Arctangent Research Articles

The Method of Probabilistic Nodes Combination in Biomechanics

Proposed method, called Probabilistic Nodes Combination (PNC), is the method of 2D curve modeling and handwriting identification by using the set of key points. Nodes are treated as characteristic points of signature or handwriting for modeling and writer recognition. Identification of handwritten letters or symbols need modeling and the model of each individual symbol or character is built by a choice of probability distribution function and nodes combination. PNC modeling via nodes combination and parameter γ as probability distribution function enables curve parameterization and interpolation for each specific letter or symbol. Two-dimensional curve is modeled and interpolated via nodes combination and different functions as continuous probability distribution functions: polynomial, sine, cosine, tangent, cotangent, logarithm, exponent, arc sin, arc cos, arc tan, arc cot or power function.

A Federated Learning Framework Based on Incremental Weighting and Diversity Selection for Internet of Vehicles

With the rapid increase of data, centralized machine learning can no longer meet the application requirements of the Internet of Vehicles (IoV). On the one hand, both car owners and regulators pay more attention to data privacy and are unwilling to share data, which forms the isolated data island challenge. On the other hand, the incremental data generated in IoV are massive and diverse. All these issues have brought challenges of data increment and data diversity. The current common federated learning or incremental learning frameworks cannot effectively integrate incremental data with existing machine learning (ML) models. Therefore, this paper proposes a Federated Learning Framework Based on Incremental Weighting and Diversity Selection for IoV (Fed-IW&DS). In Fed-IW&DS, a vehicle diversity selection algorithm was proposed, which uses a variety of performance indicators to calculate diversity scores, effectively reducing homogeneous computing. Also, it proposes a vehicle federated incremental algorithm that uses an improved arctangent curve as the decay function, to realize the rapid fusion of incremental data with existing ML models. Moreover, we have carried out several sets of experiments to test the validity of the proposed Fed-IW&DS framework’s performance. The experimental results show that, under the same global communication round and similar computing time, the Fed-IW&DS framework has significantly improved performance in all aspects compared to the frameworks FED-AVG, FED-SGD, FED-prox & the decay functions linear, square curve and arc tangent. Specifically, the Fed-IW&DS framework improves the Acc (accuracy), loss (loss), and Matthews correlation coefficient (MCC) by approximately 32%, 83%, and 66%, respectively. This result shows that Fed-IW&DS is a more reliable solution than the common frameworks of federated learning, and it can effectively deal with the dynamic incremental data in the IoV scenario. Our findings should make a significant contribution to the field of federated learning.

A generic inequality for basic special functions

We introduce a generic inequality of the form and show that many special functions such as gamma and polygamma functions, Riemann zeta function, beta and incomplete beta functions, Gauss and confluent hypergeometric functions, elliptic integrals and error function satisfy such type of the inequality. We also obtain new inequalities of this type for inverse trigonometric functions and .

Degree of Pelvic Rotation in the Coronal Plane on Postoperative Radiographs Obtained after Total Hip Arthroplasty.

There are many published cup anteversion measurements for postoperative total hip arthroplasty (THA), including Liaw's, Lewinnek's, and Murray's methods. However, most measurements ignore the potential pelvic rotation on radiographs except in Liaw's method. Without considering pelvic rotation, clinicians can miscalculate cup anteversion. Therefore, we aimed to quantify the mean degree of pelvic rotation. Herein, we collected 388 radiographs of 98 postoperative THA hips of 77 patients and measured pelvic rotation by measuring h, the horizontal displacement of the sacrococcygeal junction associated with the upper pole of the symphysis pubis, and ssd, the distance between the sacrococcygeal junction and pubic symphysis. The angle θ of pelvic rotation was defined as θ = arc sin (h/ssd) × (180°/π). The mean degree of pelvic rotation was then calculated. The standard deviation of h was 7.84 mm, and the mean ssd was 158 mm. The potential pelvic rotation was 2.50°. The p-values from the paired t-test were all >0.05 when interobserver and intraobserver errors were assessed. This is the first study to quantify the potential pelvic rotation in the coronal plane on postoperative plain radiographs. The potential pelvic rotation was too large to be neglected during the measurement of cup anteversion.

Fixed-time prescribed performance fault compensation control of uncertain pure-feedback nonlinear systems

This study is primarily focused on the issue of low-complexity global fixed-time prescribed performance fault compensation control for uncertain nonlinear systems with actuator failures, and extra disturbance. First of all, a new fixed-time function in view of inverse trigonometric function is proposed and used to define the state errors. Then, proper root-type error conversion functions and smooth orientation functions are linked to design the continuous control signals in the state feedback controller. In addition, based on the idea of proof by contradiction, it is shown that the state errors converge to a predictable compact aggregate at a definite rate at a fixed time. Meanwhile, the boundedness of any closed-loop signals might be guaranteed. The simulation results and algorithm comparisons are delivered to demonstrate the effectiveness and superiority of the developed control strategy at last.

Taylor’s series expansions for real powers of two functions containing squares of inverse cosine function, closed-form formula for specific partial Bell polynomials, and series representations for real powers of Pi

Abstract In this article, by virtue of expansions of two finite products of finitely many square sums, with the aid of series expansions of composite functions of (hyperbolic) sine and cosine functions with inverse sine and cosine functions, and in the light of properties of partial Bell polynomials, the author establishes Taylor’s series expansions of real powers of two functions containing squares of inverse (hyperbolic) cosine functions in terms of the Stirling numbers of the first kind, presents a closed-form formula of specific partial Bell polynomials at a sequence of derivatives of a function containing the square of inverse cosine function, derives several combinatorial identities involving the Stirling numbers of the first kind, demonstrates several series representations of the circular constant Pi and its real powers, recovers Maclaurin’s series expansions of positive integer powers of inverse (hyperbolic) sine functions in terms of the Stirling numbers of the first kind, and also deduces other useful, meaningful, and significant conclusions and an application to the Riemann zeta function.

Kepler in search of the “Anaclastic”

This article was written as a tribute to Enrique Tirapegui who passed away on 10 March 2020. We were working on these issues with him. He will always be remembered.We describe Kepler’s attempt to discover the law of refraction. Its very elegant analysis can be seen as a numerical search for the solution of a differential equation. Kepler’s arguments and the reasons for his failure are discussed. This analysis, which precedes Descartes’ solution of the inverse tangent problem posed by Florimond de Beaune, is an opportunity to briefly recall some aspects of the history of differential equations. It is also an opportunity to propose a pedagogical activity that consists of introducing the law of refraction to students in the wake of Kepler and Descartes. This article is intended to be pedagogical. It is self-consistent and all the properties of the hyperbola necessary to follow Kepler’ arguments are demonstrated using elementary geometry. It will also be the material for the chapter of a book intended to revisit the elementary laws of physics using Euclidean geometry. This project started with Enrique Tirapegui. This work was presented at the XVIII Instabilities and Nonequilibrium Structures Workshop in tribute to Enrique held in December 2021 in Valparaiso, Chile.

Disturbance Rejection Control for Underwater Free-Floating Manipulation

This article presents a disturbance observer-based control (DOBC) framework for an autonomous underwater biomimetic vehicle-manipulator system (UBVMS) to implement underwater manipulation under unknown external disturbances. First, a fast arctangent nonsingularity terminal sliding mode controller (FANTSMC) with the inverse trigonometric function approach law is proposed to achieve the control of the UBVMS. Then, an improved high-order observer is proposed to track unknown disturbances including external disturbances and the manipulator compensation errors in a finite time. Finally, the DOBC framework, which consists of the FANTSMC and the disturbance observer, is applied to the underwater manipulation. Comparative simulations and practical underwater manipulation are conducted to validate the performance of our proposed method in disturbance tracking and underwater manipulation realization.

Relationship of Impaired Lumbar Spine-Hip Coordination During Sit To Stand and Stand To Sit with Functional Disability in Chronic Nonspecific Low Back Pain Patients.

This study aimed to investigate the relationships of lumbar spine-hip discoordination during sit-to-stand (STD) and stand-to-sit (SIT) with pain and functional disability in chronic nonspecific low back pain (CNLBP) patients. A cross-sectional observational study was conducted in a biomechanics laboratory of the physical therapy department located at the School of Rehabilitation Sciences, Iran University of Medical Sciences (Tehran-Iran). A total of 16 CNLBP patients (men 9, female 7) aged 18-40 years (mean 31.48) were selected according to our eligibility criteria. Furthermore, ten reflective markers were placed on the spinous processes of T12 and S2, posterior and anterior superior iliac spines, greater trochanters, and lateral epicondyles. The patients were instructed to perform STD and SIT tasks at a preferred speed without using their hands. Relative phase angle was used as an indicator of coordination and was identified as the inverse tangent of angular displacement/angular velocity. Moreover, the relative phase angle between the lumbar spine and right and left hip joints was measured by subtracting the phase angle of the hip joint from the lumbar spine joint. The ratios of the total movements of the lumbar spine to the total movements of the right and left hip joints were also calculated in the sagittal plane. Finally, Pearson correlation coefficients (r) were utilized to assess the association between variables. The results of this study indicated that kinematic parameters of the pain had statistically significant direct relationships with functional disability in CNLBP participants during STD and SIT with r values ranging from 0.57 (Pvalue = 0.021) to 0.85 (Pvalue<0.001) and 0.54 (Pvalue=0.053) to 0.82 (Pvalue<0.001), respectively. Out of the results of this study, it could be stated that pain and functional disability play a major role in lumber spine-hip discoordination, and it altered the movement ratio in CNLBP patients during STD and SIT. In clinical practice, clinicians should improve lumber spine-hip discoordination in patients with CNLBP since there is a linear relationship between kinematic parameters of the pain and functional disability in patients with CNLBP.

Hysteresis-based approach to estimate constituent properties of SiC/SiC ceramic matrix composites

Hysteresis loops relate with composite’s constituent properties and damage state. In this paper, a systematic approach to estimate the constituent properties of SiC/SiC composite using hysteresis-based damage parameters was developed. Four cases including small debonding energy (SDE), large debonding energy (LDE), long matrix crack spacing (LMCS), and short matrix crack spacing (SMCS) were considered for the hysteresis analysis. Closed-form solution for the unloading/reloading inverse tangent modulus (UITM/RITM), hysteresis width, interface debonding and slip ratio (IDR/ISR) were derived from the hysteresis theory. Effects of peak stress, interface shear stress (ISS), interface debonding energy (IDE), and matrix crack spacing (MCS) on the damage parameters were analyzed. Experimental hysteresis loops of unidirectional, cross-ply and 2D plain-woven SiC/SiC composites were characterized using the developed hysteresis-based damage parameters. The composite’s constituent properties of first matrix cracking stress (FMCS), ISS and IDE were estimated.

The Method of Probabilistic Nodes Combination in Handwritten Character Reconstruction

Proposed method, called Probabilistic Nodes Combination (PNC), is the method of 2D curve modeling and handwriting identification by using the set of key points. Nodes are treated as characteristic points of signature or handwriting for modeling and writer recognition. Identification of handwritten letters or symbols need modeling and the model of each individual symbol or character is built by a choice of probability distribution function and nodes combination. PNC modeling via nodes combination and parameter γ as probability distribution function enables curve parameterization and interpolation for each specific letter or symbol. Two-dimensional curve is modeled and interpolated via nodes combination and different functions as continuous probability distribution functions: polynomial, sine, cosine, tangent, cotangent, logarithm, exponent, arc sin, arc cos, arc tan, arc cot or power function.

A novel approach to reduce both front and rear side power losses in PERC solar cells using different combinations of transparent metal oxides

This paper aims to check the TMOs (transition metal oxides) influence on the front side passivation of c-Si PERC solar cells. This simulation work compares three different configurations with standard size c-Si wafers. Configuration (1) consists of SiNx/NiOx at the front and Al2O3/MoOx/SiNx at the rear, configuration (2) SiNx/WOx at the front side and Al2O3/MoOx/SiNx at the rear side, and configuration (3) SiNx/TiO2 at the front side and Al2O3/MoOx/SiNx at a rear side of c-silicon wafer simulated through Sunsolvesoftware. These configurations are selected after comparing the thickness effect of TMOs through the Wafer Ray tracer calculator. On the front side, thicknesses of NiOx, WOx, and TiO2 are tested, ranging from 10 nm to 100 nm, using a wafer ray tracer, from which the three best thicknesses are selected. All three selected configurations have an efficiency of 21.36%, 21.08%, and 20.95%, respectively. We have also compared all possible parameters such as reflections, absorptions, transmissions, open-circuit voltage, short circuit current, fill factor and efficiency.The front side losses were reduced due to TMO's thin layers. TMOs are used as transparent conductive oxide layers at the front side of the c-Si PERC solar cell to reduce the losses of the front side as supportive thin layers with ARC SiNx.

Calculation of the value of the functions of the complex variable with by an interval argument, we will design in the hyperbolic form

Information about the interval numbers presented in the classical form, the CENTER-RADIUS system and in the hyperbolic form is given. Rules for the transition from one of the forms of representation of interval numbers to others are proposed. Information is given on complex interval numbers, the real and imaginary parts of which are presented in hyperbolic form. The rules for performing basic arithmetic operations with these numbers and the calculation of interval values of power, exponential, logarithmic functions, direct and inverse trigonometric functions, direct and inverse hyperbolic functions are described. For functions of a complex variable, information about their real and imaginary parts is given. The list of functions corresponds to the functions of a complex variable included in the EXCEL system. Relationships are obtained for determining the real and imaginary parts of the secant, cosecant, tangent and cotangent functions for circular trigonometric and hyperbolic functions, which were absent in the most common reference literature. It is shown that the operations of multiplication, division and raising to an integer power are most appropriate to perform with complex interval numbers, which are defined in hyperbolic form. The operation of calculating the root of degree n from an interval complex number presented in hyperbolic form is most expediently performed using the CENTER-RADIUS system in combination with the hyperbolic form of representing the interval number. Relationships are obtained that make it possible to obtain a function of an interval complex variable equivalent to the original one and suitable for further work with complex functions and numbers presented in hyperbolic form and in the CENTER-RADIUS system. Examples illustrating the application of the proposed technique are given.

Reliability and Validity Analysis of Pelvic Sagittal Inclination Calculated by Inverse Cosine Function Method on Pelvic Anteroposterior Radiographs.

ObjectiveEvaluation of sagittal pelvic tilt is significant for hip surgeons. However, the accurate measurement of pelvic sagittal inclination (PSI) is still a challenge. The objective of this study is to propose a new method for measurement of PSI from pelvic anteroposterior radiograph based on the inverse cosine function obtained from individualized pelvic model.MethodsCollecting the imaging data of 30 patients with both pelvic CT and full‐length spine radiographs. Establishing pelvic model by customized 3D reconstruction software. The length of three groups of longitudinal and transverse line segments (A′p and B′) were measured from full‐length spine anteroposterior radiographs. The corresponding anatomical parameters, including A, B, b, ∠α, ∠γ, were measured and calculated on the same patient's pelvic model. The estimated PSI (ePSI) based on three groups of anatomical landmarks, including ePSI‐1, ePSI‐2, and ePSI‐3, were calculated by equation, ePSI=arccosA′pb*B′−∠α, and compared with the actual PSI (aPSI) measured by Surgamap software. For the reliability and validation evaluation, three observers measured these parameters in two rounds. Intra‐class correlation and inter‐class correlation were both calculated. Bland–Altman method was used to evaluate the consistency between the estimated PSI (ePSI) and the actual PSI (aPSI).ResultsePSI‐1 and ePSI‐2 showed excellent intra‐observer reliability (0.921–0.997, p < 0.001) and inter‐observer reliability (0.801–0.977, p < 0.001). ePSI‐3 had a fair inter‐observer reliability (0.239–0.823, p < 0.001). ePSI‐1 showed the strongest correlation with aPSI (r = 0.917, p < 0.001). Mean (maximum) absolute difference of ePSI‐1, ePSI‐2, and ePSI‐3 is 2.62° (7.42°), 4.23° (13.78°), and 7.74° (31.47°), respectively. The proportion of cases with absolute difference less than 5° in three groups were 86.7% (ePSI‐1), 66.7% (ePSI‐2), 56.7% (ePSI‐3).ConclusionThis new method based on inverse cosine function has good reliability and validity when used in the evaluation of PSI on pelvic anteroposterior radiographs.

An Optimized SIFT-OCT Algorithm for Stitching Aerial Images of a Loblolly Pine Plantation

When producing orthomosaic from aerial images of a forested area, challenges arise when the forest canopy is closed, and tie points are hard to find between images. The recent development in deep leaning has shed some light in tackling this problem with an algorithm that examines each image pixel-by-pixel. The scale-invariant feature transform (SIFT) algorithm and its many variants are widely used in feature-based image stitching, which is ideal for orthomosaic production. However, although feature-based image registration can find many feature points in forest image stitching, the similarity between images is too high, resulting in a low correct matching rate and long splicing time. To counter this problem by considering the characteristics of forest images, the inverse cosine function ratio of the unit vector dot product (arccos) is introduced into the SIFT-OCT (SIFT skipping the first scale-space octave) algorithm to overcome the shortfalls of too long a matching time caused by too many feature points for matching. Then, the fast sample consensus (FSC) algorithm was introduced to realize the deletion of mismatched point pairs and improve the matching accuracy. This optimized method was tested on three sets of forest images, representing the forest core, edge, and road areas of a loblolly pine plantation. The same process was repeated by using the regular SIFT and SIFT-OCT algorithms for comparison. The results showed the optimized SIFT-OCT algorithm not only greatly reduced the splicing time, but also increased the correct matching rate.

Quadrature phase detection based on a laser self-mixing interferometer with a wedge for displacement measurement

We report a novel and simple method to generate a pair of self-mixing signals with quadrature phase difference in He-Ne laser by introducing a wedge into the external cavity. A wedge phase-shift laser self-mixing effect theory is developed to explain the phase-shift phenomenon, and the relationship between the phase difference and the distance from the center of the iris diaphragm to the center of the spot is given. Then a simple and effective wedge-quadrature laser self-mixing interferometer is established for displacement measurement by the quadrature phase detection method. Experimental results show that the max error is 182 nm for a target displacement of 20 μm. This structure provides a useful exploration for phase-shift signal generation and simplifies the self-mixing signal demodulation process with only a single four-quadrant inverse tangent operation, which has potential application in real-time displacement measurement.

A novel current sensor indicator enabled WAFTR model for tool wear prediction under variable operating conditions

The health indicators (HIs) were extracted from the current sensor to represent the tool wear progression. The extracted HIs were found poorly correlated with the progression of tool wear as the raw current sensor signal was susceptible to the influence of other parts and structures in the machine tool. Hence, this paper proposed a novel current sensor-based HI that utilized the mean of inverse hyperbolic cosine function fitted to an envelope of the current signal to improve the correlation. Using the extracted HIs, many bespoke machine learning (ML) models have been developed by researchers. However, these models have many hyperparameters, difficult to interpret and especially poor prediction accuracy has been observed under variable operating conditions. This study overcame these issues by proposing a Weibull Accelerated Failure Time Regression (WAFTR) model, which combines process parameters data with HI for improving the prediction accuracy under variable operating conditions. This model mapped a functional relationship with tool wear in the form of probability density function to identify best HIs and acceleration/deacceleration factors which makes it interpretable. The acceleration/deacceleration factors are useful to deaccelerate the tool wear evolution by controlling the specific values of the machining parameters.

A blind spot in undergraduate mathematics: The circular definition of the length of the circle, and how it can be turned into an enlightening example

We highlight the fact that in undergraduate calculus, the number pi is defined via the length of the circle, the length of the circle is defined as a certain value of an inverse trigonometric function, and this value is defined via pi, thus forming a circular definition. We present a way in which this error can be rectified. We explain that this error is instructive and can be used as an enlightening topic for discussing different approaches to mathematics with undergraduate students.

Improper Integrals Involving Powers of Inverse Trigonometric and Hyperbolic Functions

Three classes of improper integrals involving higher powers of arctanh, arctan, and arcsin are examined using the recursive approach. Numerous explicit formulae are established, which evaluate these integrals in terms of π, ln2, the Riemann zeta function, and the Dirichlet beta function.

New Masjed Jamei–Type Inequalities for Inverse Trigonometric and Inverse Hyperbolic Functions

In this paper, we establish two new inequalities of the Masjed Jamei type for inverse trigonometric and inverse hyperbolic functions and apply them to obtain some refinement and extension of Mitrinović–Adamović and Lazarević inequalities. The inequalities obtained in this paper go beyond the conclusions and conjectures in the previous literature. Finally, we apply the main results of this paper to the field of mean value inequality and obtain two new inequalities on Seiffert-like means and classical means.