Logarithmic Function Research Articles

In this study, we investigate the interplay between player speed and ball control in soccer. We present a novel pitch control algorithm that quantifies the probability of a player gaining possession at any location on the field. Our model accounts for the heterogeneity of player speeds by measuring performance during matches and assigning each player a specific top speed. We then compare the pitch control percentages derived from our approach with those from classical models, which assume uniform top speeds for all players, and analyze the results across different player roles (defenders, midfielders, and forwards). Our findings reveal a positive correlation between a player's top speed and their accumulated pitch control, with certain players benefiting more from this relationship. However, this positive correlation is constrained by the role of the player in the team, with defenders achieving the highest accumulated pitch control despite not being the fastest. Furthermore, our methodology supports team-level analysis, identifying which teams gain the greatest advantage from their players' top speeds, and extends to comparisons between the first and second halves of matches. Our model also enables exploration of how changes in top speed may affect pitch control at both the individual and team levels. To facilitate this, we introduce the stamina factor, a parameter that adjusts a player's top speed. We find that the impact of the stamina factor on pitch control follows a logarithmic function, with the scaling factor quantifying the potential benefits of increased speed. Interestingly, the influence of the stamina factor varies significantly by player position. Overall, our approach provides valuable insights into which teams or players could benefit most from improvements in physical performance.

Piecewise logarithmic Chebyshev cardinal functions: Application for nonlinear integral equations with a logarithmic singular kernel

Increasing the yield per unit area is crucial for achieving stable growth in China’s cotton production. Based on the transcendental logarithmic production function model and using panel data from eight major cotton-producing provinces in China from 1990 to 2022, this paper measures the elasticity of climate factors and factor inputs and calculates the contribution rate of each factor influencing cotton yield increase. The results show that accumulated temperature positively impacts cotton yield, while precipitation and sunshine duration have negative effects. Climate factors contribute 7.95% to yield growth. Among input factors, agricultural machinery and labor inputs positively affect yield, whereas fertilizer input negatively affects it. Factor inputs contribute 44.21% to yield improvement. Technological progress also plays a role in enhancing cotton yield. Finally, the paper suggests improving meteorological disaster forecasting, optimizing input structures, and promoting agricultural research and technology services based on local conditions.

Background:During the COVID-19 pandemic in Peru, primary health care services redistributed personnel to hospitals. In this context, Antenatal care (ANC) could have been limited in the general population, especially among indigenous communities.Objective:To determine differences in compliance with adequate ANC based on ethnic groups in Peru prior to and throughout the COVID-19 pandemic.Methods:Observational, retrospective study, secondary analysis of the Demographic and Family Health Survey conducted in the course the period 2017–2022. The sample consisted of women of childbearing age from 15 to 49 years and considered the variables of ethnicity, ANC, and sociodemographic factors. For statistical analysis, generalized linear models from the Poisson family utilizing a logarithmic link function were used. Crude and adjusted Prevalence Ratios were calculated along with their 95% confidence intervals.Results:A total of 20,165 records were reviewed. Before the pandemic, the frequency of inadequate ANC was 57.91%, and during the pandemic, it increased to 64.91%. Compared to mestizos, belonging to the Quechua ethnic group (PR: 1.08; 95% CI: 1.04–1.13), Aymara (PR: 1.21; 95% CI: 1.12–1.30), natives of the Amazon (PR: 1.15; 95% CI: 1.04–1.27), and those belonging to other races (PR: 1.15; 95% CI: 1.02–1.29) had a higher prevalence of inadequate ANC. During the pandemic, only Aymara women maintained a higher prevalence of inadequate ANC (PR: 1.18; 95% CI: 1.06–1.31).Conclusions:Before the pandemic, there were ethnic differences in the prevalence of inadequate ANC, and during the pandemic, only Aymara pregnant women maintained a higher prevalence of inadequate ANC.

This study investigates the approximation of the natural logarithm function ln(x) for integer values x=2,3,…,100 using two approaches: an infinite geometric series and a Riemann integral–based method developed in this research. The study first proves the recursive formula of ln(x) through the Riemann integral using mathematical induction, establishing a theoretical foundation for the method. Numerical evaluations are then carried out, with the Riemann integral computed both analytically and numerically using partition values n = 10,100,1000,10000,100000,1000000. The two approaches are compared in terms of accuracy and computational efficiency, with accuracy measured using the Mean Absolute Percentage Error (MAPE). The results show that while the geometric series provides higher accuracy, the proposed Riemann integral method demonstrates superior execution speed and serves as a fundamental basis for developing more advanced numerical integration techniques.

Spatiotemporal traffic data play a pivotal role in the operation of Intelligent Transportation Systems (ITS). In practice, missing values and anomalous observations are prevalent in field-collected measurements, undermining their effectiveness and utility in subsequent ITS applications. In order to improve the data quality, numerous low-rank tensor completion (LRTC) approaches have been developed, demonstrating promising performance. However, the high accuracy of existing LRTC approaches often depends on manually tuning key model parameters to accommodate varying missing data scenarios and diverse datasets. To overcome the practical limitation, this study propose a nonconvex LRTC model that incorporates an adaptive weighted nuclear norm framework for differentiated treatment of traffic data singular values. Within the framework, a heuristic logarithmic weighting function is employed to reduce manual hyperparameter tuning while maintaining effective regularization across diverse traffic scenarios. To further enhance robustness against corrupted data, the model is extended to a robust variant by integrating an anomaly-tolerant module based on the l 1 norm, capable of simultaneously eliminating anomalous observations and imputing missing values. Extensive experiments demonstrate the effectiveness, robustness, and adaptivity of the proposed method. In general, the proposed method achieves up to a 10 % and 12 % performance improvement in missing data imputation and missing and corrupted data recovery tasks respectively compared to the existing nonconvex LRTC method.

One-cycle control (OCC) is prized in power converter applications for its rapid dynamic response and effective disturbance suppression. While its core principle relies on the symmetry of the volt-second value of the inductor in each cycle, recent research shows that embedding a composite function can significantly expand the stable parameter domain of conventional OCC. This paper seeks to identify the optimal function for this enhancement. The logarithmic and arc-tangent functions are selected based on the required characteristics and analyzed using a state-space average model. Analysis of the stability boundaries demonstrates that with ln(u+1) embedded, the stability region of uref is effectively enlarged to more than 4uin. With tan−1u embedded, the stability region of uref is effectively enlarged to infinity. Therefore, embedding tan−1u can achieve optimal results, so it is considered the optimal function. This conclusion is conclusively validated by both simulation and experimental results.

The growing urgency for transportation network development in seasonally frozen regions brings attention to two critical symmetrical factors: cyclic loading and freeze–thaw cycles. In saline soil areas, these symmetrical mechanical and environmental processes, along with varying salt content, significantly affect soil mechanical properties, posing considerable challenges for engineering design. In this study, the dynamic triaxial tests were conducted on a type of carbonate saline soil considering four factors, including moisture content, salt content, freeze–thaw cycle and confining pressure, and the variations in dynamic parameters, including dynamic strength and dynamic elastic modulus, with the above four factors were studied, and the influential mechanisms of four factors were fully discussed. The results demonstrated that the variations in dynamic strength (τd) versus vibration cycles (NF) were better fitted by logarithmic functions than by a linear one. An increase in moisture content, salt content, and freeze–thaw cycle all reduced the τd and dynamic elastic modulus (Ed); in addition, the Ed decreased significantly when the dynamic axial strain was less than 0.2%, and then stabilized with further increases in dynamic axial strain. The dynamic parameters of saline soil became nearly constant after undergoing five freeze–thaw cycles, and increased significantly with increasing confining pressure. Moreover, the relationship between the maximum dynamic elastic modulus (Edmax) and the four factors could be described by power functions. These findings could provide certain references for addressing the combined effects of symmetrical cyclic loading and freeze–thaw cycles in subgrade design for saline soil regions.

To investigate the failure characteristics and high-strain-rate mechanical response of polyvinyl alcohol-modified alkali-activated materials (PAAMs) under static and dynamic impact loads, quasi-static and uniaxial impact compression tests were performed on AAMs with varying PVA content. These tests employed a universal testing machine and an 80 mm diameter split Hopkinson pressure bar (SHPB). Digital image correlation (DIC) was then utilized to study the surface strain field of the composite material, and the crack propagation process during sample failure was analyzed. The experimental results demonstrate that the compressive strength of AAMs diminishes with higher PVA content, while the flexural strength initially increases before decreasing. It is suggested that the optimal PVA content should not exceed 5%. When the strain rate varies from 25.22 to 130.08 s−1, the dynamic compressive strength, dissipated energy, and dynamic compressive increase factor (DCIF) of the samples all exhibit significant strain rate effects. Furthermore, the logarithmic function model effectively fits the dynamic strength evolution pattern of AAMs. DIC observations reveal that, under high strain rates, the crack mode of the samples gradually transitions from tensile failure to a combined tensile–shear multi-crack pattern. Furthermore, the crack propagation rate rises as the strain rate increases, which demonstrates the toughening effect of PVA on AAMs.

Normal concrete exhibits poor resistance to chloride penetration, often leading to reinforcement corrosion and premature structural failure. In contrast, ultra-high-performance concrete (UHPC) demonstrates superior resistance to corrosion caused by chloride salts. The chloride diffusion behaviour of UHPC was investigated via long-term immersion (LTI) and rapid chloride migration (RCM) tests. Additionally, this study presents the first development of a time-dependent diffusion model for UHPC under chloride corrosion, as well as the proposal of a performance-based design method for calculating the protective layer thickness. Results show that the incorporation of steel fibers reduced the chloride diffusion coefficient (D) by 37.9%. The free chloride content (FCC) in UHPC increased by 92.0% at 2 mm after 300 d of the action of LTI. D decreased by up to 91.0%, whereas the surface chloride concentration (Cs) increased by up to 92.5% under the action of LTI. The time-dependent models of D and Cs followed power and logarithmic functions, respectively. An increase in UHPC surface temperature, relative humidity, and tensile stress ratio significantly diminishes the chloride resistance of UHPC. The minimum UHPC protective layer thicknesses required for UHPC-HPC composite beams with design service lives of 100 years, 150 years, and 200 years are 30 mm, 37 mm, and 43 mm, respectively.

Predefined-time cluster output synchronization of fuzzy neural networks with a new logarithmic function

This study aimed to compare the timeliness of single role versus multirole physician-staffed prehospital models and construct an optimised capability-based case allocation map for Sydney, Australia. We retrospectively compared response, scene, and total prehospital intervals over an 11-year period. Generalized linear regression models with log function were used to control for confounders. An optimized case allocation map was derived from response time data. For the single role service 672 helicopter responses were compared with 289 road and 208 helicopter multirole responses. Multirole patients were typically closer to their base (median 18.2 km vs. 23.4 km, p = 0.004). Response interval was shorter for the single role service (median 18 min vs. 24 min, p < 0.001). Scene and total prehospital intervals were shorter for the single role service (24 min vs. 32 min, p < 0.001 and 70 min vs. 80 min, p < 0.001, respectively). On multivariate analysis, multirole allocation was significantly associated with longer scene intervals (IRR = 1.176, [95% CI 1.133, 1.221], p < 0.001) and longer total prehospital intervals (IRR = 1.402 [95% CI 1.315, 1.495], p < 0.001). The optimised case allocation map indicates multirole road response is faster for a 9 km radius circle eccentrically centred on their base. All other locations are more rapidly served by the single role helicopter model. A single role model is associated with a shorter response, scene, and total prehospital intervals compared with a multirole model. Real-world response time data rather than distance can be used to optimise case allocation when response capabilities are not equivalent.

Matrix-product states (MPS) have proven to be a versatile ansatz for modeling quantum many-body physics. For many applications, and particularly in one-dimension, they capture relevant quantum correlations in many-body wave functions while remaining tractable (polynomial scaling) to store and manipulate on a classical computer. This has motivated researchers to also apply the MPS ansatz to machine learning (ML) problems where capturing complex correlations in datasets is also a key requirement. Here, for the first time, we develop an MPS-based algorithm, MPSTime, for learning a joint probability distribution underlying a time-series dataset, and show how it can be used to tackle important time-series ML problems, including classification and imputation. MPSTime can efficiently learn complicated time-series probability distributions directly from data, requires only moderate maximum MPS bond dimension χmax (with values for our applications ranging between χmax=20−160), and can be trained for both classification and imputation tasks under a single logarithmic loss function. Using synthetic and publicly available real-world datasets—spanning applications in medicine, energy, and astronomy—we demonstrate performance competitive with state-of-the-art ML approaches, but with the key advantage of encoding the full joint probability distribution learned from the data, which is useful for analyzing and interpreting its underlying structure. This manuscript is supplemented with the release of a publicly available code package that implements our approach, and can be used to reproduce all presented results. The effectiveness of the MPS-based ansatz for capturing complex correlation structures in time-series data makes it a powerful foundation for tackling challenging time-series analysis problems across science, industry, and medicine.

Deuterium storage metals are widely used in ion sources as excellent providers of deuterium ions. Many researchers have explored the cathode spot erosion characteristics of metal deuteride cathodes in vacuum arcs. This study establishes a two-dimensional (2D) axisymmetric swirl model to investigate how arc current functions affect the formation and evolution process of cathode spots in metal deuteride cathodes under vacuum arc conditions. The model incorporates fluid dynamic equations, energy equations, and deuterium diffusion equations. The direct current (DC) model is used as the control group, and multiple Gaussian and logarithmic Gaussian distribution current models are compared. The simulation results indicate that when the time-integrated results of the current function are similar, the evaporation-induced erosion rate under a logarithmic Gaussian distribution current function is much lower than that under a Gaussian distribution current function, specifically by one order of magnitude. Moreover, within the logarithmic Gaussian distribution model, a current function with an earlier peak yields a larger deuteride release amount and a smaller evaporation erosion rate, making it a superior current function model. Additionally, when the current reaches over 60% of the total arc current, which corresponds to the cathode surface temperature exceeding 3500 K (with temperature as the underlying driver), evaporation on the cathode spot crater surface increases significantly.

The interpolative contraction and its numerous generalizations motivated this study on its relation to previously known classes of contractive mappings. We will prove that by using of the concave property of the logarithmic function (or Jensen’s inequality), it is possible to relate some observed classes of interpolative contraction to their classical analogues. The theoretical results are substantiated with several examples.

ABSTRACTIn this paper, we define the harmonic higher order Gauss Fibonacci numbers, which are the complex counterparts of higher order Gauss Fibonacci numbers and have been previously studied in the literature. Due to the absence of a known closed form for harmonic Fibonacci numbers, we also generalize their complex counterparts with respect to a certain parameter . The computational analysis in this case is significantly more complex and intricate. Furthermore, for varying values of , we can derive several sequences of harmonic complex Fibonacci‐type numbers. Using the difference operator and its properties, we give formulas for these numbers. Given their complexity, we present examples using Maple to verify our results. We define hyperharmonic higher order Gauss Fibonacci numbers, encompassing harmonic higher order Gauss Fibonacci numbers. Lastly, we give generating functions for harmonic and hyperharmonic higher order Gauss Fibonacci numbers via the generalized ‐logarithm function.

ABSTRACT Diet and lifestyle are modulators of low-grade chronic inflammation, which has been associated with depressive symptoms. This study aimed to examine the association between the dietary inflammatory index (DII) and the risk of depressive symptoms among university students. Data were from 675 students of both sexes, aged between 16 and 25 years old, who entered a Brazilian public university in 2016 and 2017 and were followed up until 2018. The inflammatory potential of the diet was assessed by the DII, which was estimated based on 39 dietary parameters obtained from 24-hour dietary recalls. The presence of depressive symptoms was determined by the Patient Health Questionaire-9, and the cutoff point adopted was ≥ 10 to classify the individuals as ‘with’ and ‘without’ depressive symptoms. Generalized linear models with binomial distribution and logarithmic linkage function estimated the relative risk (RR) and confidence intervals (95%CI) of the association between DII and depressive symptoms. The risk for depressive symptoms during follow-up increased for those eating a more inflammatory diet at baseline, reaching an increased risk of 28% in the second year of follow-up (RR = 1.28, 95%CI = 1.06; 1.56). A pro-inflammatory diet contributed to the increased risk of depressive symptoms among university students.

This article presents two-dimensional analysis of thermoelastic stresses on voids/holes in an anisotropic infinite plane or half-plane by the boundary element method (BEM). In the boundary integral equation, an extra domain integral due to thermal loading is present that needs to be transformed onto boundaries. For a finite domain, additional line integrals are needed along the branch-cuts of logarithmic complex function; however, this cannot work for unbounded infinite plane or half-plane. In the present work, the domain integral is analytically transformed onto the hole boundaries of the infinite/semi-infinite plane, yet no additional line integrals are involved. This article reports the first successful work to analyze general thermoelasticity in an infinite/semi-infinite anisotropic plane containing voids/holes by the BEM. In the modeling, only the boundaries of voids/holes need to be discretized. A few examples are presented in the end for illustrating the veracity of all presented formulations.

The article analyzes methodological approaches to overcoming students’ learning losses in the topic “Logarithmic Function” through interactive learning tools. In the context of the pandemic, remote and blended learning, as well as war and constant stress, there has been a significant decline in students’ motivation and knowledge, especially in mathematics, where the abstractness of concepts complicates the process of mastering the material. The literature review highlights considerable learning losses in understanding logarithmic functions, difficulties in grasping the definition of logarithms, graphical interpretation of functions, and applying knowledge to practical problems. Traditional teaching methods are insufficient to overcome these losses. The aim of the study is to develop a unified system and methodological recommendations to address learning losses through the use of interactive technologies. A comprehensive approach is proposed, combining knowledge diagnostics, identification of typical learning losses, construction of individual learning trajectories, use of interactive technologies and digital tools (GeoGebra, PhET, Google Forms, AhaSlides), problem-based and case tasks, differentiated reinforcement, self-analysis, and reflection. The use of interactive technologies promotes increased engagement and motivation, development of logical thinking, and the formation of practical skills in applying logarithms. The developed model allows consideration of students’ individual needs and the creation of adaptive learning trajectories. Further research involves experimental verification of the effectiveness of interactive technologies in reducing learning losses and improving learning outcomes in the topic “Logarithmic Function.” The proposed approaches can also be adapted for other abstract mathematical topics, increasing their universality and practical value for pedagogical practice

This study proposes two novel subclasses of analytic and bi-univalent functions constructed using cosine and logarithmic functions. The primary aim is to establish sharp estimates for the initial coefficients in their Taylor--Maclaurin series. Furthermore, the Fekete-Szegö inequality is investigated for functions within these subclasses and Several corollaries are obtained by assigning specific values to the associated parameters. These results contribute to the advancement of geometric function theory and provide a basis for future research in related function classes.