Quadratic Function Research Articles

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

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

Inventory Optimization Model with Quadratic Demand Pattern and Non-instantaneous Deterioration Rate with Mixed Cash and Advance Payment Scheme

Objectives: To develop a mathematical model for cost minimization by examining a scenario involving a seasonal product retail market, evaluating the outcome, and providing solutions to unanswered queries of existing models are the highlights of this present work. Methods: 1. Using dynamic differential equations with initial and boundary conditions, we mathematically model a retail market scenario for perishable commodities and determine the retailer's total cost. 2. We consider a quadratic function of time to precisely describe the maintenance cost and demand function. Also, a combination of cash and advance payment schemes is implemented in our model. 3. Our model was numerically formulated using a dataset deduced from earlier research. We have included a sensitivity analysis and a numerical example with a graphical depiction of the total cost function that highlights the most and least influential factors. 4. Based on the Hessian matrix condition and the calculus derivative approach, we developed an algorithm to calculate the optimal cost. Findings: This model provides retailers with insights for effective stock management while optimizing the cost of retailing perishable commodities with non-linear demand patterns. Sensitivity analysis showed that the two factors that have the biggest impact on retailers' overall costs are non-linear demand and holding cost parameters. The smaller optimal inventory cycle time and the large value of time at which a shortage happens raise the overall cost. Additionally, because of higher demand, the time-dependent deterioration rate has the least impact on overall cost. Additionally, a three-dimensional graph of cost minimization is offered to illustrate the convexity of total cost. Novelty: Implementation of Quadratic time-dependent demand and non-linear holding costs under a mixed cash advance payment structure, along with time-dependent deterioration and partial backlog shortages. Keywords: Quadratic Demand pattern, Non­instantaneous decay rate, Cash and Advance Payment scheme, Partial Backlog Shortages

Initialization-free Lie-bracket Extremum Seeking

Stability results for extremum seeking control in Rn have predominantly been restricted to local or, at best, semi-global practical stability. Extending semi-global stability results of extremum-seeking systems to unbounded sets of initial conditions often demands a stringent global Lipschitz condition on the cost function, which is rarely satisfied by practical applications. In this paper, we address this challenge by leveraging tools from higher-order averaging theory. In particular, we establish a novel second-order averaging result with global (practical) stability implications. By leveraging this result, we characterize sufficient conditions on cost functions under which uniform global (i.e., under any initialization) practical asymptotic stability can be established for a class of extremum-seeking systems acting on static maps. Our sufficient conditions include the case when the gradient of the cost function, rather than the cost function itself, satisfies a global Lipschitz condition, which covers quadratic cost functions. Our results are also applicable to vector fields that are not necessarily Lipschitz continuous at the origin, opening the door to non-smooth Lie-bracket ES dynamics. We illustrate all the results via different analytical and/or numerical examples.

Characterization of Micro-seismic Activity in Northern Cyprus Using Complexity and Corner Frequency Methods

Cyprus is an island country located in the eastern Mediterranean, to the south of Türkiye and the western of Syria and Lebanon, and is a popular tourist destination. Due to being surrounded by seas on all four sides, meticulous planning of rescue, assistance, and evacuation plans is necessary in the face of disasters such as earthquakes and tsunamis. Tectonically, the southern part of the island is controlled by the Cyprus Arc, while the northern part is dominated by the Kyrenia Range. The demand for raw materials for construction and industry is met through controlled quarry blasting operations carried out by open-pit quarry companies in the districts of Kyrenia and Nicosia. As a result, both natural and artificial seismic events occur in the region, and these quakes are documented in seismic catalogs by seismology centers. However, due to the low energy content of micro-seismic events and the inadequacy of seismic stations on the island, the source types of these seismic events can be misidentified in the catalogs. In this context, the study focuses on 122 seismic events with magnitudes between 0.9≤Ml≤2.7 that occurred in Northern Cyprus during the January 2018 - December 2021 period (4 years). The seismic events recorded by the station LFK, operated by Boğaziçi University Kandilli Observatory and Earthquake Research Institute Regional Earthquake-Tsunami Monitoring Center (KOERI-RETMC), were classified using Linear and Quadratic Discriminant Functions based on complexity and corner frequency methods. According to the results obtained, 10 of the 122 seismic events were identified as natural, and 96 were determined to be artificial, resulting in a general success rate of 86.89%. However, classification results for 16 seismic events were inconclusive with the methods used. As a result, more detailed secondary analyses should be conducted to accurately determine the source types of micro-seismic events, and the seismic catalogs should be updated accordingly.

The effect of self-efficacy and mathematical anxiety on students' mathematical communication skills

This study aims to identify and examine the effect of self-efficacy and mathematical anxiety on students' mathematical communication skills. This study used a quantitative method with a survey design. The population of this study was students of class X at SMA Negeri 2 Cianjur and the sample used was students of class X-D to X-G, totaling 34 people. The sampling technique was carried out by purposive sampling. Data regarding self-efficacy and mathematical anxiety using a questionnaire each amounted to 20 statements and data regarding students' mathematical communication abilities used a test which amounted to 5 description questions with quadratic function material. The data processing technique used is multiple linear regression tests, by fulfilling the prerequisite tests including the normality test, homoscedasticity test, linearity test and multicollinearity test. Based on the research results, it was found that the P value was 0.074, meaning > 0.05, then H0 was accepted and H1 was rejected. This means that there is no influence between self-efficacy and mathematical anxiety on students' mathematical abilities.

Prediction research on the contribution degree of solar energy to the coefficient of heating performance of carbon dioxide heat pump

The carbon dioxide heat pump (CDHP) offers efficient heating over a wide temperature range, but its heating energy efficiency under low-temperature conditions decreases obviously due to a correlation with the evaporation temperature, limiting the widespread use of CDHP in cold regions. To address this, we implemented a passive solar house with short-term heat storage capacity to raise the evaporation temperature and studied it by experimental and theoretical analysis. We introduced the concept of contribution degree (K) of solar energy to the heating performance coefficient of CDHP. The prediction models for the evaporation temperature, heating performance coefficient, and K were constructed gradually. The study optimized the operation mode of solar-assisted CDHP using the evaporation temperature partition theory and presented a method to calculate the limit distribution range of K. The result shows that the order of importance level of the significant items on evaporation temperature was ambient temperature (Ta) > direct solar irradiance on the inclined surface (Gb) > scattering solar irradiance on the inclined surface (Gr) > GḃTa > Ta2 > Gb2. The predicted limit distribution range of K was [0.00, 8.17]. The study optimized the operation mode of solar-assisted CDHP into three categories. It was observed that K changes positively with any single variable according to the law of a composite function composed of a natural exponential function and a quadratic function. Our study suggests a feasible development direction and offers a new theoretical and engineering basis for overcoming the large-scale application limitations of CDHP in cold regions.

Observer-based guaranteed-cost bipartite time-varying formation tracking control for multiagent systems subject to communication delays

The paper centers on the guaranteed-cost bipartite time-varying formation tracking control for multiagent systems with nonuniform time-varying communication delays under the framework of a distributed observer. An integral quadratic cost function is presented, founded on formation tracking and observation error. Firstly, distributed state observers, using the relative measurement output between neighbors, are established so that the leader and follower can accurately estimate each agent's unmeasurable state. Subsequently, a bipartite formation tracking protocol is proposed, incorporating estimation information and nonuniform time-varying communication delays. This protocol aims to achieve bipartite time-varying formation tracking for multiagent systems. Next, a new augmented Lyapunov-Krasovskii functional with delay-product and triple integral terms is constructed. By integrating the second-order Bessel-Legendre inequality technique, the delay-dependent reciprocally convex combination inequality method, and the free weight matrix technique, a new guaranteed-cost bipartite time-varying formation tracking criterion is developed, offering less conservatism. Compared to certain existing delay conditions, the stability conditions assure a greater allowable upper bound of delay. Finally, a numerical example is conducted to confirm the validity and superiority of the theoretical findings.

Arboreal Galois groups for quadratic rational functions with colliding critical points

Arboreal Galois groups for quadratic rational functions with colliding critical points

Solitons, lump and interactional solutions of the (3+1)-dimensional BLMP equation in incompressible fluid

Based on the Hirota bilinear method, we systematically investigate the (3+1)-dimensional Boiti-Leon-Manana-Pempinelli (BLMP) equation in incompressible fluids and main results include: (1) the formulas of N-kink-soliton solutions and the bound states of multi solitons are all presented, (2) the lump solution is derived by the positive quadratic function method, (3) the interactional solutions are given, i.e., one lump interacts with one- and two-kink-soliton, (4) some special periodic solutions are discussed, i.e., lump-periodic solutions and homoclinic breather solutions.

Experimental study on mechanical transfer regularity and step distance of individuals under different collision impulse

In the process of emergency evacuation, the fall of an individual may lead to the instability of surrounding pedestrians. The evacuation routes would be obstructed by fallen pedestrians, thereby diminishing the evacuation efficiency and increasing the overall crowd risk. The purpose of this study is to explore the balance-recovery process of pedestrians under collision. The study conducted a series of experiments to obtain the total step distance and number of steps taken by individuals during the balance-recovery process under different impulses. The impulse applied by the unstable individual to the obstacle ahead for balance recovery was also measured. The gait characteristics and mechanical principles of the individuals in the balance-recovery process after collision force were analyzed. The results indicate that individuals can recover balance within a 90 cm step distance if the impulse received is in the range of 20–160 N s. The relationship between the step distance of an individual and the impulse received can be expressed as a quadratic function. When stepping is prohibited, the impulse applied by the individual to the obstacle ahead is approximately 1.85 times the impulse he received. However, if stepping is allowed, the dispersion of the impulse applied to the obstacle gradually decreases with increasing step space. Additionally, logistic regression models were constructed to calculate individual instability probabilities for different conditions based on the experimental data. These results can serve as a foundation method for evaluating the individual and group instability risk during emergency evacuation scenarios.

Relaxed mixed convex combination lemmas: Application to stability analysis of time-varying delay systems

Stability analysis of time-varying delay systems is a fundamental issue, and reducing the conservatism of stability criteria is the key. The high-order Bessel-Legendre inequality together with appropriate Lyapunov–Krasovskii functional (LKF) has the potential to reduce conservatism, but the reciprocally convex and quadratic correlation terms of time-varying delay appear in LKF derivative. This paper proposes the relaxed mixed convex combination lemmas, in which the reciprocally convex combination lemma introduces delay square terms in the LKF derivative, and the quadratic convex combination lemma relaxes the negativity condition for quadratic polynomial functions. The less conservative stability criteria of systems with time-varying delay satisfying two cases are derived, making the delay upper bound larger, and the design degree of freedom higher. Finally, two well-known numerical examples are presented to demonstrate the superiority of the lemmas and criteria over the previous works.

Identification methods for ordinal potential differential games

This paper introduces two new identification methods for linear quadratic (LQ) ordinal potential differential games (OPDGs). Potential games are notable for their benefits, such as the computability and guaranteed existence of Nash Equilibria. While previous research has analyzed ordinal potential static games, their applicability to various engineering applications remains limited. Despite the earlier introduction of OPDGs, a systematic method for identifying a potential game for a given LQ differential game has not yet been developed. To address this gap, we propose two identification methods to provide the quadratic potential cost function for a given LQ differential game. Both methods are based on linear matrix inequalities (LMIs). The first method aims to minimize the condition number of the potential cost function’s parameters, offering a faster and more precise technique compared to earlier solutions. In addition, we present an evaluation of the feasibility of the structural requirements of the system. The second method, with a less rigid formulation, can identify LQ OPDGs in cases where the first method fails. These novel identification methods are verified through simulations, demonstrating their advantages and potential in designing and analyzing cooperative control systems.

Exploring Weight Functions: A Family of One-Point Methods for Multiple Roots

In this paper we introduce a family of one-point methods for multiple roots using weight functions. We analyze the necessary conditions for the weight function to achieve third-order accuracy, and then present a sequence of rational weight functions for a complex quadratic function. The resulting members of the family exhibit Julia sets that align with the perpendicular bisector of the roots. We compare selected members with existing methods based on basin of attraction analysis, demonstrating their competitive performance. Additionally, we investigate the influence of root multiplicity on the fractal structures of basin of attraction for equations with both simple and multiple roots.

Overt and Covert Participation in an Argumentative whole-class Discussion: Spread of Ideas about Quadratic Functions

AbstractMathematical ideas are developed and spread during argumentative whole class discussions between a teacher and her students. The goal of the current study is to characterize how ideas about quadratic functions emerge and are spread during a whole-class discussion among ninth graders. To this end, we recorded both discussions between pairs of students and whole-class discussions led by the teacher. We used the approach of documenting collective activities as our methodological lens. The findings show that mathematical ideas about quadratic functions, like positive and negative range, increase and decrease range, minimum or maximum point, intersection with the axes, and more were spread in two parallel layers. Students participated in an overt layer in which they uttered their ideas in a public discussion. At the same time, they also uttered mathematical ideas privately with their peers in a covert layer. That is, whole-class discussions are not identical for all participants in that the covert layer turns these discussions into a unique experience.

An economic-environmental energy management system design for MT-HVDC networks via a semi-definite programming approximation with robust analysis

This research deals with the economic-environmental dispatch problem (EEDP) in multi-terminal high-voltage direct-current (MT-HVDC) systems by proposing a convex approximation based on semi-definite programming (SDP). The exact formulation of the EEDP corresponds to a non-convex, nonlinear programming problem due to the presence of a nonlinear quadratic constraint regarding the products between voltage variables. The thermal plants' economic and objective functions are modeled using typical quadratic functions. The SDP approach allows reaching a convex approximation that ensures the global optimal solution for each objective function independently or the construction of the optimal Pareto front via the weighting-factor optimization methodology. The proposed SDP approach also considers uncertainties in the demand and in the available power of renewable sources, which makes it robust. The main contribution of this research is a multi-period analysis that includes large-scale renewable generation sources and a robust analysis regarding demand and variations in renewable generation. Numerical results in two MT-HVDC systems demonstrate the effectiveness of the proposed SDP approach when compared to combinatorial optimization algorithms. All numerical simulations were carried out using the CVX convex disciplined tool, with the help of the SEDUMI and SDPT solvers, in the MATLAB programming environment.

Krylov subspace model order reduction of linear dynamical systems with quadratic output

This paper presents Krylov subspace model order reduction of linear dynamical systems with quadratic output. To reach this goal, the quadratic transfer function with two variables is taken into consideration. The Krylov subspaces are established at the finite and the infinite frequency points where both the one-sided and the two-sided projection cases are discussed. The moment matching is accordingly studied and the quadratic transfer function of the reduced system resulting from the two-sided projection case can match more moments. Finally, numerical results illustrate the performance of the proposed Krylov model order reduction methodology.

Mathematics E-learning Packets for Technical and Livelihood Education (TLE) Automotive Servicing

This study aimed to develop and evaluate Mathematics e-learning packets tailored for Technical and Livelihood Education (TLE) Automotive Servicing students at a public national high school, in the City Schools Division of Marikina for the School Year 2023-2024. The impetus for the study was based on the increasing number of learners who are having difficulty solving basic mathematical problems. The research addressed the following questions: (1) identifying Mathematics 9 topics suitable for e-learning packets based on literature review, (2) evaluating the developed packets by automotive and mathematics teachers according to DepEd guidelines concerning content quality, instructional quality, technical quality, assessment, visuals, and plan layout, (3) assessing any significant differences between the evaluations of the two groups of respondents, and (4) gathering comments and suggestions for improvement. Utilizing a descriptive research method, the study involved evaluations from 15 mathematics teachers and 15 automotive teachers from two public national high schools in Marikina City, Philippines. Data were analyzed using Weighted Mean and the Mann-Whitney U Test. Key findings include (1) prioritized topics under the Matatag Curriculum: perpendicular and parallel lines, relations and functions, quadratic functions and equations, direct and inverse variations, trigonometric ratios, and probability of simple events; (2) both groups rated the e-learning packets highly acceptable, with grand weighted means of 3.50 and 3.60, respectively; (3) no significant difference was found between the evaluations of the two groups; (4) respondents praised the packets' adequacy, organization, relevance, and interactivity, offering suggestions to incorporate more visuals, expand topics and activities, add video demonstrations, and involve industry professionals. In conclusion, the developed e-learning packets were well-received and deemed effective, with recommended enhancements aimed at further enriching the learning experience and bridging theoretical knowledge with practical application.

Enhanced low-light image fusion through multi-stage processing with Bayesian analysis and quadratic contrast function

This manuscript introduces an innovative multi-stage image fusion framework that adeptly integrates infrared (IR) and visible (VIS) spectrum images to surmount the difficulties posed by low-light settings. The approach commences with an initial preprocessing stage, utilizing an Efficient Guided Image Filter for the infrared (IR) images to amplify edge boundaries and a function for the visible (VIS) images to boost local contrast and brightness. Utilizing a two-scale decomposition technique that incorporates Lipschitz constraints-based smoothing, the images are effectively divided into distinct base and detail layers, thereby guaranteeing the preservation of essential structural information. The process of fusion is carried out in two distinct stages: firstly, a method grounded in Bayesian theory is employed to effectively combine the base layers, so effectively addressing any inherent uncertainty. Secondly, a Surface from Shade (SfS) method is utilized to ensure the preservation of the scene's geometry by enforcing integrability on the detail layers. Ultimately a Choose Max principle is employed to determine the most prominent textural characteristics, resulting in the amalgamation of the base and detail layers to generate an image that exhibits a substantial enhancement in both clarity and detail. The efficacy of our strategy is substantiated by rigorous testing, showcasing notable progressions in edge preservation, detail enhancement, and noise reduction. Consequently, our method presents significant advantages for real-world applications in image analysis.

Development of Mathematics Tests for Higher Order Thinking Skills on Quadratic Equations and Functions

This research aims to produce valid and reliable phase E mathematics higher-order thinking Skills (HOTS) questions on quadratic equations and functions. The research conducted by researchers in development research with the development model used, namely the type of development studies with two stages, namely preliminary and formative evaluation. The subjects of this study were students of class X SMAN 2 Karimun. In this study, the average score of the assessment of the three aspects (material, construct, and language) by the three validators was 86.16%, with a very valid category. In contrast, the average of all statements in the student response questionnaire was 88.33% with very good criteria and had a reliability value of 0.89 with a high category. This study concluded that the resulting 22 questions of Higher Order Thinking Skills (HOTS) mathematics phase E on the material of equations and quadratic functions developed have met the valid and reliable criteria.

Bipedalism or bipedalisms: The os coxae of StW 573.

There has been a long debate about the possibility of multiple contemporaneous species of Australopithecus in both eastern and southern Africa, potentially exhibiting different forms of bipedal locomotion. Here, we describe the previously unreported morphology of the os coxae in the 3.67 Ma Australopithecus prometheus StW 573 from Sterkfontein Member 2, comparing it with variation in ossa coxae in living humans and apes as well as other Plio-Pleistocene hominins. Statistical comparisons indicate that StW 573 and 431 resemble humans in their anteroposteriorly great iliac crest breadth compared with many other early australopiths, whereas Homo ergaster KNM WT 15000 surprisingly also has a relatively anterioposteriorly short iliac crest. StW 573 and StW 431 appear to resemble humans in having a long ischium compared with Sts 14 and KNM WT 15000. A Quadratic Discriminant Function Analysis of morphology compared with other Plio-Pleistocene hominins and a dataset of modern humans and hominoids shows that, while Lovejoy's heuristic model of the Ardipithecus ramidus os coxae falls with Pongo or in an indeterminate group, StW 573 and StW 431 from Sterkfontein Member 4 are consistently classified together with modern humans. Although clearly exhibiting the classic "basin shaped" bipedal pelvis, Sts 14 (also from Sterkfontein), AL 288-1 Australopithecus afarensis, MH2 Australopithecus sediba and KNM-WT 15000 occupy a position more peripheral to modern humans, and in some analyses are assigned to an indeterminate outlying group. Our findings strongly support the existence of two species of Australopithecus at Sterkfontein and the variation we observe in os coxae morphology in early hominins is also likely to reflect multiple forms of bipedality.