Quadratic Function Research Articles

Decision-making Process of Mathematics Teacher in Responding to Student's Mistake

Students lack deep understanding and their learning needs, make it difficult for teachers to respond effectively to student errors. The case study investigated the decision-making process of teachers who respond to the student’s mistakes with solutions to quadratic function problems. This type of research is qualitative research with a descriptive approach. The results are revealed based on the decision-making stages: (1) generating ideas, the subject designed strategies using the GeoGebra application, and developing problems to train students’ HOTS skills. (2) Clarifying ideas, the subject considered students’ ability to understand the problem more easily if it is presented in visual form. (3) Assessing the fairness of ideas, the subject believed that when students were trained from carefully correcting their work to carefully proofreading their work, they learned from their mistakes and prevented them from repeating it. In conclusion, teacher can make good decisions by constructing and assessing ideas so that students can understand their mistakes. Keywords: decision making, mathematics teacher, quadratic function

A High-Dimensional Cramér–von Mises Test

The Cramér–von Mises test provides a useful criterion for assessing goodness of fit in various problems. In this paper, we introduce a novel Cramér–von Mises-type test for testing distributions of high-dimensional continuous data. We establish an asymptotic theory for the proposed test statistics based on quadratic functions in high-dimensional stochastic processes. To estimate the limiting distribution of the test statistic, we propose two practical approaches: a plug-in calibration method and a subsampling method. Theoretical justifications are provided for both techniques. Numerical simulation also confirms the convergence of the proposed methods.

Numerical simulation for spray spatial distribution of swirl nozzle and its target dustfall area prediction.

The droplet breakup and distribution of the internal flow field and external spray field of a nozzle were obtained under different pressures. The thickness of the liquid film increased with pressure as a quadratic function. The maximum value of 0.281mm at 2MPa decreased to 0.172mm at 10MPa, equivalent to 38.8% decrease. The MATLAB was used to obtain the particle size distribution characteristics of the droplets under different pressures. At 2MPa, droplet breakup dominated the axial interval [0-700mm]. With the increase of pressure, D50 distribution as a whole continues to decrease, 2-6MPa change, the particle size reduction is larger, every increase of 2MPa reduced by about 15%. 6-10MPa change, the particle size reduction is smaller, every increase of 2MPa reduced by about 8%. A model to predict the optimal dust reduction areas under varying pressures was developed. The prediction results indicate that at pressures of 5MPa and 9MPa, the target dust removal areas were [344mm, 710mm] and [424mm, 942mm] along the axial direction, respectively.

Exact and Data-Driven Lump Wave Solutions for the (3+1)-Dimensional Hirota–Satsuma–Ito-like Equation

In this paper, the lump wave solutions for (3+1)-dimensional Hirota–Satsuma–Ito-like (HSIl) equation are constructed by employing the Hirota bilinear method and quadratic function approach, and the corresponding propagation behaviors and nonlinear dynamical properties are also investigated. At the same time, the physics informed neural network (PINN) deep learning technique is employed to study the data-driven solutions for the HSIl equation from the derived lump wave solutions. The machine learning results show high effectiveness and accuracy, providing new techniques for discussing more nonlinear dynamics of lump waves and discovering new lump wave solutions.

A Diathesis-Stress Approach to Understanding Partner Maltreatment Perpetration: A Longitudinal Study.

Despite evidence suggesting that partner maltreatment-a concept which represents a wide array of negative, destructive, and abusive behaviors in romantic relationships-is likely to fluctuate over time, the longitudinal trajectory of partner maltreatment is unclear. This study aims to identify the average trajectory of partner maltreatment over a 5-month period before applying an attachment-informed diathesis-stress framework to explain points of escalation or de-escalation in partner maltreatment perpetration. Two hundred and eight individuals completed 5 monthly assessments of partner maltreatment as well as an assessment of adult attachment and perceived stress at baseline. A nonlinear (i.e., quadratic function) trajectory provided optimal fit for partner maltreatment perpetration over time. An interaction between attachment avoidance and perceived stress predicted the trajectory of partner maltreatment perpetration. Specifically, higher attachment avoidance and higher perceived stress predicted a small initial decline followed by a steep increase in partner maltreatment perpetration. No significant associations were found for the interaction between attachment anxiety and stress and partner maltreatment. The findings suggest that perpetration of partner maltreatment fluctuates over time, and that an attachment-informed diathesis-stress framework can be applied to explain levels of partner maltreatment perpetration and points of escalation and de-escalation over time.

Hebbian learning from first principles

Recently, the original storage prescription for the Hopfield model of neural networks – as well as for its dense generalizations – has been turned into a genuine Hebbian learning rule by postulating the expression of its Hamiltonian for both the supervised and unsupervised protocols. In these notes, first, we obtain these explicit expressions by relying upon maximum entropy extremization à la Jaynes. Beyond providing a formal derivation of these recipes for Hebbian learning, this construction also highlights how Lagrangian constraints within entropy extremization force network’s outcomes on neural correlations: these try to mimic the empirical counterparts hidden in the datasets provided to the network for its training and, the denser the network, the longer the correlations that it is able to capture. Next, we prove that, in the big data limit, whatever the presence of a teacher (or its lacking), not only these Hebbian learning rules converge to the original storage prescription of the Hopfield model but also their related free energies (and, thus, the statistical mechanical picture provided by Amit, Gutfreund and Sompolinsky is fully recovered). As a sideline, we show mathematical equivalence among standard Cost functions (Hamiltonian), preferred in Statistical Mechanical jargon, and quadratic Loss Functions, preferred in Machine Learning terminology. Remarks on the exponential Hopfield model (as the limit of dense networks with diverging density) and semi-supervised protocols are also provided.

The bullwhip effect, market competition and standard deviation ratio in two parallel supply chains

The bullwhip effect widely exists in supply chains and shows its significance for the competitiveness of enterprises in supply chains. In this study, we analyze the bullwhip effect in two parallel supply chains with competing products, each one consisting of a supplier and a retailer. A model is detailed for measuring the bullwhip effect in which the demand of retailers follows a similar vector autoregressive model (VAR-like) process. The results show that the bullwhip effect can be characterized as a quadratic function of the standard deviation ratio. The impact of market competition on the bullwhip effect of the supply chain may have the opposite result, which depends on some parameters, including lead time and market competition in the parallel supply chain. The parameters have asymmetric influence on bullwhip effect. Compared with VAR(1) and AR(1) model, the empirical results show that our VAR(1)-like model is closer to reality. Furthermore, we discuss the conclusion of research and its inspiration for supply chain management.

Asymptotic Results for the Estimation of the Quadratic Score of a Clustering

In cluster analysis one often finds several partitions of a data set using different clustering methods and algorithms set with a variety of hyperparameters and tunings. The number of clusters K is one of the most relevant of such hyperparameters. Cluster selection is the task of choosing the desired partitions. The Bootstrap Quadratic Scoring is a recently introduced method where the cluster selection is performed by optimizing a score attached to a partition that is based on the quadratic discriminant function. Previously, we proposed the estimation of this cluster score via bootstrap resampling and investigated the proposed estimator based on numerical experiments and real data applications. However, that earlier work did not provide theoretical guarantees. In this paper, we fill that gap. We study the asymptotic behavior of the scoring method and show that the proposed estimator converges to well-defined population counterparts.

THE RELATIONSHIP BETWEEN LEARNING STYLES AND UNDERSTANDING MATHEMATICAL CONCEPTS IN QUADRATIC FUNCTIONS

The research teka-teki is that students are experiencing difficulties in understanding the concept of quadratic functions. The aim of the study is to determine the relationship between learning styles and mathematical concept understanding in solving quadratic function problems among 9th grade students at SMP Negeri 3 Ulunoyo. This research adopts indah ambang quantitative approach with indah ambang correlational study type. The population in this study consists of 20 9th grade students from SMP Negeri 3 Ulunoyo, and the sampling technique used is besaran sampling. The instruments used are questionnaires and tests of mathematical concept understanding. Based on the research results, indah ambang correlation coefficient of 0.64 was obtained, indicating indah ambang positive and strong correlation. The coefficient of determination is 40.96%, showing that the contribution of learning styles to concept understanding is 40.96%, with the remaining contribution attributed to other factors. The calculated t-value is 3.50324, and to determine the critical t-value at indah ambang 0.05% significance taraf, the t-table value is 2.100922. Since the calculated t-value (3.50324) is greater than the t-table value (2.100922), it can be concluded that there is indah ambang relationship between learning styles and mathematical concept understanding among 9th grade students at SMP Negeri 3 Ulunoyo, with the correlation taraf falling into the moderate category. Therefore, the researcher suggests (1) students should recognize their own learning styles and optimize them to facilitate better learning and achieve optimal performance, and (2) teachers are encouraged to adapt their teaching styles to meet the needs of all students, not just those with specific learning styles.

Development of mathematical modeling of a mobile robot's motion control system

This study focuses on developing a mathematical model for precise control and stabilization of unmanned aerial vehicles (UAVs) in various spatial conditions. Addressing the problem of achieving precise control and stability, the proposed solution designs a control system based on a linear-quadratic controller (LQR) and simulates it using a proportional-derivative (PD) controller implemented in Matlab/Simulink. The results demonstrate high precision and stability in controlling the UAV motion parameters – roll, pitch, yaw, and altitude. This important level of performance is achieved due to the adaptivity of the LQR-based control system, which optimizes control actions according to the unsteady dynamics of the UAV. The integration of the PD controller improves responsiveness and stability, providing precise motion control over a range of spatial states. These features effectively solve the problem by handling the complex dynamics of the UAV and providing precise control. The results are explained by the ability of the LQR to provide optimal control laws that minimize deviations using a quadratic cost function, while the PD controller quickly corrects errors and responds to disturbances. The benefits of this approach include a significant reduction in control errors by about 25–30 %, increased response speed to external disturbances, and reduced computational latency due to efficient processing compared to more resource-intensive methods such as model predictive control. The developed mathematical model can be applied in practice in conditions requiring robust real-time control and adaptation to dynamic changes in the environment. It is especially suitable for industries such as logistics, surveillance, and environmental monitoring, providing an effective and optimal solution for stabilizing and controlling the motion of UAVs in various spatial states. This approach improves the performance of UAVs and expands their capabilities in various operating conditions.

The effect of seasonal temperature difference on estimated wood density in Populus ussuriensis Kom. trees by the resistance drilling method

ABSTRACT Wood density is a main determinant of wood quality assessment for wood end-use and plantation trees. The resistance drilling method (RST) is one of the rapid assessment methods of wood density for standing trees. As the season changes, the water within trees occurs to freeze and is expected to indirectly affect RST results. However, the impact of ambient temperature is unknown. To investigate the effect of seasonal temperature difference on estimated wood density, the RST of Populus ussuriensis Kom. trees was examined across different seasons. A good linear regression model between average wood density and RST was established at normal temperature during summer, with a coefficient of determination of 0.708. The average RST of sapwood and heartwood increased by 6% to 22% from autumn (−5–0°C) to winter (−20–−10°C) compared to normal temperature in summer. Correspondingly, the average wood density estimated by RST increased by 4% to 16% from autumn to winter relative to normal temperature. The quadratic polynomial functions described the relationship between relative RST, relative wood density and seasonal temperature, with coefficients of determination over 0.922. Seasonal temperature had a significant effect on wood density, which was overestimated by RST in winter. This overestimation should be considered in the RST application.

Dynamic output feedback model predictive control for polytopic uncertain systems under decode and forward cooperative relay protocol

This paper addresses the robust model predictive control (RMPC) problem for a class of polytopic uncertain systems under the decode and forward cooperative relay protocol (DFCP). In order to combat signal fading from sensors to controllers and improve communication reliability, spatial diversity is achieved through cooperative terminal relay signals. At each transmission instant, the terminal only selects a better signal for transmission and a switched system is constructed. The aim of this paper is to design a set of desired dynamic output feedback controllers in the framework of RMPC such that the switched system with the underlying DFCP is exponentially stable. By taking the influence of the randomness of DFCP derived from a three nodes model composed of DF relay, a new objective function based on the mathematical expectation of the traditional quadratic function of MPC is established. Then, the orthogonal complement technique is applied to handle the inherent couplings of variables to formulate a solvable optimisation problem. Moreover, the sufficient conditions are provided to guarantee the recursive feasibility of the MPC strategy and the stability of the established closed-loop system in terms of the average dwell time (ADT) method. Finally, three experiments are given to illustrate the effectiveness of the proposed method.

Dynamical transition in controllable quantum neural networks with large depth

Understanding the training dynamics of quantum neural networks is a fundamental task in quantum information science with wide impact in physics, chemistry and machine learning. In this work, we show that the late-time training dynamics of quantum neural networks with a quadratic loss function can be described by the generalized Lotka-Volterra equations, leading to a transcritical bifurcation transition in the dynamics. When the targeted value of loss function crosses the minimum achievable value from above to below, the dynamics evolve from a frozen-kernel dynamics to a frozen-error dynamics, showing a duality between the quantum neural tangent kernel and the total error. In both regions, the convergence towards the fixed point is exponential, while at the critical point becomes polynomial. We provide a non-perturbative analytical theory to explain the transition via a restricted Haar ensemble at late time, when the output state approaches the steady state. Via mapping the Hessian to an effective Hamiltonian, we also identify a linearly vanishing gap at the transition point. Compared with the linear loss function, we show that a quadratic loss function within the frozen-error dynamics enables a speedup in the training convergence. The theory findings are verified experimentally on IBM quantum devices.

Output-feedback control design for Takagi-Sugeno fuzzy systems through Lyapunov functions depending polynomially on the states

This paper proposes a new strategy based on linear matrix inequalities (LMIs) for the synthesis of state- and output-feedback controllers through parallel distributed compensation for continuous-time Takagi-Sugeno fuzzy systems. The main novelty of the proposed design technique is the use of homogeneous polynomial Lyapunov functions of degree larger than two on the states, generalizing the results based on quadratic functions. Parametrized in terms of a constant matrix, those homogeneous polynomial Lyapunov functions provide less conservative results in terms of stability and decay rate of trajectories when dealing with membership functions whose time-derivatives have unknown bounds. The synthesis procedure is formulated in terms of a locally convergent iterative algorithm, where a set of LMIs defined in an augmented parameter space is solved at each iteration. Numerical examples are used to compare the proposed method with quadratic stabilizability techniques available in the literature, illustrating the advantages of higher degree Lyapunov functions.

A robust approach to Gaussian process implementation

Abstract. Gaussian process (GP) regression is a flexible modeling technique used to predict outputs and to capture uncertainty in the predictions. However, the GP regression process becomes computationally intensive when the training spatial dataset has a large number of observations. To address this challenge, we introduce a scalable GP algorithm, termed MuyGPs, which incorporates nearest-neighbor and leave-one-out cross-validation during training. This approach enables the evaluation of large spatial datasets with state-of-the-art accuracy and speed in certain spatial problems. Despite these advantages, conventional quadratic loss functions used in the MuyGPs optimization, such as root mean squared error (RMSE), are highly influenced by outliers. We explore the behavior of MuyGPs in cases involving outlying observations and, subsequently, develop a robust approach to handle and mitigate their impact. Specifically, we introduce a novel leave-one-out loss function based on the pseudo-Huber function (LOOPH) that effectively accounts for outliers in large spatial datasets within the MuyGPs framework. Our simulation study shows that the LOOPH loss method maintains accuracy despite outlying observations, establishing MuyGPs as a powerful tool for mitigating unusual observation impacts in the large data regime. In the analysis of US ozone data, MuyGPs provides accurate predictions and uncertainty quantification, demonstrating its utility in managing data anomalies. Through these efforts, we advance the understanding of GP regression in spatial contexts.

Analysis of Optimal Water Levels in the Great Lakes Based on A Comprehensive Satisfaction Function

The purpose of this paper is to analyze the optimal water levels in the Great Lakes to meet the needs of multiple stakeholders and thus achieve sustainable development in the region. The Weight-Function-Synthesis-Optimization (WFSO) model was used in this study. By classifying the water level needs of different interest groups, their respective satisfaction functions are established, and the overall satisfaction function is formed by weighting. In addition, the quadratic polynomial function is used to fit the optimal monthly water level time series of Lake Ontario, and the fitting effect is good, fully reflecting the periodic and seasonal changes in the water level of Lake Ontario. By considering the priorities of various stakeholders, this paper defines an overall satisfaction metric, "allsa," and identifies a time series of monthly optimal water levels for Lake Ontario and beyond. The ideal water level was determined to be 74.9 meters. The study results suggest that maintaining this water level can maximize the satisfaction of ecosystems, transportation, real estate, terminals, and hydropower companies, and promote sustainable development of the region.

Quadratic Frequency Dispersion in the Oscillations of Intermediate-mass Stars

Asteroseismology, the study of stellar vibration, has met with great success, shedding light on stellar interior structure, rotation, and magnetism. Prominently known as δ Scutis, the intermediate-mass main-sequence oscillators that often exhibit rapid rotation and possess complex internal stratification are important targets of asteroseismic study. δ Scuti pulsations are driven by the κ (opacity) mechanism, resulting in a set of acoustic modes that can be challenging to interpret. Here, we apply machine learning to identify new patterns in the pulsation frequencies of δ Scuti stars, discovering resonances spaced according to quadratic functions of integer mode indices. This unusual connection between mode frequencies and indices suggests that rotational influence may play an important role in determining the frequencies of these acoustic oscillations.

CT imaging analysis of the C7 pedicle and lateral mass in children aged 0–14 years

Background and objectiveThere is currently a lack of anatomical research on the C7 pedicle and lateral mass in children aged 0–14 years, and experience with pediatric lower cervical spine surgeries remains limited. This study aims to investigate the anatomical structure of the C7 pedicle and lateral mass in a large pediatric sample to understand their growth patterns, providing imaging and anatomical references for pediatric lower cervical spine surgeries and the design of pediatric C7 screws. MethodsWe measured 12 parameters, including the width, length, height, and angles of the C7 pedicle and lateral mass in children aged 0–14 years. The t-test was used to analyze bilateral and gender differences, and least squares regression was applied for curve fitting analysis. ResultsThe size parameters of the C7 pedicle and lateral mass generally increased with age, while angular parameters fluctuated within specific ranges. The average values for each parameter between ages 0–14 were as follows: D1: 2.78–5.28 mm, D2: 18.15–30.54 mm, D3: 2.44–6.01 mm, angle A: 38.05–44.48°; D4: 6.81–10.94 mm, D5: 5.37–11.23 mm, D6: 8.07–13.28 mm, D7: 1.94–4.88 mm, D8: 5.67–9.39 mm, angle C: 18.50–28.80°, angle D: 43.23–52.01°, angle E: 33.74–44.96°. Bilateral analysis showed no significant differences for most parameters across most age groups, with differences mainly observed in angle A, angle C, D6, and D8. Gender differences were primarily noted in length parameters across different age groups, especially in the 3–4, 6–7, 9–11, and 13–14 age groups. Regression analysis indicated that most parameters followed cubic function curves, while a few followed power or quadratic function curves. ConclusionThis study provides detailed CT imaging anatomy of the C7 pedicle and lateral mass in children aged 0–14 years. Preoperative thin-slice CT scans and careful measurements of key parameters are essential for pediatric lower cervical spine surgeries. The findings offer valuable imaging and anatomical references for pediatric posterior cervical fixation surgery and screw design.

Stability Analysis of Matrices and Single-Parameter Matrix Families on Special Regions

This paper proposes an efficient method for determining the D-stability of matrices using Kelley’s cutting-plane method. The study examines the D-stability of matrices in symmetric regions of the complex plane, defined by quadratic matrix inequalities (QMI) and polynomial functions. Using Kelley’s cutting-plane method, we present an iterative algorithm for determining the D-stability of a given matrix. Further, the robustness of single-parameter matrix families is analyzed, and a method is proposed for considering their D-stability. Through examples, we indicate the possible applicability of the proposed approach in addressing stability problems in linear systems.

Optimal shift control for new dual-clutch full-powershift transmission of heavy-duty tractor based on genetic algorithm–particle swarm optimization

The calculation of a linear quadratic regulator (LQR) control weighting matrix determines whether it can achieve optimal control. In order to solve the problem of complex calculation and low efficiency of this matrix, this paper proposed genetic algorithm–particle swarm optimizatio algorithm (GAPSO) combining genetic algorithm and particle swarm algorithm to solve the weighting matrix of LQR clutch oil pressure control strategy, so as to control the design efficiency and precision of the strategy. Subsequently, the shifting quality of a tractor was improved. A dynamic model of the shifting process was established using a comprehensive indicator that combined jerk and sliding friction work as a quadratic function. An LQR-weighted matrix calculated using GAPSO enabled optimal control strategy designing for clutch hydraulic pressure. Furthermore, the shift process from the 11th to 12th gear was analyzed via simulations during plowing operations. The simulation results show that the genetic operation of crossover and variation is introduced into PSO, and the optimization ability of PSO is improved effectively. Compared with the clutch oil pressure control strategy calculated by GAPSO with LQR weighted matrix and LQR clutch oil pressure control strategy calculated by PSO with LQR weighted matrix, the clutch heat load is slightly increased, the smoothness of shifting is greatly improved, and the shifting quality of tractors is improved. Overall, this study provides valuable insights for designing and computationally analyzing optimal clutch LQR control.