-in-variables(EIV)model in which all the elements in the observation vector and the coefficient matrix are contaminated with random errors.However,in many geodetic applications,some elements are error-free and some random observations appear repeatedly in different positions in the augmented coefficient matrix.It is called the linear structured EIV(LSEIV)model.Two kinds of methods are proposed for the LSEIV model from functional and stochastic modifications.On the one hand,the functional part of the LSEIV model is modified into the errors-in-observations(EIO)model.On the other hand,the stochastic model is modified by applying the Moore-Penrose inverse of the cofactor matrix.The algorithms are derived through the Lagrange multipliers method and linear approximation.The estimation principles and iterative formula of the parameters are proven to be consistent.The first-order approximate variance-covariance matrix(VCM)of the parameters is also derived.A numerical example is given to compare the performances of our proposed three algorithms with the STLS approach.Afterwards,the least squares(LS),total least squares(TLS)and linear structured weighted total least squares(LSWTLS)solutions are compared and the accuracy evaluation formula is proven to be feasible and effective.Finally,the LSWTLS is applied to the field of deformation analysis,which yields a better result than the traditional LS and TLS estimations.

Design,preparation,application of advanced array structured materials and their action mechanism analyses for high performance lithium-sulfur batteries

Lithium-sulfur battery(LSB)has brought much attention and concern because of high theoretical specific capacity and energy density as one of main competitors for next-generation energy storage systems.The widely commercial application and development of LSB is mainly hindered by serious“shuttle effect”of lithium polysulfides(Li PSs),slow reaction kinetics,notorious lithium dendrites,etc.In various structures of LSB materials,array structured materials,possessing the composition of ordered micro units with the same or similar characteristics of each unit,present excellent application potential for various secondary cells due to some merits such as immobilization of active substances,high specific surface area,appropriate pore sizes,easy modification of functional material surface,accommodated huge volume change,enough facilitated transportation for electrons/lithium ions,and special functional groups strongly adsorbing Li PSs.Thus many novel array structured materials are applied to battery for tackling thorny problems mentioned above.In this review,recent progresses and developments on array structured materials applied in LSBs including preparation ways,collaborative structural designs based on array structures,and action mechanism analyses in improving electrochemical performance and safety are summarized.Meanwhile,we also have detailed discussion for array structured materials in LSBs and constructed the structure-function relationships between array structured materials and battery performances.Lastly,some directions and prospects about preparation ways,functional modifications,and practical applications of array structured materials in LSBs are generalized.We hope the review can attract more researchers'attention and bring more studying on array structured materials for other secondary batteries including LSB.

Evolutionary game dynamics of combining two different aspiration-driven update rules in structured populations

In evolutionary games,most studies on finite populations have focused on a single updating mechanism.However,given the differences in individual cognition,individuals may change their strategies according to different updating mechanisms.For this reason,we consider two different aspiration-driven updating mechanisms in structured populations:satisfied-stay unsatisfied shift(SSUS)and satisfied-cooperate unsatisfied defect(SCUD).To simulate the game player’s learning process,this paper improves the particle swarm optimization algorithm,which will be used to simulate the game player’s strategy selection,i.e.,population particle swarm optimization(PPSO)algorithms.We find that in the prisoner’s dilemma,the conditions that SSUS facilitates the evolution of cooperation do not enable cooperation to emerge.In contrast,SCUD conditions that promote the evolution of cooperation enable cooperation to emerge.In addition,the invasion of SCUD individuals helps promote cooperation among SSUS individuals.Simulated by the PPSO algorithm,the theoretical approximation results are found to be consistent with the trend of change in the simulation results.

Structured Multi-Head Attention Stock Index Prediction Method Based Adaptive Public Opinion Sentiment Vector

The present study examines the impact of short-term public opinion sentiment on the secondary market,with a focus on the potential for such sentiment to cause dramatic stock price fluctuations and increase investment risk.The quantification of investment sentiment indicators and the persistent analysis of their impact has been a complex and significant area of research.In this paper,a structured multi-head attention stock index prediction method based adaptive public opinion sentiment vector is proposed.The proposedmethod utilizes an innovative approach to transform numerous investor comments on social platforms over time into public opinion sentiment vectors expressing complex sentiments.It then analyzes the continuous impact of these vectors on the market through the use of aggregating techniques and public opinion data via a structured multi-head attention mechanism.The experimental results demonstrate that the public opinion sentiment vector can provide more comprehensive feedback on market sentiment than traditional sentiment polarity analysis.Furthermore,the multi-head attention mechanism is shown to improve prediction accuracy through attention convergence on each type of input information separately.Themean absolute percentage error(MAPE)of the proposedmethod is 0.463%,a reduction of 0.294% compared to the benchmark attention algorithm.Additionally,the market backtesting results indicate that the return was 24.560%,an improvement of 8.202% compared to the benchmark algorithm.These results suggest that themarket trading strategy based on thismethod has the potential to improve trading profits.

Effectiveness of structured teaching program on knowledge and attitude toward foot care

Objective:The study was aimed to assess the knowledge and attitude toward foot care and evaluate the effectiveness of structured teaching program(STP)on the knowledge and attitude of patients with diabetes mellitus regarding foot care.Materials and Methods:A preexperimental research design was used in this study,which was conducted in medical-surgical wards of a secondary hospital in Northeast India,Nagaland.A total of 30 subjects were recruited by consecutive sampling technique.A closed-ended structured questionnaire and 5-point Likert scale was used to assess knowledge and attitude toward foot care.A STP was administered after which posttest was done after 24 h.Results:The majority(86.6%)of the participants in the pretest had inadequate knowledge and 80%had negative attitudes toward foot care.After STP,70%had moderately adequate knowledge and 53.3%had a positive attitude regarding foot care.The structured teaching program was effective in improving the knowledge and attitude at P<0.01 which is highly significant and there was a significant association between the pretest level knowledge and duration of diabetes at P<0.01.Conclusion:STP was effective in enhancing the level of knowledge and attitude toward foot care among diabetic inpatients.This type of interaction can be organized at regular intervals to reinforce knowledge and attitude toward foot care among inpatients.

Systematic Review of Community Type 2 Diabetes Structured Health Education (CT2DSHE)

Aim: This paper aims to evaluate disparities of type 2 diabetes structured health education programmes that is utilised within the communities. Design: systematic review, (a type of secondary research design) aiming to summarize the results of prior primary research studies on available evidence Community type 2 diabetes structured education (CT2DSHE). Methods: Research question: Type 2 diabetic structured health education within a community how effective is it? Qualitative Systematic review, defined as a way to get reliable and objective picture of current available evidence on the specific topic—(CT2DSHE), (Denscombe, 2021) through reflexivity synthesis of available data as an example. This is valuable in time constraints such as project assignments that must be met within specific time and also to bring together available evidence together [1]. Results: This review has shown that CT2DSHE is effective with seven out of the eleven authors supporting, three authors against and one was neutral, further showed that knowledge and skills acquired can last longer with patient activation improved among T2DM patients ideal for sustaining their self-management of T2DM. Conclusion: This research provides suggestive answers to the research question: “Type 2 diabetic structured health education within a community how effective is it?”, This has demonstrated CT2DSHE effectiveness in knowledge acquisition and improving T2DM awareness among T2DM patients, whilst evidencing long effects beyond the study times of 3 - 9 months period in relation to patient activation. Also Identified diabetes education self-management on newly diagnosed (DESMOND) patient as CT2DSHE program for recommendation. Patient or Public Contribution: This work aspires to contribute to CT2DSHE in these areas;Influencing policy decision-making for community diabetes care within the UK and world at large., Contributing to already vast knowledge on diabetes self-management and reasons why?, Influencing educators on how CT2DSHEP are designed, delivered by putting the patient at the Centre and bringing different perspectives on CT2DSHEP in one place that is serving users time of having to consult several resources especially busy clinicians [2] [3].

Application of Structured Health Education and Combined 5A Nursing Model in Self-control of Vulnerable Chronic Diseases: Take Coronary Heart Disease as an Example

Objective:To explore the intervention effect of the Structured Health Education course and 5A nursing model for self-control of elderly patients with coronary heart disease.Methods:Using the random sampling method,124 elderly CAD patients admitted to the First Affiliated Hospital of Bengbu Medical University were randomly divided into an experimental group and a control group.The control group line routine health education,experimental group take structured health education combined with 5A nursing before and after the intervention using a coronary heart disease assessment questionnaire,coronary heart disease self-control scale evaluation of two groups of intervention,compare two groups before and after intervention blood pressure,blood sugar,body mass index,lipid index level and complications within 8 months after discharge.Results:After the course intervention,the disease cognition and self-behavior of the experimental group were higher than that of the control group,and the differences were statistically significant(all P<0.1).Conclusion:This course is suitable for elderly patients with coronary heart disease.The 5A model improves the cognitive and management ability of elderly patients to a certain extent,which is worthy of clinical application.

Numerical Simulation of Super-Resolution Structured Illumination Microscopy (SIM) Using Heintzmann-Cremer Algorithm with Non-Continuous Spatial Frequency Support

We report a comprehensive numerical study of super resolution (SR) structured illumination microscopy (SIM) utilizing the classic Heintzmann-Cremer SIM process and algorithm. In particular, we investigated the impact of the diffraction limit of the underlying imaging system on the optimal SIM grating frequency that can be used to obtain the highest SR enhancement with non-continuous spatial frequency support. Besides confirming the previous theoretical and experimental work that SR-SIM can achieve an enhancement close to 3 times the diffraction limit with grating pattern illuminations, we also observe and report a series of more subtle effects of SR-SIM with non-continuous spatial frequency support. Our simulations show that when the SIM grating frequency exceeds twice that of the diffraction limit, the higher SIM grating frequency can help achieve a higher SR enhancement for the underlying imaging systems whose diffraction limit is low, though this enhancement is obtained at the cost of losing resolution at some lower resolution targets. Our simulations also show that, for underlying imaging systems with high diffraction limits, however, SR-SIM grating frequencies above twice the diffraction limits tend to bring no significant extra enhancement. Furthermore, we observed that there exists a limit grating frequency above which the SR enhancement effect is lost, and the reconstructed images essentially have the same resolution as the one obtained directly from the underlying imaging system without using the SIM process.

Analysis of the Application Effect of Structured Healthcare Education in Brittle Diabetic Patients

Objective:To explore the application effect of structured healthcare education in patients with brittle diabetes mellitus.Methods:188 brittle diabetic patients admitted to our hospital from May 2021 to December 2023 were selected as the study subjects,and were divided into the control group(n=94)and the observation group(n=94)according to the random number table method.The control group used conventional nursing intervention and the observation group used structured healthcare education.The general information,glycemic indexes,self-efficacy,compliance,and nursing satisfaction of patients in the two groups were observed.Results:There was no statistical significance in the basic information of the two groups of patients(P>0.05);after the intervention,the fasting plasma glucose,2-hour postprandial blood glucose,and HbA1c of the patients in the observation group were lower than those of the control group(P<0.001);after the intervention,the self-efficacy scores of the patients in the two groups increased,and the scores of the observation group were significantly higher than those of the control group(P<0.001);the total adherence rate of the patients in the observation group(90/95.75%)was significantly higher than that of the control group(80/90.10%)(χ^(2)=6.144,P<0.05);and the total satisfaction rate of patients in the observation group(92/97.87%)was significantly higher than that of the control group(78/82.98%)(χ^(2)=12.042,P<0.05).Conclusion:In patients with brittle diabetes mellitus,structured healthcare education can effectively control patients’blood glucose levels,improve patients’self-efficacy and adherence,and enhance patient satisfaction.

Enhancing electromagnetic wave absorption with core‐shell structured SiO_(2)@MXene@MoS_(2) nanospheres

Material composition and structural design are important factors influencing the electromagnetic wave(EMW)absorption performance of materials.To alleviate the impedance mismatch attributed to the high dielectric constant of Ti_(3)C_(2)T_(x)MXene,we have successfully synthesized core‐shell structured SiO_(2)@MXene@MoS_(2)nanospheres.This architecture,comprising SiO_(2) as the core,MXene as the intermediate layer,and MoS_(2) as the outer shell,is achieved through an electrostatic self‐assembly method combined with a hydrothermal process.This complex core‐shell structure not only provides a variety of loss mechanisms that effectively dissipate electromagnetic energy but also prevents self‐aggregation of MXene and MoS_(2) nanosheets.Notably,the synergistic combination of SiO_(2) and MoS_(2) with highly conductive MXene enables the suitable dielectric constant of the composites,ensuring optimal impedance matching.Therefore,the core‐shell structured SiO_(2)@MXene@MoS_(2) nanospheres exhibit excellent EMW absorption performance,featuring a remarkable minimum reflection loss(RL_(min))of−52.11 dB(2.4 mm).It is noteworthy that these nanospheres achieve an ultra‐wide effective absorption bandwidth(EAB)of 6.72 GHz.This work provides a novel approach for designing and synthesizing high‐performance EMW absorbers characterized by“wide bandwidth and strong reflection loss.”

Rapid drop in ductility of the bimodal-structured Mg-15Gd binary alloy during early aging

A bimodal-structured Mg^(-1)5Gd binary alloy with 45%volume fraction of elongated grains and 55%of dynamically recrystallized(DRXed)grains is fabricated by the extrusion process.The precipitating behavior correlating with the evolution of mechanical properties is systematically characterized during the subsequent aging treatment at 200°C.The extruded alloy presents an outstanding strength with tensile yield strength of 466 MPa and ultimate tensile strength of 500 MPa at peak aging condition,while the elongation drops from 9.2%in extrusion state to 3.1%.It is found there obviously exist a rapidly decreasing range of ductility at the early stage of aging.Just during this time,the nano precipitates form preferentially at lamellar dislocation boundaries(LDBs)within the elongated grains,but there is no dense and uniform precipitation in the matrix.The results suggest that the low elongation in the aged Mg^(-1)5Gd alloy is mainly attributed to the nano precipitates prior formed at the LDBs with a high density in the elongated grains.The related mechanism has been clarified.

Reversible Mn^(2+)/Mn^(4+)double-electron redox in P3-type layer-structured sodium-ion cathode

The balance between cationic redox and oxygen redox in layer-structured cathode materials is an important issue for sodium batteries to obtain high energy density and considerable cycle stability.Oxygen redox can contribute extra capacity to increase energy density,but results in lattice instability and capacity fading caused by lattice oxygen gliding and oxygen release.In this work,reversible Mn^(2+)/Mn^(4+)redox is realized in a P3-Na_(0.65)Li_(0.2)Co_(0.05)Mn_(0.75)O_(2)cathode material with high specific capacity and structure stability via Co substitution.The contribution of oxygen redox is suppressed significantly by reversible Mn^(2+)/Mn^(4+)redox without sacrificing capacity,thus reducing lattice oxygen release and improving the structure stability.Synchrotron X-ray techniques reveal that P3 phase is well maintained in a wide voltage window of 1.5-4.5 V vs.Na^(+)/Na even at 10 C and after long-term cycling.It is disclosed that charge compensation from Co/Mn-ions contributes to the voltage region below 4.2 V and O-ions contribute to the whole voltage range.The synergistic contributions of Mn^(2+)/Mn^(4+),Co^(2+)/Co^(3+),and O^(2-)/(O_n)^(2-)redox in P3-Na_(0.65)Li_(0.2)Co_(0.05)Mn_(0.75)O_(2)lead to a high reversible capacity of 215.0 m A h g^(-1)at 0.1 C with considerable cycle stability.The strategy opens up new opportunities for the design of high capacity cathode materials for rechargeable batteries.

Construction of core@double-shell structured energetic composites with simultaneously enhanced thermal stability and safety performance

The poor thermal stability and high sensitivity severely hinder the practical application of hexanitrohexaazaisowurtzitane(CL-20).Herein,a kind of novel core@double-shell CL-20 based energetic composites were fabricated to address the above issues.The coordination complexes which consist of natural polyphenol tannic acid(TA) and Fe~Ⅲ were chosen to construct the inner shell,while the graphene sheets were used to build the outer shell.The resulting CL-20/TA-Fe~Ⅲ/graphene composites exhibited simultaneously improved thermal stability and safety performance with only 1 wt% double-shell content,which should be ascribed to the intense physical encapsulation effect from inner shell combined with the desensitization effect of carbon nano-materials from outer shell.The phase transition(ε to γ) temperature increased from 173.70 ℃ of pure CL-20 to 191.87℃ of CL-20/TA-Fe~Ⅲ/graphene composites.Meanwhile,the characteristic drop height(H_(50)) dramatically increased from 14.7 cm of pure CL-20 to112.8 cm of CL-20/TA-Fe~Ⅲ/graphene composites,indicating much superior safety performance after the construction of the double-shell structure.In general,this work has provided an effective and versatile strategy to conquer the thermal stability and safety issues of CL-20 and contributes to the future application of high energy density energetic materials.

Vacancy engineering mediated hollow structured ZnO/ZnS S-scheme heterojunction for highly efficient photocatalytic H_(2) production

Designing a step-scheme(S-scheme)heterojunction photocatalyst with vacancy engineering is a reliable approach to achieve highly efficient photocatalytic H_(2)production activity.Herein,a hollow ZnO/ZnS S-scheme heterojunction with O and Zn vacancies(VO,Zn-ZnO/ZnS)is rationally constructed via ion-exchange and calcination treatments.In such a photocatalytic system,the hollow structure combined with the introduction of dual vacancies endows the adequate light absorption.Moreover,the O and Zn vacancies serve as the trapping sites for photo-induced electrons and holes,respectively,which are beneficial for promoting the photo-induced carrier separation.Meanwhile,the S-scheme charge transfer mechanism can not only improve the separation and transfer efficiencies of photo-induced carrier but also retain the strong redox capacity.As expected,the optimized VO,Zn-ZnO/ZnS heterojunction exhibits a superior photocatalytic H_(2) production rate of 160.91 mmol g^(-1)h^(-1),approximately 643.6 times and 214.5 times with respect to that obtained on pure ZnO and ZnS,respectively.Simultaneously,the experimental results and density functional theory calculations disclose that the photo-induced carrier transfer pathway follows the S-scheme heterojunction mechanism and the introduction of O and Zn vacancies reduces the surface reaction barrier.This work provides an innovative strategy of vacancy engineering in S-scheme heterojunction for solar-to-fuel energy conversion.

Speckle structured illumination endoscopy with enhanced resolution at wide field of view and depth of field

Structured illumination microscopy(SIM)is one of the most widely applied wide field super resolution imaging techniques with high temporal resolution and low phototoxicity.The spatial resolution of SIM is typically limited to two times of the diffraction limit and the depth of field is small.In this work,we propose and experimentally demonstrate a low cost,easy to implement,novel technique called speckle structured illumination endoscopy(SSIE)to enhance the resolution of a wide field endoscope with large depth of field.Here,speckle patterns are used to excite objects on the sample which is then followed by a blind-SIM algorithm for super resolution image reconstruction.Our approach is insensitive to the 3D morphology of the specimen,or the deformation of illuminations used.It greatly simplifies the experimental setup as there are no calibration protocols and no stringent control of illumination patterns nor focusing optics.We demonstrate that the SSIE can enhance the resolution 2–4.5 times that of a standard white light endoscopic(WLE)system.The SSIE presents a unique route to super resolution in endoscopic imaging at wide field of view and depth of field,which might be beneficial to the practice of clinical endoscopy.

Complex-amplitude Fourier single-pixel imaging via coherent structured illumination

We propose a method of complex-amplitude Fourier single-pixel imaging(CFSI)with coherent structured illumination to acquire both the amplitude and phase of an object.In the proposed method,an object is illustrated by a series of coherent structured light fields,which are generated by a phase-only spatial light modulator,the complex Fourier spectrum of the object can be acquired sequentially by a single-pixel photodetector.Then the desired complex-amplitude image can be retrieved directly by applying an inverse Fourier transform.We experimentally implemented this CFSI with several different types of objects.The experimental results show that the proposed method provides a promising complex-amplitude imaging approach with high quality and a stable configuration.Thus,it might find broad applications in optical metrology and biomedical science.

Acute pancreatitis:Structured report template of magnetic resonance imaging

Acute pancreatitis(AP)is a common acute abdomen disease of the digestive system.It has a potentially fatal risk because of its variable severity and various complications.With the widespread application of the Revised Atlanta Classification,new requirements for AP imaging reports are introduced.Experts in abdominal radiology and pancreatology in the United States published the first structured computed tomography reporting template for AP in 2020.However,there is no corresponding structured magnetic resonance imaging(MRI)reporting template globally.Therefore,this article focuses on the structured MRI report of AP images from our pancreatitis imaging center,which is intended to improve the systematic understanding of this disease and standardize the writing of MRI structured reports.In the meantime,we aim to promote the clinical diagnosis and assessment of MRI efficacy for AP and its multiple complications.It is further intended to facilitate academic exchanges and scientific research between different medical centers.

Pressure drop of structured packing in pilot column and comparison to common correlations

Packed columns are widely used in the chemical industry such as absorption,stripping,distillation,and extraction in the production of e.g.organic chemicals,and pharmaceuticals.Pressure loss and pressure drop correlations are of special interest when it comes to the hydrodynamic properties of a column.The pressure loss across the column is of interest in the design phase when the size of the blower to drive the gas stream through the column has to be decided.The loading point and flooding point are also influenced by the pressure loss and the area of operation is determined from these points.This work examines four different correlations on pressure drop.The correlations are(i)Ergun’s equation(1952),(ii)an improved version of Ergun’s equation by Stichlmair,Bravo,and Fair(1989),(iii)an equation developed by Billet and Schultes(1999),and(iv)an equation by Rocha,Bravo,and Fair(1993).The complexity of the correlations is increasing in the mentioned order,Ergun’s equation being the simplest one.This study investigates if the more complicated correlations give better predictions to pressure drop in packed columns.This is determined by comparing the correlations to experimental data for pressure drop in a packed column with 8.2 m of structured packing using water as the liquid and atmospheric air as the gas.Seven experiments were carried out for determining the pressure drop in the column with liquid flows varying from 0 to 500 kg·h^(-1).At constant liquid flow,the gas flow was varied from approximately 10 to 70 kg·h^(-1).The pressure drop across the non-wetted column was best described by the correlation by Rocha et al.while the pressure drop for liquid flows from 100 to 500 kg·h^(-1)was,in general,best described by Stichlmair’s equation.For an irrigated column,the highest deviation was a predicted pressure drop 69.6%lower than measured.The best prediction was 0.1%higher than the measured.This study shows,surprisingly,that for a system of water and atmospheric air,complicated correlations on pressure drop determination do not provide better estimates than simple equations.