Third-order Function Research Articles

Shape Functions Development for Beam-Column Element with Semi-Rigid Connections in Second-Order Steel Frame Analysis

The objective of this paper is to provide a novel method for developing the shape functions of a beam-column element with semi-rigid connection ends, thereby establishing a static analysis method for semi-rigid steel frames. This method takes into account the influence of the P-Delta effect, according to the finite element method based on displacement (FEM). The shape function is established directly from a third-order Hermitian displacement function polynomial combined with the bending element deflection differential equation. The linear elastic stiffness matrix, the geometric stiffness matrix of a semi-rigid connection beam-column, and the equilibrium equation of the element in a local coordinate system are simultaneously obtained by applying Castigliano’s theorem (Part 1) for elastic deformation potential energy expression. The computational program was developed using Matlab software, and the calculation results are verified against published research results, showing that the derived shape functions and the steel frame analysis method are reliable and trustworthy. In addition, this article also derives stiffness matrices and an equivalent nodal load vector for specific cases where the semi-rigid connection is fully rigid (FR) or a pin connection. The derived shape functions are polynomial expressions with coefficients that are simply calculated from the connection stiffness and the geometric and material characteristics of the element, making them highly convenient to use. Doi: 10.28991/CEJ-2025-011-01-021 Full Text: PDF

K-Nearest Neighbors with Third-Order Distance for Flooding Attack Classification in Optical Burst Switching Networks

Optical burst switching (OBS) is a network architecture that combines the advantages of packet and circuit switching techniques. However, OBS networks are susceptible to cyber-attacks, such as flooding attacks, which can degrade their performance and security. This paper introduces a novel machine learning method for flooding attack detection in OBS networks, based on a third-order distance function for k-nearest neighbors (KNN3O). The proposed distance is expected to improve detection accuracy due to higher sensitivity with respect to the distance difference between two points. The developed method is compared with seven other machine learning methods, namely standard KNN, KNN with cosine distance (KNNC), multi-layer perceptron (MLP), naive Bayes classifier (NBC), support vector machine (SVM), decision tree (DT), and discriminant analysis classifier (DAC). The methods are further assessed using five metrics: accuracy, precision, recall, F1-score, and specificity. The proposed method achieved an accuracy of 99.3%, outperforming the original KNN, MLP, and SVM, which achieved accuracies of 99%, 76.4%, and 94.7%, respectively. The results show that KNN3O is the best method for flooding attack detection in OBS networks, as it achieves the highest scores in all five metrics.

Higher-order stochastic averaging for investigating a vehicle suspension system with nonlinear damping and stiffness

The paper deals with a quarter-car model with nonlinear damping and stiffness under white noise base excitation using the higher-order stochastic averaging method for analyzing approximate responses. Recently, a novel higher-order averaging procedure has been developed to find analytically the first-, second-, and third-order stationary joint probability density functions (PDF) of amplitude and full phase by solving the corresponding Fokker-Planck-Kolmogorov (FPK) equation, and it will be extended to the nonlinear quarter-car model. Accordingly, the mean-square responses such as the displacement, and velocity of the sprung mass can be obtained analytically. The influences of excitation intensity on the dynamic responses, as well as the variations of linear and nonlinear damping, are analyzed. A very satisfactory agreement is found between the accuracy of the solutions corresponding to higher-order stochastic averaging and that of Monte Carlo simulation.

Human-derived food shrinks home ranges and alters resource selection of mammals at the urban-wild interface

Protected lands are an important source of food, shelter, and reproductive opportunities for wildlife, especially in urban landscapes. When urban development abuts the edges of undisturbed ecosystems, synanthropic species can alter their foraging behaviors and movement to utilize human-supplemented resources throughout the urban-wild interface. Therefore, urban development on the edges of protected lands can have pronounced effects on animal movement and ecosystem functions. Iconic urban adaptive mesopredators such as northern raccoons (Procyon lotor) and Virginia opossums (Didelphis virginiana) often benefit from human-supplemented food sources such as unsecured garbage, pet food, and fresh water when available. To investigate how urban edges affect the movements of urban-adapted omnivores within conservation lands, we estimated home ranges and third-order resource selection of 27 raccoons and 12 opossums with GPS collars throughout the protected areas of northern Key Largo, FL, USA between April 2022–October 2023. The proportion of urban development in an individual's home range was the most influential factor associated with home range size, followed by species and sex. Individuals with greater proportions of residential neighborhoods and commercial areas in their home ranges exhibited smaller home ranges. Third-order resource selection functions identified both mesopredator species using residential and commercial land use areas more than they were available on the landscape. These results indicate that urban areas attract urban-adapted mesopredators from protected areas and result in smaller home ranges in the face of abundant human-derived food. Reduced home ranges on edges can support higher densities of animals, which may increase rates of disease transmission, especially when the urban borders support populations of feral domestic species. Shifting foraging behaviors from the protected areas to urban edges could have cascading downward effects if seed-dispersing roles are diluted. As urbanization increases and the distance between wild lands and human disturbance decreases, it is increasingly important to study the mechanisms of how urban development on the edges of protected areas affect the movement of wildlife.

The shadows of quintessence non-singular black hole

In 2022, the Event Horizon Telescope (EHT) collaboration has reported the first observations of Sagittarius A*(SgrA*). Applying the EHT observational results, we find out constraints on non-singular Hayward parameter of regular dark energy black hole. Considering these constraints and different thin disk accretion, we present a detailed investigation into influence of different dark energy and Hayward parameters on shadows from non-singular Hayward black holes. In the first second-order attenuation function model, corresponding shadow radius and peak for observed intensity from direct image decrease with increasing dark energy parameter and Hayward parameter. However, for the lensing ring and photon ring, corresponding peak become bigger as dark energy parameter increase in case of fixed Hayward parameter. In the second third-order attenuation function model, significantly different from model 1, above two rings completely overlay on the direct image, resulting in two distinct peaks in the observed intensity. As increase of Hayward and dark energy parameters, the difference between the two peaks decreases, and shadows and observed intensity decrease. In the final inverse trigonometric function attenuation model, the result shows corresponding lensing ring as well as photon ring can be distinguished within the superposition region, and the superposition region becomes larger. With the increase of the dark energy parameter, the shadow radius exhibits a decreasing trend, while observed intensity increases. However, with the increase of the Hayward parameter, both decreases. Compared with the first two models, the shadow radius becomes smaller, but the observed intensity becomes larger, making the bright ring wider and brighter. Therefore, different accretion models and non-singular Hayward parameters can give rise to interesting and distinguish characteristic for the black hole shadow and rings.

Shadow and photon ring of black hole in asymptotically safe gravity

In this paper, we discuss the influence of thin disk accretion and asymptotically safe (AS) gravity correction parameters on the shadow and photon ring of black hole. For the thin disk accretion, the dark region is the shadow of the black hole, and the bright photon ring is composed of direct image, lensing ring, and photon ring. For the specific intensity of the radiation source of the accretion disk, we consider three different emission profile models. For the second-order attenuation function model in which the emission starts from the innermost circular orbit, direct image, lensing ring, and photon ring can be clearly distinguished. The direct image contributes most of the brightness, and the lensing ring contributes a small portion, while the contribution of the photon ring can almost be ignored. And the observed corresponding intensity peak decreases with the increase of the AS gravity parameter, that is, the corresponding brightness of the photon ring darkens as correction parameter increases. For the third-order attenuation function model in which the emission begins at the radius of the photon sphere, lensing ring and photon ring are superimposed on the direct radiation. Thus a new extreme value of the observed intensity emerges, and the extreme value increases with the increase of the AS gravity parameter, which leads to a brighter observed photon ring. For the anti-trigonometric attenuation function model in which the radiation starts from the event horizon, the superposition range of lensing ring and photon ring on the direct radiation becomes larger, which makes photon ring wider. The smaller the AS gravity parameter, the more difficult it is to distinguish between the lensing ring and photon ring, and the brighter the photon ring turns. In short, the results show that the shadow radius decreases with the increase of the AS correction parameter. For different AS gravity correction parameters, the light intensities of emission source, especially emission profiles of the observed intensity are significantly different, resulting in obvious differences in observed emission intensity between the shadow of the black hole and the bright photon ring of the black hole.

Fundamental frequencies of cracked FGM beams with influence of porosity and Winkler/Pasternak/Kerr foundation support using a new quasi-3D HSDT

In this study, we have introduced, for the first time, a novel integral quasi-3D higher-order shear deformation theory (HSDT) employing a third-order shape function. This approach is employed to analyze the free vibration characteristics of a cracked porous functionally graded material (FGM) beam supported on a three-parameter elastic foundation (Winkler/Pasternak/Kerr). This new Quasi HSDT introduces a stretching effect that surpasses the capabilities of FSDT and other HDST. The employed shape function satisfies the conditions of shear stress nullity at both the higher and lower facets without the need for correction factors. The study incorporates a mathematical model representing Winkler/Pasternak/Kerr foundation types into the Hamiltonian to derive the equations of motion. The FGM beam studied in this paper is assumed to be composed of materials with a distribution that varies according to a power law along its height. Our results are compared with previous studies and we reinforce our findings with a parametric study assessing the impact of crack attributes on the natural frequencies of the FG plate. This study presents an advanced integral quasi-3D HSDT, applied for the first time, to analyze the behavior of FG beams resting on a three-parameter foundation.

Spline retracker: a geometrical retracking algorithm for coastal and open ocean altimetry

Satellite altimetry has enhanced the understanding of ocean dynamics through high-rate sampling and global coverage. However, land contamination and bad reflection effects limit its accuracy. We present a geometrical method for retracking altimetry waveforms in coastal areas. Our method follows a geometrical assumption related to the symmetrical reciprocal motion of the radar pulse. Based on this assumption, the altimetry waveform is modelled as a continuous and differentiable third-order spline function, and the symmetry point of this function is considered as the retracking gate. The spline retracking algorithm is validated against the tide gauges at Onsala, Halmstad, and Muscat stations in Sweden and Oman, and its performance is compared with existing retracking algorithms. Our results showed a remarkable reduction of 50-91% in the unbiased-Root-Mean-Squared-Error (ubRMSE) and an increase of at least 13% in correlation coefficients when compared with other algorithms in Swedish coast. This algorithm presented equivalent results with the threshold and improved threshold retrackings in Muscat station, based on Jason-2 measurements. However, along the Jason-3 pass, our spline method showed a considerable reduction of 80% in ubRMSE and the minimum increase of 42% in correlation coefficients than the empirical algorithms. This method also outperformed the ALES algorithm in most cases.

The theory of fifth-order Stokes waves in a linear shear current

In this study, a new set of fifth-order Stokes wave solutions, incorporating the effects of a linear shear current, is derived by using the perturbation method originally proposed for pure waves that was recently published. The present solutions are checked against the existing experimental data, the third-order stream function solutions, as well as the numerical results. The comparisons demonstrate that the present solutions are more accurate in describing the velocity distributions during wave propagation, especially in strong following currents and negative vorticity conditions. Subsequently, the present solutions are used to investigate the fluid particle trajectories for different wave–current interaction conditions. The results indicate that the background vorticity can alter the patterns of fluid particle trajectories and the direction of Stokes drifts.

Dynamic behavior of risers under nonlinear oceanic environmental loading

The riser system acts as the vital link between the subsea blow-out preventer and the drilling platform. Affected by factors like top tension and marine environmental forces, the riser undergoes deformation and wear, carrying the risk of environmental pollution and financial losses upon failure. Hence, this study examines the riser's dynamic response to marine environmental loading. Initially, the motion differential equation for the riser system under the influence of nonlinear oceanic load is deduced using the principle of minimum potential energy and the variational method for extremum seeking. Subsequently, a nonlinear wave-current load model based on the Morrison equation is established, and the resulting equation is discretized into a finite element model using third-order Hermite interpolation function and the Galerkin weighted residual method. Finally, the dynamic response of the riser is scrutinized employing the Newmark numerical integration method. The study also investigates the impact of both oceanic environmental parameters and drilling parameters on the riser’s dynamic behavior. Comparative analysis of the numerical results reveals that the maximum displacement of the riser occurs at the middle section, whereas the maximum deflection angle is observed at the end of the riser. The periodicity of the deflection angle response is influenced by the position of the riser, showing a trend of decreasing and then increasing from the middle section towards the ends. Notably, the top tension and the velocity of the surface tidal current significantly affect the dynamic behavior of the riser. The findings of this study provide a theoretical foundation for the assessment of riser reliability and the determination of operational parameters.

Some Results on Third-Order Differential Subordination and Differential Superordination for Analytic Functions Using a Fractional Differential Operator

In this study, we explore the implications of a third-order differential subordination in the context of analytic functions associated with fractional differential operators. Our investigation involves the consideration of specific admissible classes of third-order differential functions. We also extend this exploration to establish a dual principle, resulting in a sandwich-type outcome. We introduce these admissible function classes by employing the fractional derivative operator DzαSN,Sϑz and derive conditions on the normalized analytic function f that lead to sandwich-type subordination in combination with an appropriate fractional differential operator.

Phase field modeling with large driving forces

There is growing interest in applying phase field methods as quantitative tools in materials discovery and development. However, large driving forces, common in many materials systems, lead to unstable phase field profiles, thus requiring fine spatial and temporal resolution. This demands more computational resources, limits the ability to simulate systems with a suitable size, and deteriorates the capability of quantitative prediction. Here, we develop a strategy to map the driving force to a constant perpendicular to the interface. Together with the third-order interpolation function, we find a stable phase field profile that is independent of the magnitude of the driving force. The power of this approach is illustrated using three models. We demonstrate that by using the driving force extension method, it is possible to employ a grid size orders of magnitude larger than traditional methods. This approach is general and should apply to many other phase field models.

Detecting the damage of concrete subjected to fatigue load coupled with freeze-thaw cycles using alternating current electric impedance spectroscopy

The coupled effect of fatigue load and freeze-thaw cycles may cause damage to concrete structures. This paper explores the potential of alternating current (AC) electric impedance spectroscopy (EIS) technology to characterize the micro-damage of concrete under the mentioned coupled effect. Firstly, the mass loss rate (ML) and the relative dynamic elastic modulus (RDEM) evolution were employed to evaluate damage of concrete subjected to freeze-thaw cycles coupled with fatigue load. Comparatively, the degradation was also described in terms of ACEIS. The functional correlations between impedance parameters and concrete damage under the coupled effect were identified using the maximum entropy theory. The results show that the impedance parameters, RCCP and RCP, follow a consistent three-stage trend with ML and RDEM as the number of freeze-thaw cycles and fatigue load time increase. RCCP and RCP, however, are more vulnerable to early damage and can be used to faster recognize concrete damage caused by the coupled effect. Compared to RCP, RCCP is a more stable indicator for characterizing concrete damage, as it is less affected by external factors. The relationships between impedance parameters (RCCP and RCP) and traditional indicators (ML and RDEM) at stage II can be quantified as a third-order power function.

A single-variable proof of the omega SPT congruence family over powers of 5

In 2018, Liuquan Wang and Yifan Yang proved the existence of an infinite family of congruences for the smallest parts function corresponding to the third-order mock theta function omega (q). Their proof took the form of an induction requiring 20 initial relations, and utilized a space of modular functions isomorphic to a free rank 2 {mathbb {Z}}[X]-module. This proof strategy was originally developed by Paule and Radu to study families of congruences associated with modular curves of genus 1. We show that Wang and Yang’s family of congruences, which is associated with a genus 0 modular curve, can be proved using a single-variable approach, via a ring of modular functions isomorphic to a localization of {mathbb {Z}}[X]. To our knowledge, this is the first time that such an algebraic structure has been applied to the theory of partition congruences. Our induction is more complicated, and relies on sequences of functions which exhibit a somewhat irregular 5-adic convergence. However, the proof ultimately rests upon the direct verification of only 10 initial relations, and is similar to the classical methods of Ramanujan and Watson.

Higher-order averaging procedure for performance analysis of a mono-stable Duffing piezoelectric energy harvesting system under white noise excitation

This work is concerned with the energy harvesting of mono-stable Duffing oscillators with piezoelectric coupling under white noise excitation. This is the first time that the higher-order averaging procedure is extended to find the first-, second- and third-order approximate stationary joint probability density functions of amplitude and full phase by solving the corresponding Fokker-Planck-Kolmogorov (FPK) equation. Then the mean-square electric voltage and mean output power are derived through the approximate relation between the electric voltage and the mechanical states. The analytical results are verified by Monte Carlo simulation. The effects of system parameters, such as excitation intensity, mechanical damping term, nonlinear stiffness coefficients, and the resistance ratio of the resistive–capacitive circuit on the mean-square displacement and mean output power are discussed in detail. It has been shown that the averaging procedure of the averaging method of coefficients in FPK equation can be considered as a particular case of the proposed higher-order averaging procedure and the higher-order approximate solutions are much more accurate than the one obtained by the averaging method of coefficients in FPK equation.© 2022 The Authors. Published by Elsevier Ltd.

Application of two variable hermite splines in digital image processing

During the research, the use of piece-polynomial methods in digital processing of images was considered. The Hermite spline function is chosen from piece-polynomials as a mathematical model in digital processing of signals, and the construction of a two-variable third-order Hermite spline function is presented. An image restoration algorithm was developed based on the constructed mathematical model.

High-accuracy lunar global brightness-temperature mapping using third-order Fourier fitting and co-kriging interpolation

The use of microwave radiation data collected by lunar orbiters to infer the lunar brightness-temperature (TB) distribution is of great importance to lunar scientific exploration. The TB data acquired by China's Chang'E-2 passive microwave radiometer (MRM) is one of the basic materials for lunar temperature mapping. Core issues of lunar microwave TB mapping include data fitting at different latitudes, outlier removal, and integrated spatial interpolation considering influencing factors. We applied a third-order Fourier function to fit the lunar TB data, which effectively removes outliers considering the fitting characteristic and three-sigma rule (FCTSR). Because the lunar surface temperature is influenced by a variety of factors such as topography and albedo, we then built a multi-factor co-kriging interpolation method to perform lunar TB mapping accurately. The cross-validation shows that, in terms of the mean-absolute-error and root-mean-square-error, the multi-factor co-kriging interpolation improves the mapping accuracy by 30%–60% compared to inverse distance interpolation and ordinary kriging interpolation. The analysis shows that topography and albedo are the most important factors influencing the lunar TB, and the TB maps at 37 GHz reveal cold spots that can be considered as younger craters. It is concluded that the proposed data fitting, denoising and spatial interpolation methods significantly improve the lunar TB mapping. The results and scientific data can also provide a basic energy map for lunar roving path planning and subsequent lunar exploration.

Study of the Green Coronal Line with Altitude from Out-of-Eclipse Observations during Solar Cycle 24

The results of studies of the coronal emission line λ = 5303 Å (Fe XIV) for the period of solar cycle 24 are presented. The spectral data were obtained with an out-of-eclipse Lyot coronagraph at the Mountain Astronomical Station of the Pulkovo Observatory, Russian Academy of Sciences (near Kislovodsk). As a result of processing of out-of-eclipse observations, a database of three types of daily coronal maps of green line intensity I5303 was created with a height distribution h from 1R☉ to 1.38R☉ (R☉ is the radius of the Sun). Irregularities along the λ = 5303 Å line were found and identified, which were associated with the configuration of magnetic fields in the solar corona above active regions. The length of the green line from the position angle of the Sun was calculated. We have shown that the time distribution of the line length in the polar regions has two maxima, which coincide with the times of the reversal of the polar magnetic field on the Sun. The maximum values of the average length of the coronal line along the entire limb occur in 2012–2014. For different phases (the rise, the period of maximum, the decline, and the minimum solar activity) of this solar cycle and for different regions of solar activity, dependences of the height variations of the I5303 values were plotted and studied. The regression equations for these fitting curves are presented. The variation in I5303 with height for the polar regions is most likely determined by a logarithmic function, and the approximating trend curves for the remaining latitudinal zones are determined by a third-order power function.

Stress wave propagation analysis of 2D-FGM axisymmetric finite hollow thick cylinder

In this paper, a numerical axisymmetric elastic stress wave propagation analysis of the 2D-FGM hollow thick finite cylinder has been performed. The new accurate material distribution model based on Mori-Tanaka scheme and third-order transition function for 2D-FGMs has been implemented in the graded finite element method with higher-order quadrilateral elements. A non-uniform impulsive pressure has been applied on the one-third of the top inner cylinder wall. Consequently, different wave velocities and reflections from the boundaries have been observed due to the graded distribution of material along with two directions. Moreover, the effects of two-dimensional material property distributions on the stress wave propagations and displacements through the cylindrical wall at different locations have been studied systematically. Normalized effective stresses based on von-Mises criteria has been performed in comparison with the internal impulsive pressure amplitude to provide a measure of the cylinder wall response to the mechanical impact loading, the effective stresses are nearly 2.5 times the internal pressure loading amplitude, closed to the impact location. According to the results, 2D material property gradations has effected the stresses significantly. Alternatively, desired stress values can be achieved through optimal design and appropriate choice of property distribution.

Shadow and photon ring of black hole in asymptotically safe gravity

In this paper, we focus on discussing the influence of thin disk accretion and asymptotically safe (AS) gravity correction parameter on the shadow and photon ring of black holes. For the thin disk accretion, the dark region is the shadow of the black hole, and the bright photon ring is composed of Direct image, lensing ring and Photon ring. For the specific intensity of the radiation source of the accretion disk, we consider three different emission profile models. For the second-order attenuation function model in which emission starts from the innermost circular orbit, Direct image, lensing ring and Photon ring can be clearly distinguished. The Direct image contributes most of the brightness, and the lensing ring contributes a small portion, while the contribution of the Photon ring can almost be ignored. And the peak value of the corresponding observed intensity decreases with the increase of the AS gravity parameter, that is, the corresponding brightness of the photon ring darkens as correction parameter increases. For the third-order attenuation function model in which the emission begins at the radius of the photon sphere, lensing ring and Photon ring are superimposed on the direct radiation. Thus a new extreme value of the observed intensity emerges, and the extreme value increases with the increase of the AS gravity parameter, which leads to observed photon ring brighter. For the anti-trigonometric attenuation function model in which the radiation starts from the event horizon, the superposition range of lensing ring and Photon ring on the direct radiation becomes larger, which makes photon ring wider. The smaller the AS gravity parameter is, the more difficult it is to distinguish the lensing ring and Photon ring, and the photon ring gets brighter. In short, the results show that the shadow radius decreases with the increase of the AS correction parameter. For different AS gravity correction parameters, the light intensity of emission source, especially emission profiles of the observed intensity are significantly different, resulting in obvious differences for the shadow and bright photon ring of the black hole.