Extreme Value Theorem Research Articles (Page 1)

Assessing conceptual understanding in mathematics remains a persistent challenge for educators, as traditional assessment methods often prioritize procedural fluency over the complexity of connections between mathematical ideas. Consequently, these methods frequently fail to capture the depth of students’ conceptual understanding. This paper addresses this gap by developing and applying a novel rubric based on the Structure of Observed Learning Outcomes (SOLO) Taxonomy, designed to classify student responses according to demonstrated knowledge capacity and cognitive complexity. The rubric introduces transitional levels between the main SOLO categories and includes provisions for evaluating unconventional solutions, enabling a more nuanced assessment of student work based on knowledge depth and integration. The rubric was constructed through an analysis of the conceptual knowledge components required to solve each problem, validated by expert review, and guided by criteria aligned with SOLO level classifications. It also incorporates qualitative feedback to justify each SOLO level assignment. Using this rubric, the study analyzed responses from 57 first-year undergraduate students—primarily chemistry and computer science majors at a private university in the Philippines—to test items on linear approximations and the Extreme Value Theorem. Interrater reliability was established through weighted Cohen’s kappa coefficients (0.659 and 0.667 for the two items). The results demonstrate the rubric’s capacity to differentiate levels of conceptual understanding and reveal key patterns in student thinking, including reasoning gaps, reliance on symbolic manipulation, and misconceptions in mathematical logic. These findings underscore the value of the SOLO Taxonomy in evaluating complex and relational thinking and offer insights for enhancing calculus instruction. By emphasizing the interconnectedness of mathematical ideas, the study highlights the potential of conceptually oriented assessments to foster deeper learning and improve educational outcomes. Furthermore, the rubric’s adaptability suggests its applicability beyond calculus, supporting a broader shift toward concept-focused assessment practices in higher education.

This article discusses the Second Mean Value Theorem for integrals by presenting a comprehensive mathematical proof using a deductive-mathematical approach that involves the Extreme Value Theorem and the Comparison Theorem. Given a continuous function and an integrable function that does not change sign on the interval , it is proven that there exists at least one point such that: \[ \int_a^b f(x)g(x)\,dx = f(c) \int_a^b g(x)\,dx \] The article also provides various examples of the theorem’s application, including numerical computations using the Newton-Raphson method to determine the value of in certain cases. In addition, case studies are presented that link the theorem to modeling in probability, economics, and engineering, thereby demonstrating its relevance in data analysis and dynamic systems. The results of this study not only enrich the theoretical foundation of integral analysis but also offer practical contributions to problem solving in various disciplines.

Safety Analysis at Unsignalized T Intersection Using PET and Extreme Value Theorem

This paper addresses the low-complexity fixed-time prescribed performance control problem for uncertain full-vehicle active suspension systems. Different from existing results that ignore the dynamics of the actuator, this paper considers a hydraulic actuator in the controller design. To address the nonlinearities of hydraulic active suspension systems, a new low-complexity fixed-time prescribed performance control method is proposed. In this method, the function approximators (e.g. neural networks (NNs) and fuzzy systems (FSs)) are not needed, which means that the heavy computational costs are avoided. Furthermore, by developing a fixed-time performance function, the proposed method can ensure that the suspension motions converge to the prescribed range within a finite time. Based on the Lyapunov theorem and Extreme Value Theorem, the stability of the closed-loop suspension system is strictly proved. Finally, the comparative simulation results show that the proposed method effectively improves the attitude stability and ride comfort of the suspension system.

Uncertain fractional differential equation (UFDE) is a useful tool for studying complex systems in uncertain environments. The mathematical characteristics of solution of an UFDE are also widely used in various fields. In this paper, we give the extreme value theorems of solution of Caputo–Hadamard UFDE and applications. A numerical algorithm for obtaining the inverse uncertainty distributions (IUDs) for extreme values of solution of Caputo–Hadamard UFDE is presented; the stability and feasibility of the proposed algorithm are validated by numerical experiments. As an application of extreme value theorems in uncertain financial market, the pricing formulas of the American option based on the new uncertain stock model are given. Besides, the algorithms for computing the price of the American option without explicit pricing formulas based on the Simpson's rule are designed. Finally, the price fluctuation of the American option is illustrated by numerical experiments.

The primary goal of this study is to expand the application of the extreme value theorem by developing the modeling of extreme values using non-linear normalization. The issue of estimating the extreme value index (the non-zero extreme value index) under power and exponential normalization is addressed in this study. Under exponential normalization, counterparts of the Hill estimators for the extreme value index estimators under linear normalization are proposed based on the characteristics of the extreme value index, threshold, and the data itself. In addition, based on the generalized Pareto distributions, more condensed and flexible Hill estimators are proposed under power and exponential normalization. These proposed estimators assist us to choose the threshold more flexibly and getting rid of data waste. The R-package runs a thorough simulation analysis to examine the effectiveness of the suggested estimators.

SAR target classification is an important application in SAR image interpretation. In practical applications, the battlefield is open and dynamic, and the SAR target classification model often encounters the targets of unknown classes. However, most of the existing SAR target classification methods follow the close-set assumption. It makes them only classify several fixed classes of targets and can’t deal with the targets from unknown classes. To this end, this paper proposes a novel SAR target classification method. This method can not only classify the targets from known classes and search targets from unknown classes but also incrementally update the classification model with these unknown class targets. Specifically, an autoencoder improved by MS-SSIM (multi-scale structural similarity) loss is utilized to extract targets’ features, and it can better utilize the structural information in SAR images. Next, the classifier based on EVT (Extreme Value Theorem) is established, which can classify the known class targets and search the unknown class targets. Then, we perform improved model reduction on the established classifier. This operation could speed up the model and prepare for incremental learning. Finally, after manually labeling those unknown class targets, the classifier is updated with these data in incremental form. Experimental results on the MSTAR (Moving and Stationary Target Automatic Recognition) dataset indicate that, compared with the state-of-the-art methods, our proposed method has better performance in open set recognition and incremental learning.

Abstract Sequential Monte Carlo (SMC) is a powerful technique originally developed for particle filtering and Bayesian inference. As a generic optimizer for statistical and nonstatistical objectives, its role is far less known. Density‐tempered SMC is a highly efficient sampling technique ideally suited for challenging global optimization problems and is implementable with a somewhat arbitrary initialization sampler instead of relying on a prior distribution. SMC optimization is anchored at the fact that all optimization tasks (continuous, discontinuous, combinatorial, or noisy objective function) can be turned into sampling under a density or probability function short of a norming constant. The point with the highest functional value is the SMC estimate for the maximum. Through examples, we systematically present various density‐tempered SMC algorithms and their superior performance vs. other techniques like Markov Chain Monte Carlo. Data cloning and k‐fold duplication are two easily implementable accuracy accelerators, and their complementarity is discussed. The Extreme Value Theorem on the maximum order statistic can also help assess the quality of the SMC optimum. Our coverage includes the algorithmic essence of the density‐tempered SMC with various enhancements and solutions for (1) a bi‐modal nonstatistical function without and with constraints, (2) a multidimensional step function, (3) offline and online optimizations, (4) combinatorial variable selection, and (5) noninvertibility of the Hessian.This article is categorized under: Statistical and Graphical Methods of Data Analysis > Monte Carlo Methods Algorithms and Computational Methods > Stochastic Optimization Algorithms and Computational Methods > Integer Programming

Abstract This paper establishes a defect model of thermoelectric materials that the oblique multi‐field loads are applied along an arbitrary direction with the crack initiated from a circular hole, and gives both analytical solutions and finite element method (FEM) simulation. On the one hand, in virtue of the conformal mapping function and the Cauchy integral formula, the analytical solutions of electric current density, energy flux, and stress are obtained. Furthermore, the thermoelectric field intensity factors (TEIFs) and the stress intensity factors are also attained. Most notably, we apply the extreme value theorem of multivariate function to explore the effect of the ratio of the radius of the hole, R, and crack length, L, on the fields’ intensity factors of thermoelectric materials under oblique multi‐field loads. Importantly, the maximum value of the mode‐II stress intensity factor (SIF) is obtained when . On the other hand, the finite element simulation is used to explore the fracture behaviors of thermoelectric materials. The analytical solutions and the finite element simulation results are consistent. The effects of the oblique multi‐field loads on the thermoelectric field and the stress field at the crack tip are detailed. The concentration phenomenon of the thermoelectric field and the stress field at the crack tip are influenced by the radius of the circular hole, the crack length, and the directions of the loads.

Day-ahead interval optimization of combined cooling and power microgrid based on interval measurement

Extremes of Lévy-driven spatial random fields with regularly varying Lévy measure

Proper timeliness is vital for a lot of real-world computing systems. Understanding the phenomena of extreme workloads is essential because unhandled, extreme workloads could cause violation of timeliness requirements, service degradation, and even downtime. Extremity can have multiple roots: (1) service requests can naturally produce extreme workloads; (2) bursts could randomly occur on a probabilistic basis in case of a mixed workload in multiservice systems; (3) workload spikes typically happen in deadline bound tasks.Extreme Value Analysis (EVA) is a statistical method for modeling the extremely deviant values corresponding to the largest values. The foundation mathematics of EVA, the Extreme Value Theorem, requires the dataset to be independent and identically distributed. However, this is not generally true in practice because, usually, real-life processes are a mixture of sources with identifiable patterns. For example, seasonality and periodic fluctuations are regularly occurring patterns. Deadlines can be purely periodic, e.g., monthly tax submissions, or time variable, e.g., university homework submission with variable semester time schedules.We propose to preprocess the data using time series decomposition to separate the stochastic process causing extreme values. Moreover, we focus on the case where the root cause of the extreme values is the same mechanism: a deadline. We exploit known deadlines using dynamic time warp to search for the recurring similar workload peak patterns varying in time and amplitude.

Rocks are one of the major surface features of Mars. The accurate characterization of the chemical and mineralogical composition of Martian rocks would yield significant evolutionary information about relevant geological processes and exobiological exploration. Many existing rock recognition systems generally assume that all testing classes are known during training. Over real planetary surfaces, the autonomous recognition system is likely to encounter an unknown category of rock that is crucial to the performance of the rock classification task. Therefore, we develop an open-set Martian rock-type classification framework based on their spectral signatures, with the subgoal of new/unknown rock-type recognition and category-incremental learning for expanding the recognition model. First, the spectral signatures of rock samples are captured to characterize their mineralogical compositions and physical properties, which serves as the input of the developed framework. To further produce the highly discriminative feature representation from the original spectral signatures, a Transformer architecture integrated with contrastive learning is constructed and trained in an end-to-end manner to force instances of the same class to remain close-by while pushing those of a dissimilar class farther apart. Following this, according to the extreme value theorem (EVT), category-specific distance distribution analysis is conducted to detect and identify new/unknown types of rock samples due to the isolated characteristics of new/unknown rock samples in the latent feature space. Finally, the recognition model is incrementally updated to learn these identified "unknown" samples without forgetting previously known categories when the associated labels are progressively obtained. The multispectral camera, a duplicated payload of the counterpart onboard the Zhurong rover, is used as the multispectral sensor for capturing the spectral information of the laboratory rock dataset shared by the National Mineral Rock and Fossil Specimens Resource Center for both qualitative and quantitative evaluation. Experimental results indicate the effectiveness and robustness of the developed in situ analysis framework.

Application of extreme value theorem in modelling oil consumption of organisation of petroleum exporting countries

This paper deals with the uniform torsion of thin-walled elliptical tube. The material of the tube is nonhomogeneous and it depends on one of the curvilinear coordinates which defines the cross section of thin-walled bar with closed profile. The approximate solution for the stresses, torsion function and torsional rigidity are obtained by the application of two extreme value theorems of linearized elasticity.

Bermudan options pricing formulas in uncertain financial markets

This paper introduces the concept of softarison. Softarisons merge soft set theory with the theory of binary relations. Their purpose is the comparison of alternatives in a parameterized environment. We develop the basic theory and interpretations of softarisons. Then, the normative idea of ‘optimal’ alternatives is discussed in this context. We argue that the meaning of ‘optimality’ can be adjusted to fit in with the structure of each problem. A sufficient condition for the existence of an optimal alternative for unrestricted sets of alternatives is proven. This result means a counterpart of Weierstrass extreme value theorem for softarisons; thus, it links soft topology with the act of choice. We also provide a decision-making procedure—the minimax algorithm—when the alternatives are compared through a softarison. A case-study in the context of group interviews illustrates both the application of softarisons as an evaluation tool, and the computation of minimax solutions.

Summary. IF-THEN rules in fuzzy inference is composed of multiple fuzzy sets (membership functions). IF-THEN rules can therefore be considered as a pair of membership functions [7]. The evaluation function of fuzzy control is composite function with fuzzy approximate reasoning and is functional on the set of membership functions. We obtained continuity of the evaluation function and compactness of the set of membership functions [12]. Therefore, we proved the existence of pair of membership functions, which maximizes (minimizes) evaluation function and is considered IF-THEN rules, in the set of membership functions by using extreme value theorem. The set of membership functions (fuzzy sets) is defined in this article to verifier our proofs before by Mizar [9], [10], [4]. Membership functions composed of triangle function, piecewise linear function and Gaussian function used in practice are formalized using existing functions. On the other hand, not only curve membership functions mentioned above but also membership functions composed of straight lines (piecewise linear function) like triangular and trapezoidal functions are formalized. Moreover, different from the definition in [3] formalizations of triangular and trapezoidal function composed of two straight lines, minimum function and maximum functions are proposed. We prove, using the Mizar [2], [1] formalism, some properties of membership functions such as continuity and periodicity [13], [8].

First, we consider a stationary random field indexed by an increasing sequence of subsets of $\mathbb {Z}^{d}$ . Under certain mixing and anti–clustering conditions combined with a very broad assumption on how the sequence of spatial index sets increases, we obtain an extremal result that relates a normalized version of the distribution of the maximum of the field over the index sets to the tail distribution of the individual variables. Furthermore, we identify the limiting distribution as an extreme value distribution. Secondly, we consider a continuous, infinitely divisible random field indexed by $\mathbb {R}^{d}$ given as an integral of a kernel function with respect to a Levy basis with convolution equivalent Levy measure. When observing the supremum of this field over an increasing sequence of (continuous) index sets, we obtain an extreme value theorem for the distribution of this supremum. The proof relies on discretization and a conditional version of the technique applied in the first part of the paper, as we condition on the high activity and light–tailed part of the field.

This paper discusses the problem of suboptimal local piecewise H ∞ fuzzy control of quasi-linear spatiotemporal dynamic systems with control magnitude constraints. A Takagi-Sugeno fuzzy partial differential equation (PDE) model with space-varying coefficient matrices is first assumed to be derived for exactly describing nonlinear system dynamics. In the light of the fuzzy model, a local piecewise fuzzy feedback controller is then constructed to guarantee the exponential stability with a prescribed H ∞ disturbance attenuation level for the resulting closed-loop system, while the control constraints are also ensured. A sufficient condition on the existence of such fuzzy controller is developed by the Lyapunov direct method and an integral inequality and presented in terms of space algebraic linear matrix inequalities (LMIs) coupled with LMIs. By virtue of extreme value theorem, a suboptimal-constrained local piecewise H ∞ fuzzy control design in the sense of minimizing the disturbance attenuation level is formulated as a minimization optimization problem with LMI constraints. Finally, the proposed method is applied to solve the feedback control of a quasi-linear FitzHugh-Nagumo equation with space-varying coefficients, and simulation results show its effectiveness and merit.