Functional Limit Research Articles

Systems involving mean value formulas on trees

Abstract In this paper, we study the Dirichlet problem for systems of mean value equations on a regular tree. We deal both with the directed case (the equations verified by the components of the system at a node in the tree only involve values of the unknowns at the successors of the node in the tree) and the undirected case (now the equations also involve the predecessor in the tree). We find necessary and sufficient conditions on the coefficients in order to have existence and uniqueness of solutions for continuous boundary data. In a particular case, we also include an interpretation of such solutions as a limit of value functions of suitable two-players zero-sum games.

A method for solving the limit of multivariate composite functions

Abstract In this paper, according to the concept of multivariate composite function, the limit and continuity conditions of multivariate composite function are given, and the continuity relationship between the inner function and the outer function of the composite function at the same point is investigated. The relationship between the limit and continuity of a composite function composed of two functions at the same point is investigated by examples, while some limit results are explained from the perspective of composite functions, and the limit calculation law of multivariate composite functions is proposed. Using the consistent central limit theorem in the limit calculation law, the absolute value and limit of the interval of the multivariate composite function are obtained, and the iterative limit of the variational reduction approximation method is confirmed by combining the absolute value and limit of the interval of the multivariate composite function. On the basis of the application of limit operation of a multivariate composite function, the limit distribution of high-dimensional likelihood ratio statistics of multivariate composite function is analyzed by using data simulation analysis. The results show that both limit distributions have better simulation results when p are small, such as p = 5 or p = 10, but the traditional limit is better than the normal limit, indicating that the results of multivariate composite function limit calculations have good accuracy. A method for solving the limits of multivariate composite functions is proposed in this study, which is of great theoretical and practical significance in the study of higher mathematics.

Evidence-based position paper on physical and rehabilitation medicine professional practice for persons with COVID-19, including post COVID-19 condition: the European PRM position (UEMS PRM Section).

Although multiple factors still pose challenges to inpatient/outpatient rehabilitation for survivors of COVID-19, rehabilitation plays a key role for this patient population. This study aimed to improve Physical and Rehabilitation Medicine (PRM) physician's professional practice for persons with COVID-19-related functioning limitations, to promote functional recovery and reduce activity limitations and/or participation restrictions. A systematic review of the scientific literature was performed from December 2019 to August 2022, followed by production of recommendations through 5 Delphi rounds, by consensus among the delegates of all European countries represented in the Union of European Medical Specialists PRM Section. The systematic literature review is reported together with thirty-two recommendations resulting from the Delphi procedure. The PRM physician's role for persons with COVID-19-related limitations of functioning is to develop, foster, and monitor the implementation of an individual rehabilitation project tailored to the patient's age, previous medical and functional status, current comorbidities and complications, activity limitations and participation restrictions and personal and environmental factors. This is done by applying the concept of a multi-specialty integrated service model with multi-professional/interdisciplinary teams, providing care at all stages of COVID-19 illness. This evidence-based position paper represents the official position of the European Union through the UEMS PRM Section.

Quantitative Evaluation Model of the Quality of Remanufactured Product

Remanufacturing is a significant process for achieving carbon neutrality. However, the existing literature shows that consumer concerns regarding the quality of remanufactured products restrict the large-scale development of the remanufacturing industry. Studies on the quantitative evaluation of the quality of remanufactured products are limited. Therefore, we propose a quantitative evaluation model of the quality of remanufactured products. Extended from the formation process of the quality of remanufactured products, the quality loss function of remanufactured parts and assemblies was constructed based on Taguchi's quality concept. Subsequently, the relationship among social loss, functional limit, and quality loss of remanufactured products was assessed. Accordingly, a quality measurement model of remanufactured products was established. Moreover, the parameter estimation algorithm based on a neural network was evaluated to deal with the dynamic changes and uncertainties of remanufacturing technology and the remanufacturing market. Finally, the application of the proposed model was demonstrated through a case study in which the quality of remanufactured engines was quantitatively evaluated. The relevance of the model was verified through market feedback. This article can provide practitioners and decision-makers with new and effective tools and insights to improve the quality of remanufactured products and reduce the cognitive gaps of customers. It is expected to accelerate the high-quality development of the remanufacturing industry.

Phronesis (Practical Wisdom) and Philia (Friendship)

In Aristotle’s ethics, friendship is a complex and nuanced construct, which he termed philia. Aristotle’s philia is much broader than affinity­‑based peer relationships, as friendship is often conceived currently. He identified three types of friendship (utility, pleasure, and character), and he devoted approximately 20% of the Nicomachean Ethics to friendship. This is because close personal relationships and developing oneself as a character friend are extremely important to living well as a human. This centrality makes character or virtue friendship distinctive and highly valuable. We apply the construct of phronesis to philia to reveal important aspects of both. First, we argue that situations common to character friendships highlight the reasonable limits of the emotion regulation function of phronesis in resolving seemingly intractable ethical conflicts. Second, we suggest that these limits suggest, counterintuitively, that character friendships (even among the most virtuous actors) are still subject, under some conditions, to substantial feelings of conflict and emotional ambivalence.

The Sharp Interface Limit of an Ising Game

The Ising model of statistical physics has served as a keystone example of phase transitions, thermodynamic limits, scaling laws, and many other phenomena and mathematical methods. We introduce and explore an Ising game, a variant of the Ising model that features competing agents influencing the behavior of the spins. With long-range interactions, we consider a mean-field limit resulting in a nonlocal potential game at the mesoscopic scale. This game exhibits a phase transition and multiple constant Nash-equilibria in the supercritical regime. Our analysis focuses on a sharp interface limit for which potential minimizing solutions to the Ising game concentrate on two of the constant Nash-equilibria. We show that the mesoscopic problem can be recast as a mixed local/nonlocal space-time Allen-Cahn type minimization problem. We prove, using a Γ-convergence argument, that the limiting interface minimizes a space-time anisotropic perimeter type energy functional. This macroscopic scale problem could also be viewed as a problem of optimal control of interface motion. Sharp interface limits of Allen-Cahn type functionals have been well studied. We build on that literature with new techniques to handle a mixture of local derivative terms and nonlocal interactions. The boundary conditions imposed by the game theoretic considerations also appear as novel terms and require special treatment.

Ab. No. 140 Non-Surgical Interventions for the Prevention and Treatment of Radiation-Induced Fibrosis and Its Associated Disability in Patients With Head and Neck Cancer: An ICF Framework-Based Systematic Review

Introduction: To outline non-surgical interventions effective in preventing or treating the effects of Radiation-induced fibrosis (RIF) in Head and Neck Cancer (HNC) Survivors across domains of International Classification of Functioning, Disability and Health (ICF). RIF is a debilitating, long term sequalae of radiotherapy in HNC survivors. ICF provides an opportunity to map limitation in body functions, activity and participation with interventions but has not been used in understanding the impact of RIF in HNC survivors. Methods: Studies reporting non-surgical interventions for prevention/treatment of RIF. The review was registered in PROSPERO. Using a systematic review framework, a comprehensive search was run in six databases, independently screened, and verified by authors. Risk-of-Bias was determined using NIH quality assessment tool. Qualitative descriptive analysis was performed for the included studies. Intervention targets and outcomes were mapped to ICF domains. Results: Fifteen of 7966 retrieved studies representing data of 603 participants were included for the review. Fourteen studies were of fair and one of good quality. Seven studies representing 294 participants aimed at prevention while eight representing 309 participants aimed at treatment of RIF. Interventions studied included- exercise, electrotherapy, orthotics, swallow therapy and pharmacotherapy. Studies weren’t mutually exclusive for ICF domains. Among prevention studies, four focused on body structure, six on body function, and one on activity and participation. Among treatment studies, five addressed body structure, seven on body function, and five on activity and participation. Conclusion: Non-surgical interventions for prevention or treatment of RIF are available across ICF domains. However, the studies are limited in numbers and are of fair quality. Implications: This review provides a summary of the current state of research evidence and could be useful in providing direction to future research for prevention/treatment of RIF in HNC survivors.

Signorini problem as a variational limit of obstacle problems in nonlinear elasticity

<abstract><p>An energy functional for the obstacle problem in linear elasticity is obtained as a variational limit of nonlinear elastic energy functionals describing a material body subject to pure traction load under a unilateral constraint representing the rigid obstacle. There exist loads pushing the body against the obstacle, but unfit for the geometry of the whole system body-obstacle, so that the corresponding variational limit turns out to be different from the classical Signorini problem in linear elasticity. However, if the force field acting on the body fulfils an appropriate geometric admissibility condition, we can show coincidence of minima. The analysis developed here provides a rigorous variational justification of the Signorini problem in linear elasticity, together with an accurate analysis of the unilateral constraint.</p></abstract>

Simultaneous Model Change Detection in Multivariate Linear Regression With Application to Indonesian Economic Growth Data

In this paper, we study asymptotic model change detection in multivariate linear regression by using the Kolmogorov–Smirnov function of the partial sum process of recursive residuals. We approximate the rejection region and also the power function of the test by establishing a functional central limit theorem for the sequence of the partial sum processes of the recursive residuals of the observations. When the assumed model is true, the limit process is given by the standard multivariate Brownian motion which does not depend on the regression functions. However, when the assumed model is not true (some models change), the limit process is represented by a vector of deterministic trend plus the standard multivariate Brownian motion. The finite sample size rejection region and the power of the test are investigated by means of Monte Carlo simulation. The simulation study shows evidence that the proposed test is consistent in the sense that it attains the power larger than the size of the test when the hypothesis is not true. We also demonstrate the application of the proposed test method to Indonesian economic growth data in which we test the adequacy of three‐variate low‐order polynomial model. The test result shows that the growth of the Indonesian economy is neither simultaneously constant nor linear. The test has successfully detect the appearance of a change in the model which is mainly caused by the COVID‐19 pandemic in 2020.

A functional central limit theorem for the K-function with an estimated intensity function

A functional central limit theorem for the K-function with an estimated intensity function

CLX-155: A Novel, Oral 5-FU Prodrug Displaying Antitumor Activity in Human Colon Cancer Xenograft Model in Nude Mice

Introduction: Capecitabine is an oral prodrug of 5-FU, which interpatient pharmacokinetic (PK) variability related to liver function and severe adverse events (e.g., hand-foot syndrome, myelosuppression, and neurotoxicity) limits. CLX-155 is a novel oral 5’-DFCR prodrug involving 5’-DFCR as an intermediate for generating 5-FU, unlike capecitabine, which the liver does not metabolize. This study addresses the following research question: what is the activity of CLX-155 in a human colon cancer xenograft model in nude mice? Methods: This study involved 50 Foxn1 athymic nude female mice implanted with the human colon cancer cell line HCT116 (5 million cells per site). Investigators randomized animals into five treatment groups (N = 10): vehicle control, CLX-155 at doses of 125, 250, and 500 mg/kg/day, or capecitabine 1000 mg/kg/day. Animals received oral treatment once daily for five days a week with two days off for a total of three consecutive weeks. Investigators evaluated treatment toxicity based on body weight loss. Calculations for tumor growth inhibition involved comparing changes in tumor volume on a given day to tumor volumes on Day 1. Results: CLX-155 demonstrated statistically significant, dose-dependent tumor growth inhibition at all doses compared to vehicle control (p<0.0001). Tumor growth inhibition at Day 15 for CLX-155 treatment groups of 125, 250, and 500 mg/kg/day was 57.8%, 70.4%, and 90.6% respectively. Two animals in the CLX-155 500 mg/kg/day treatment group experienced complete tumor regression, and all animals in the CLX-155 treatment groups survived. Two animals in the CLX-155 250 and 500 mg/kg/day dosing groups experienced a decrease in body weight. In contrast, two mice in the capecitabine group exhibited clinical signs of hunchback and scaly skin, progressive weight loss, and eventual death. Conclusion: CLX-155 demonstrated comparable tumor growth inhibition to capecitabine but at a lower dose, suggesting increased potency. In addition, CLX-155 exhibited improved tolerability and fewer adverse effects. These promising results support further investigation in Phase 1 clinical trials for managing colon cancer.

Applying L’Hospital’s Rule to Evaluate the Limit of a Function and Its Practical Applications in Physics and Economics

Applying L’Hospital’s Rule to Evaluate the Limit of a Function and Its Practical Applications in Physics and Economics

Arthroscopic Management of Arthrofibrosis After Hemiarthroplasty for Proximal Humerus Fracture Dislocation

Poor functional outcomes after hemiarthroplasty for proximal humerus fractures are common, yet revision surgery is relatively rare. Arthroscopic treatment for postoperative stiffness can be considered in the setting of functional limits to glenohumeral range of motion impacting activities of daily living after adequate conservative treatment with physical therapy and in the setting of healed, well-positioned tuberosities and humeral components. This Technical Note illustrates a stepwise approach to an arthroscopic lysis of adhesions and capsular release for the treatment of arthrofibrosis of the shoulder. The advantages of this technique include an alternative approach to entering the glenohumeral joint under direct subacromial visualization and a 2-posterior portal approach to the inferior and anteroinferior capsule, which can be challenging to achieve in the setting of severe postsurgical arthrofibrosis.

SOME RESULTS ON BOX DIMENSION ESTIMATION OF FRACTAL CONTINUOUS FUNCTIONS

In this paper, we explore upper box dimension of continuous functions on [Formula: see text] and their Riemann–Liouville fractional integral. Firstly, by comparing function limits, we prove that the upper box dimension of the Riemann–Liouville fractional order integral image of a continuous function will not exceed [Formula: see text], the result similar to [Y. S. Liang and W. Y. Su, Fractal dimensions of fractional integral of continuous functions, Acta Math. Appl. Sin. E 32 (2016) 1494–1508]. Secondly, we prove that upper box dimension of multiple algebraic sums of continuous functions does not exceed the largest box dimension among them, backing up our conclusion with an appropriate example. Finally, we draw the same conclusions for the product of multiple continuous functions.

Conditional functional limit theorem for random reccurence sequence conditioned on large deviation event

Пусть $Z_n, n\geqslant 0\}$ - ветвящийся процесс в случайной среде, представляющей собой последовательность независимых одинаково распределенных случайных величин, $\{S_n, n\geqslant 1\}$ - его сопровождающее блуждание, $\xi_i$ - шаги сопровождающего блуждания. В предположении, что $\xi_1$ удовлетворяет условию Крамера и выполнены моментные условия на количество потомков одной частицы, для $Z_n$ известна асимптотика вероятностей больших уклонений $\mathbf{P}(\ln Z_n > x)$, где $x/n>\mu^*$, $\mu^*$ - некоторый параметр, зависящий от типа процесса. В работе исследуется поведение траектории процесса, совершающего такого рода уклонение. В частности, получена условная функциональная предельная теорема для траектории $(Z_{[nt]}, t\in [0,1])$, рассматриваемой при условии совершения ею большого уклонения $\ln Z_n>x$. Результат получен для более общей модели случайной рекуррентной последовательности $Y_{n+1}=A_n Y_n + B_n$, $n\geqslant 0$, где последовательность $\{A_i\}$ является последовательностью независимых одинаково распределенных величин, а $Y_0$, $B_i$, $i\geqslant 0$, вообще говоря, зависимы и имеют различное распределение, но удовлетворяют некоторым моментным условиям.

In cases of humeral diaphyseal fractures, is lateral approach surgery without radial nerve exploration as effective and safe as conventional surgery?

This study compares the efficacy and safety of lateral approach surgery with and without radial nerve dissection in treating humeral diaphyseal fractures. It assesses clinical, radiological, and complication outcomes, providing a description of the surgical methods and perioperative benefits. We retrospectively analyzed data from 71 patients admitted between May 2015 and December 2022 who underwent lateral approach surgery for humeral diaphyseal fractures. Group 1, consisting of 34 patients without radial nerve dissection, and Group 2, comprising 37 patients with radial nerve dissection, were compared. Parameters such as age, gender, fracture side (right/left), fracture type, follow-up time, surgical duration, blood loss, radiological and clinical evaluations (including Shoulder-Elbow range of motion [ROM] and Quick Disabilities of the Arm, Shoulder, and Hand score [Q-DASH]), and complications were examined. Surgical techniques and outcomes were documented. Both groups exhibited comparable distributions in age, gender, fracture types, and follow-up times (p>0.05). Group 1 demonstrated significantly lower surgical duration and blood loss compared to Group 2 (p<0.05 for both). Clinical assessment revealed satisfactory shoulder and elbow ROM within functional limits for all patients, with no instances of infection. Q-DASH scores were similar between groups. Postoperative radial nerve palsy occurred in one patient in Group 1 and three patients in Group 2, with all cases resolving uneventfully during outpatient follow-ups. Radiological assessment confirmed uneventful union in all patients. Lateral approach surgery without radial nerve dissection for humeral diaphyseal fractures offers comparable effectiveness and safety to conventional surgery, with potential perioperative advantages such as reduced operation time and blood loss.

THE EFFECTIVENESS OF THE MNEMONIC METHOD TO IMPROVE LONG-TERM MEMORY ON TRIGONOMETRIC FUNCTION LIMITS MATERIAL

This research aims to determine the effectiveness of applying the mnemonic method which is linked to long-term memory in the material on limits of trigonometric functions in basic calculus courses in the Palangka Raya University mechanical engineering education study program. The research is action research based on the Kurt Lewin model which consists of several stages including: planning, action, observation and reflection. The research subjects were 44 students taking basic calculus courses. The results of the research show that the mnemonic method is effective for improving long-term memory in Trigonometry function limits material in basic calculus courses in the Mechanical Engineering Education Study Program, Palangka Raya University. This can be seen in the average test result after being given lectures using the mnemonic method, which is 70.5 with classical completeness achieved being 88.6%. Thus, the expected classical completeness of 75% was achieved. Student responses to the use of mnemonic methods in lectures. As many as 95% of students agreed that lectures were fun. The majority of students are actively involved in lectures using this mnemonic method, where as many as 90.9% agree and it is easier for students to remember mathematical formulas in the material on limits of trigonometric functions with this mnemonic method, where as many as 97% agree. The majority of students agree that the mnemonic method needs to be used again in the next lecture material which contains many mathematical formulas, with 90.4% agreeing.

A characterization of BV and Sobolev functions via nonlocal functionals in metric spaces

We study a characterization of BV and Sobolev functions via nonlocal functionals in metric spaces equipped with a doubling measure and supporting a Poincaré inequality. Compared with previous works, we consider more general functionals. We also give a counterexample in the case p=1 demonstrating that, unlike in Euclidean spaces, in metric measure spaces the limit of the nonlocal functionals is only comparable, not necessarily equal, to the variation measure ‖Df‖(Ω).

ON TWO-SIDED UNIDIRECTIONAL MEAN VALUE INEQUALITY IN A FRÉCHET SMOOTH SPACE

The paper is devoted to a new unidirectional mean value inequality for the Fréchet subdifferential of a continuous function. This mean value inequality finds an intermediate point and localizes its value both from above and from below; for this reason, the inequality is called two-sided. The inequality is considered for a continuous function defined on a Fréchet smooth space. This class of Banach spaces includes the case of a reflexive space and the case of a separable Asplund space. As some application of these inequalities, we give an upper estimate for the Fréchet subdifferential of the upper limit of continuous functions defined on a reflexive space.

A Novel Approach to Modeling Incommensurate Fractional Order Systems Using Fractional Neural Networks

This research explores the application of the Riemann–Liouville fractional sigmoid, briefly RLFσ, activation function in modeling the chaotic dynamics of Chua’s circuit through Multilayer Perceptron (MLP) architecture. Grounded in the context of chaotic systems, the study aims to address the limitations of conventional activation functions in capturing complex relationships within datasets. Employing a structured approach, the methods involve training MLP models with various activation functions, including RLFσ, sigmoid, swish, and proportional Caputo derivative PCσ, and subjecting them to rigorous comparative analyses. The main findings reveal that the proposed RLFσ consistently outperforms traditional counterparts, exhibiting superior accuracy, reduced Mean Squared Error, and faster convergence. Notably, the study extends its investigation to scenarios with reduced dataset sizes and network parameter reductions, demonstrating the robustness and adaptability of RLFσ. The results, supported by convergence curves and CPU training times, underscore the efficiency and practical applicability of the proposed activation function. This research contributes a new perspective on enhancing neural network architectures for system modeling, showcasing the potential of RLFσ in real-world applications.