Parabolic Function Research Articles

Effectiveness of Scientific Calculators’ Usage in the Teaching and Learning of Grade 11 Parabola Functions to Improve Learner Performance

The use of scientific calculators in teaching parabola functions has been shown to have significant effect in learner performance. This study examined the effectiveness of scientific calculators’ usage in the teaching and learning of Grade 11 parabola functions to improve learner performance. Bandura’s observational learning theory underpinned this study. A non-equivalent quazi-experimental design was espoused to collect data from the experimental and control groups. A pre-test and a post-test were administered to both experimental group and control group in the same day and at the same time, to avoid contamination of results. Wilcoxon-Rank sum test was used to determine the significant difference between the two study groups. The results showed that the experimental group performed significantly better than the control groupin the post-test after the implementation of scientific calculators in teaching parabola functions. The study suggests that scientific calculators can be used to teach parabola functions to improve learner performance and advance teacher content knowledge, pedagogical content knowledge and professional development.

Fractional Wiener chaos: Part 1

AbstractIn this paper, we introduce a fractional analogue of the Wiener polynomial chaos expansion. It is important to highlight that the fractional order relates to the order of chaos decomposition elements, and not to the process itself, which remains the standard Wiener process. The central instrument in our fractional analogue of the Wiener chaos expansion is the function denoted as $${\mathcal {H}}_\alpha (x,y)$$ H α ( x , y ) , referred to herein as a power-normalised parabolic cylinder function. Through careful analysis of several fundamental deterministic and stochastic properties, we affirm that this function essentially serves as a fractional extension of the Hermite polynomial. In particular, the power-normalised parabolic cylinder function with the Wiener process and time as its arguments, $${\mathcal {H}}_\alpha (W_t,t)$$ H α ( W t , t ) , demonstrates martingale properties and can be interpreted as a fractional Itô integral with 1 as the integrand, thereby drawing parallels with its non-fractional counterpart. To build a fractional analogue of polynomial Wiener chaos on the real line, we introduce a new function, which we call the extended Hermite function, by smoothly joining two power-normalized parabolic cylinder functions at zero. We form an orthogonal set of extended Hermite functions as a one-parameter family and use tensor products of the extended Hermite functions as building blocks in the fractional Wiener chaos expansion, in the same way that tensor products of Hermite polynomials are used as building blocks in the Wiener chaos polynomial expansion.

Experimental investigation of the behavior of UHPCFST under repeated eccentric compression

This paper investigates the mechanical behavior of ultra-high-performance concrete-filled steel tubes (UHPCFST) under repeated eccentric compression. A total of 30 UHPCFST specimens are designed, fabricated, and tested. The design variables include steel tube thickness, UHPC type, loading eccentricity and load pattern. Failure modes, force-axial shortening curves, section strain distributions, lateral deflection distributions, bearing capacity and stiffness are studied. Three failure modes, i.e., steel tube bulge, compressive crush and tensile crack of the UHPC infill are observed. Specimens with larger loading eccentricity and thinner steel tube are more likely to exhibit all the three modes. Subjected to eccentric loading, the compressive strength and stiffness of the UHPCFST increase significantly with the increase of steel tube thickness and UHPC strength. In the case of repeated loading, stiffness degradation is observed. Existing formulas for the N-M curve and the eccentric compressive capacity are evaluated against the test results. A formula for eccentric compressive stiffness is derived based on the parabolic function assumption. Additionally, an empirical model is introduced to describe the force-axial shortening relationship of the UHPCFST under repeated eccentric compression, which may be applied in practical design and analysis.

Mathematical model and generation analysis for crossed helical gear system

Purpose Crossed helical gears have point contact and their surfaces are subject to high surface stress. Contact stress and root tooth stress are the most common sources of failure in crossed helical gear. This paper aims to study the load-carrying capacity and performance of crossed helical gear teeth with different gear tooth profiles. To overcome defects and reduce the sensitivity to small shaft angle changes. The combined tooth profile is designed to reduce the bending stresses, contact stresses and tooth deflection, and prevent pinion failure in the gearbox. Design/methodology/approach The principle of the method is a line contact is introduced instead of a point contact between two teeth in mesh with each other. The tooth surface of the helical gear is designed and cut by a modified tool. Higher normal pressure angles like 25° and 35° are used. The modified involute is accomplished to eliminate interference between the teeth. Engineering software packages have been applied to generate all crossed helical gears gear profiles. The modification is compounding curves consisting of an epicycloid-involute-hypocycloid gear teeth profile generated by the cutter modified. Findings The stresses in the crossed helical gear teeth profile were reduced by increasing the normal pressure angle values. Using a 35° pressure angle the enhancement percentage in contact stress and teeth fillet region will be about 29.345% and 15.421%, respectively. The best enhancement in a gear’s resistance is the epicycloid-involute-hypocycloid gear teeth profile. The enhancement was 32.610% and 18.588%. The skew in line of action in skewed helical gear will be sensitive when the crossing angles are small. Their teeth surface tends to be easily worn out; however, the wearing process will be reduced by using a proposed gear teeth profile. Practical implications The gear teeth to be modified are cut by a shaper process. The modifying rack cutter of this study can be used as a reference for creating a different helical gears sample. The helical teeth surface is modified to become an envelope of the other. This makes an original point contact change into a line contact. The epicycloid-involute-hypocycloid gear teeth profile is preferred for a higher contact ratio and a large load capacity. This work explicitly introduces a new method of kinematic consideration to improve the load capacity of crossed helical gear. Originality/value This paper showed some novel results by the unique shape of the rack-cutter designed to generate different helical angles and different gear positions. In the future, it will make valuable contributions to the further development of the dynamic performance of a crossed helical gear system through the study in the field of using asymmetric teeth profiles of helical gears with tip relief as the manner to enhance the crossed helical gear performance. Investigation of crossed helical gear by applying a predesigned parabolic function of transmission error enables the absorption of linear discontinuous functions caused by misalignment.

A Reconfigurable, Nonlinear, Low-Power, VCO-Based ADC for Neural Recording Applications.

Neural recording systems play a crucial role in comprehending the intricacies of the brain and advancing treatments for neurological disorders. Within these systems, the analog-to-digital converter (ADC) serves as a fundamental component, converting the electrical signals from the brain into digital data that can be further processed and analyzed by computing units. This research introduces a novel nonlinear ADC designed specifically for spike sorting in biomedical applications. Employing MOSFET varactors and voltage-controlled oscillators (VCOs), this ADC exploits the nonlinear capacitance properties of MOSFET varactors, achieving a parabolic quantization function that digitizes the noise with low resolution and the spikes with high resolution, effectively suppressing the background noise present in biomedical signals. This research aims to develop a reconfigurable, nonlinear voltage-controlled oscillator (VCO)-based ADC, specifically designed for implantable neural recording systems used in neuroprosthetics and brain-machine interfaces. The proposed design enhances the signal-to-noise ratio and reduces power consumption, making it more efficient for real-time neural data processing. By improving the performance and energy efficiency of these devices, the research contributes to the development of more reliable medical technologies for monitoring and treating neurological disorders. The quantization step of the ADC spans from 44.8 mV in the low-amplitude range to 1.4 mV in the high-amplitude range. The circuit was designed and simulated utilizing a 180 nm CMOS process; however, no physical prototype has been fabricated at this stage. Post-layout simulations confirm the expected performance. Occupying a silicon area is 0.09 mm2. Operating at a sampling frequency of 16 kS/s and a supply voltage of 1 volt, this ADC consumes 62.4 µW.

A reliability analysis method based on the intersection area division of hypersphere and paraboloid

In terms of the approximate failure probability of the limit state function (LSF), First Order Reliability Method (FORM) directly solves the failure probability through the most probable point (MPP), but it can only achieve high accuracy in the LSF with low linearity and non-linearity. Second Order Reliability Method (SORM) performs second-order expansion on the basis of MPP to improve reliability accuracy. However, the approximation may be wrong, and even has a large error in high-dimensional nonlinear problems. In order to get more precise approximation of SORM, this paper proposes an area partition method based on the Intersection Area Division Method (IADM) of Hypersphere and Paraboloid. IADM constructs a hypersphere and an approximate parabolic function of the LSF at MPP. The intersection area of the hypersphere and the approximate paraboloid is defined as the failure region from the geometric of the approximate paraboloid function, and the area of the failure region is calculated by the spherical integral. Finally, the approximation of the failure probability is determined by integrating the ratio of the area to the area of the hypersphere. Several examples proved that IADM is more stable compared with SORM which avoids the generation of wrong solutions and improves the accuracy.

Damage evolution and fracture characteristics of coal rock impacted by self‐excited pulse water jet under different stress loading conditions

AbstractIn this paper, based on smoothed‐particle hydrodynamics‐finite element method, numerical models of plunger squeezing water at a sinusoidal velocity were established to simulate self‐excited pulse water jet (SEPWJ). RHT constitutive model was adopted to describe the damage and failure of coal rock impacted by water jet. The morphological evolutions of broken pits and timeliness of rock‐breaking efficiency of SEPWJ and continuous water jet (CWJ) under the conditions with and without stress loadings were obtained and compared. The evolution laws of damage and stress inner coal rock induced by jet impact, and the failure mechanism were revealed. And the influences of different stress loading magnitudes on the fracture characteristics of coal rock were investigated. The results show that the morphologies of broken pits formed by self‐excited pulse jet undergo changes in a semi‐circular, U‐shaped, V‐shaped, and bullet shaped in sequence under the stress‐free loading condition. When applying one‐dimensional (1D) and 2D stress loadings, the shallow but wide broken pits with laminar main cracks along the stress loading direction and the inverted trapezoidal bowl broken pits are formed, respectively. With the increase of 1D stress, the depth and width of broken pits slightly decrease as a quadratic parabolic function and linearly increase, respectively. And the broken pit width and area both show an exponential slow decreasing trends with the increasing 2D stress. SEPWJ can induce higher stresses to cause the earlier occurrence of initial damage and the shorter duration of damage accumulation to coal rock than CWJ, which leads to a better rock‐breaking effect. The surface and deeper coal rock elements are broken mainly due to compressive shear stresses. The 2D stress loading delays the initial damage occurrence and prolongs the damage accumulation duration due to inhibitory effect of stress loading on jet impact.

Design and development of a feedback system for automatic treadmill speed adaptation

Treadmills with automatic speed adjustment offer a unique advantage: they can mimic outdoor conditions for specific training or testing protocols. This versatility makes them highly applicable in both sports and rehabilitation settings. This study presents a novel framework based on a feedback control loop system, conceived to control a treadmill belt speed using a non-invasive and low-cost sensor (Microsoft Kinect) to detect the users’ position and avoid object obstruction issues. The speed of the treadmill belt is regulated according to the user’s speed, by means of a proportional-integrative-derivative (PID) controller and a parabolic gain function. By tuning the gain function parameters, the user can customize the response of the treadmill. Position data collected during exercise using the Microsoft Kinect sensor was compared with that collected with a stereophotogrammetric motion capture system, showing promising results in terms of accuracy in position assessment. The comparison highlighted a 0.9 degree of correlation between the two systems during the running and cross-country indoor skiing tests performed. In addition, upon considering the relationship between the differences and averages of the two measures, no systematic bias was identified. The system proved to be functional for running and cross-country skiing, and it can therefore re-create similar typical characteristics of outdoor environments (e.g., speed and slope) thanks to the non-invasive user’s position detection and the customizable gain function.

Glimmer Synchronization Of Luciola Parvula Fireflies To External Stimulas

It is known that the Luciola parvula fireflies clearly emit the lights at regular time intervals and synchronization between multiple males are observed, but the details of synchronization have not yet been clarified. Presently, we took videos of the luminescence phenomenon of multiple male Luciola parvula fireflies and performed a correlation analysis on the details of the time change of the luminescence brightness of each male Luciola parvula fireflies. The 12 Luciola parvula fireflies are kept in glass wells so that each Luciola parvula fireflies can only moves in restricted areas and their luminescence is recorded by a digital movie camera at the video rate of 30 fps. According to the preliminary study, each glimmer lasts about 3 frames and thus the time series brightness is fitted by a parabolic function in order to give a precise timing of maximum brightness. First of all, time intervals between glimmers of each Luciola parvula fireflies are evaluated. Secondly, the glimmer correlation between different Luciola parvula fireflies are investigated. As a result, it was clarified that the light emission cycle of Luciola parvula fireflies is almost constant at 0.54 seconds, more than 70% of all light emissions are synchronized and the order of synchronization is almost fixed with very small delay. Consequently, it has been shown quantitatively that the glimmers of male Luciola parvula fireflies are well synchronized.

Renewable energy sources in Kyrgyzstan and energy supply to rural consumers

The study assesses the potential of renewable energy sources in Kyrgyzstan and explores their application to provide energy to rural consumers. This study used an approximation of the parabolic function of solar radiation change, statistical processing of data on the average annual water flow of small rivers, as well as calculation of the volume of manure produced and its processing into biogas and bio fertilisers to assess their potential in agriculture and environmental impact. Kyrgyzstan, located between 40 and 68° north latitude, has evenly distributed solar radiation, small rivers and biomass, which have significant renewable resources. The distribution of solar radiation on the territory corresponds to the normal law of the monthly average mathematical expectation of 175.79 kWh/(m²*month) and a standard deviation of 92.44 kWh/(m²*month). On average, each square metre of a solar power plant can produce 0.451 kWh of energy. The intervals of average water discharge between the small rivers of Kyrgyzstan follow a power law distribution with a mathematical expectation of 3.112 m³/s and a standard deviation of 2.46 m³/s. With a natural water flow rate of 0.652 m³/s, a low-pressure micro-hydroelectric power plant (with a water head of 1 to 2 m) can generate up to 8.95 kW of power. The total consumption of biogas by an average farm in Kyrgyzstan and its consumption for heating raw materials in a bioreactor during the cold season ranges from 16.34 to 18.93 kg/hour. This demand for biogas is met by producing domestic feedstock (animal manure) using compact biogas plants with reactors of up to 20 m³. These facts indicate that the use of the above-mentioned renewable energy sources in Kyrgyzstan can provide autonomous power supply to remote rural consumers and contribute to solving existing environmental problems, as well as energy-saving

Rapid variable-step computation of dynamic convolutions and Volterra-type integro-differential equations: RK45 Fehlberg, RK4

We introduce a novel, time-efficient adaptive Runge-Kutta computational scheme tailored for systematically solving linear and nonlinear Volterra-type Integro-Differential Equations (VTIDEs). This scheme is particularly effective for equations featuring a specific class of convolution memory, K(t,s)=aκ(t−s) within convolution integrals, where a,κ∈R. Such equations frequently arise in fields like viscoelasticity, population dynamics, epidemiology, control systems, financial mathematics, and neuroscience, where current states are influenced by historical data, especially in dynamical systems within physics. Our proposed scheme demonstrates remarkable computational efficiency, achieving a cost-effective computation time of O(Nt) for Nt iterations, a significant improvement over existing schemes for VTIDE, which typically exhibit complexities of O(Ntlog⁡Nt) and O(Nt(log⁡Nt))2. The key feature of our scheme lies in its ability to elegantly and efficiently handle convoluted integrals of varying complexity such as those including nonlinearities and derivatives. Specifically, our approach eliminates the need for backstage integration, allowing the main Runge-Kutta scheme to operate efficiently with only a few data points at each time step. This not only saves time but also prevents potential data storage issues while rendering numerical implementation straightforward and tractable. Moreover, this article introduces flexible discretizations for integration and differentiation, accommodating unequally spaced data points. We implement a fast RK45-Fehlberg algorithm (TE-RK45-Fehlberg), which we use to solve three examples of nonlinear VTIDE presented in this study. We further demonstrate the effectiveness of our approach by using an example, where we employ a novel parabolic step size generator function to efficiently generate step sizes for a computationally fast RK4 scheme (Flex-RK4). For all examples tackled using TE-RK45-Fehlberg and Flex-RK4, we generate plots that compare numerical data with exact solutions, providing insight into the accuracy and reliability of our approach. We also show how the scheme provides straightforward ways of transforming it into implicit forms and obtaining stability curves.

Equations for efficient cycle-by-cycle computation of fatigue crack retardation and acceleration due to amplitude changes

The paper presents an efficient methodology for computation of the evolution of the fatigue crack retardation and acceleration effects due to crack closure during variable amplitude loading. The presented approach can lead to a very effective tool for estimation of residual fatigue life of engineering structures. Conventionally, consideration of variable-amplitude loading is either computationally too demanding, such as in finite element modelling, or challenging to unify the strip-yield model with another computational code. The simple analytical formulae available in literature are based on residual stress, which does not describe correctly the delayed retardation effect and in some cases it can be non-conservative. This work presents simple analytical equations describing the development of the crack closure based on the results of the strip-yield model for amplitude changes at the load ratio R = 0.1. The parabolic function enables simple computation of the maximum crack closure occurring after an overload without running any simulation. The equations also enable computation of crack closure behaviour following amplitude changes with the consideration of various cyclic material properties reflected in the tuneable ratio between the monotonic and cyclic plastic zone sizes.

Long-time asymptotic behavior for the Hermitian symmetric space derivative nonlinear Schrödinger equation

Abstract Resorting to the spectral analysis of the 4 × 4 matrix spectral problem, we construct a 4 × 4 matrix Riemann–Hilbert problem to solve the initial value problem for the Hermitian symmetric space derivative nonlinear Schrödinger equation. The nonlinear steepest decent method is extended to study the 4 × 4 matrix Riemann–Hilbert problem, from which the various Deift–Zhou contour deformations and the motivation behind them are given. Through some proper transformations between the corresponding Riemann–Hilbert problems, the basic Riemann–Hilbert problem is reduced to a model Riemann–Hilbert problem, by which the long-time asymptotic behavior to the solution of the initial value problem for the Hermitian symmetric space derivative nonlinear Schrödinger equation is obtained with the help of the asymptotic expansion of the parabolic cylinder function and strict error estimates.

Retrospective determination of exposure temperature of standing trees during wildfires with solid-state NMR

Timber plantations across the world are suffering from the effects of increasingly frequent wildfires, which potentially degrade the wood of affected trees, depending on the exposure temperature and time. However, it is rather complicated to determine the exact temperature of the fire, or the temperature to which the wood was exposed. This study aimed to determine the exposure temperature of wood retrospectively through solid-state NMR analysis. Models were developed from softwood and hardwood samples exposed to defined temperatures, which successfully linked the NMR signal to the exposure temperature. Various fit equations were developed to link the half-width or peak area of the NMR signal to the exposure temperatures. Hard- and softwoods displayed noticeable differences: a linear function best described the half-width in the higher temperature region for Pine and Eucalyptus, whereas a parabolic function for the peak area of Eucalyptus yielded the best correlation to the entire temperature range. This non-destructive and direct method offers a valuable evaluation method to determine, if wood in burnt trees is degraded and can be processed. An informed choice can be made on the decision to use, or discard burnt wood.

Reflection Coefficient for Two‐Dimensional Propagation of Fast Magnetosonic Waves Emerging One‐Dimensional Mesoscale Density Boundaries

AbstractIn recent studies, two‐dimensional propagation model of fast magnetosonic (MS) waves has been proposed to interpret the satellite observations of MS waves knocking into a density boundary. Although the theoretical model is able to capture the main properties of the two‐dimensional propagation of MS waves, quantitative description about the MS wave behaviors has not been given yet. Here, with the assumption of a parabolic function for the potential function near its minimum, we solve the wave equation only with a potential function to obtain the reflection coefficients. It is found that the wave equation with a potential function can describe the full reflection and full transmission of MS waves rather well. Furthermore, the first‐order derivative term in the wave equation is utilized to modify the reflection coefficient when the minimum of the potential function is near zero. Our result is helpful for further understanding the two‐dimensional propagation of MS waves.

Ballast Settlement Accumulation in Zones with Unsupported Sleepers

The high influence of impact and vibration on the behavior of crushed stone and ballast materials has been known for a long time. The zones with unsupported sleepers, which are always present in transition zones, crossings, welds, etc., are typically characterized by impact interaction, ballast full unloading, and additional preloading. However, no studies on ballast layer settlements consider impact vibration loading. Moreover, the influence of the cyclic loading on the ballast settlement intensity is considered ambiguously, with both decelerating and accelerating trends. The comprehensive literature review presents the influence of factors on settlement intensity. The present study aims to estimate the long-term processes of sleeper settlement accumulation depending on the loading factors: impact, cyclic loading, and preloading. The typical for a void zone ballast loading pattern was determined for various void sizes and the position along the track by using a model of vehicle-track interaction that was validated by experimental measurements. The loading patterns were parametrized with four parameters: maxima of the cyclic loading, impact loading, sleeper acceleration, and minimal preloading. A specially prepared DEM simulation model was used to estimate the ballast settlement intensity after initial settlement stabilization for more than 100 loading patterns of the void zone cases. The settlement simulation results clearly show that even a low-impact loading pattern causes many times increased settlement intensity than ordinary cyclic loading. Moreover, the initial preloading in the neighbor-to-void zones can cause even a decrease in the settlement intensity compared to the full ordinary or partial unloading. A statistical analysis using a machine learning approach and an analytic one was used to create the model for the intensity prediction regarding the loading patterns. The analytic approach demonstrates somewhat lower prediction quality, but it allows to receive plausible and simple analytic equations of the settlement intensity. The results show that the maximal cyclic loading has a nonlinear influence on the settlement intensity that corresponds to the 3–4 power function, and the impact loading is expressed by the linear to parabolic function. The ballast’s minimal preloading contributes to the reduction of the settlement intensity, especially for high cyclic loadings that are typical for neighbor-to-void zones. The results of the present study could be used for the complementing of the present phenomenological equations with the new factors and further application in the algorithms of the settlements accumulation prediction.

Parabola or second-degree polynomial function exploration in initial teacher training: Integrating intuition and didactic situations

This work is the result of a master’s investigation in Brazil, which discusses the teaching of the parabola in the initial training of mathematics teachers. Our theoretical framework addresses the relationship between intuition and the dialectics of the theory of didactic situations, which supports the analysis of the results of this study Its objective was to identify and register categories of intuitive reasoning manifested by the subjects when solving a didactic situation involving the parabola with GeoGebra software contribution. The methodology adopted was didactic engineering, experimented with eight students in initial training at a Brazilian public university, among 6<sup>th</sup> and 9<sup>th</sup> undergraduate semesters. The posterior analysis and validation allowed us to verify the need to discuss the parabola, articulating its geometric, algebraic, and analytical views, as well as to reinforce the importance of its teaching with the use of technology.

Characterizations of parabolic Muckenhoupt classes

This paper extends and complements the existing theory for the parabolic Muckenhoupt weights motivated by one-sided maximal functions and a doubly nonlinear parabolic partial differential equation of p-Laplace type. The main results include characterizations for the limiting parabolic A∞ and A1 classes by applying an uncentered parabolic maximal function with a time lag. Several parabolic Calderón–Zygmund decompositions, covering and chaining arguments appear in the proofs.

On behavior analysis of solutions for the coupled higher-order WKI equation

A coupled higher-order WKI equation is proposed and its 4 × 4 matrix Lax pair is derived. On the basis of spectral analysis of the 4 × 4 matrix spectral problems related to the coupled higher-order WKI equation, the basic Riemann–Hilbert problem is established by using the inverse scattering transformation. Resorting to the nonlinear steepest descent method, the solution of the basic Riemann–Hilbert problem is transformed to the solution of the model Riemann–Hilbert problem after several contour deformation. Finally, we explicitly obtain the long-time asymptotics of the solution to the Cauchy problem of the coupled higher-order WKI equation in the solitonless sector region with the aid of the parabolic cylinder function.

Coherent long-term body-size responses across all Northwest Atlantic herring populations to warming and environmental change despite contrasting harvest and ecological factors.

Body size is a key component of individual fitness and an important factor in the structure and functioning of populations and ecosystems. Disentangling the effects of environmental change, harvest and intra- and inter-specific trophic effects on body size remains challenging for populations in the wild. Herring in the Northwest Atlantic provide a strong basis for evaluating hypotheses related to these drivers given that they have experienced significant warming and harvest over the past century, while also having been exposed to a wide range of other selective constraints across their range. Using data on mean length-at-age 4 for the sixteen principal populations over a period of 53 cohorts (1962-2014), we fitted a series of empirical models for temporal and between-population variation in the response to changes in sea surface temperature. We find evidence for a unified cross-population response in the form of a parabolic function according to which populations in naturally warmer environments have responded more negatively to increasing temperature compared with those in colder locations. Temporal variation in residuals from this function was highly coherent among populations, further suggesting a common response to a large-scale environmental driver. The synchrony observed in this study system, despite strong differences in harvest and ecological histories among populations and over time, clearly indicates a dominant role of environmental change on size-at-age in wild populations, in contrast to commonly reported effects of fishing. This finding has important implications for the management of fisheries as it indicates that a key trait associated with population productivity may be under considerably less short-term management control than currently assumed. Our study, overall, illustrates the need for a comparative approach within species for inferences concerning the many possible effects on body size of natural and anthropogenic drivers in the wild.