limit function
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2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Xianyong Huang ◽  
Bicheng Yang

By the use of the weight functions, the symmetry property, and Hermite-Hadamard’s inequality, a more accurate half-discrete Mulholland-type inequality involving one multiple upper limit function is given. The equivalent conditions of the best possible constant factor related to multiparameters are studied. Furthermore, the equivalent forms, several inequalities for the particular parameters, and the operator expressions are provided.


Author(s):  
A.G. Chentsov

Nonlinear differential game (DG) is investigated; relaxations of the game problem of guidance are investigated also. The variant of the program iterations method realized in the space of position functions and delivering in limit the value function of the minimax-maximin DG for special functionals of a trajectory is considered. For every game position, this limit function realizes the least size of the target set neighborhood for which, under proportional weakening of phase constraints, the player interested in a guidance yet guarantees its realization. Properties of above-mentioned functionals and limit function are investigated. In particular, sufficient conditions for realization of values of given function under fulfilment of finite iteration number are obtained.


2021 ◽  
Vol 2 (4) ◽  
pp. 539-548
Author(s):  
Kado Kado

The paradigm shift from traditional didactic instruction to technology-enriched teaching and learning environments significantly benefits learners. Educational technology can visualize abstract mathematical concepts contextually and graphically and allow learners to actively construct this knowledge. This study aims to ascertain the efficacy of a computer-assisted instruction method using GeoGebra in further developing the concept of the function limit for grade XI students. This study employed a quasi-experiment static-group comparison design with 60 students from Gongzim Ugyen Dorji Central School at Haa in Bhutan. The students were divided into two equal groups. Group ‘A’ used the GeoGebra software, while group ‘B’ used the conventional method to learn the limit of the function. The data was collected through a Conceptual Knowledge Test of Limit Function. In addition, an independent sample t-test was employed using the Statistical Package for the Social Sciences (SPSS 22.0). This study demonstrated that students who were taught using GeoGebra outperformed those who learned through conventional methods. The results confirmed that GeoGebra software could enhance and significantly improve students’ conceptual understanding of the limit of the function.


2021 ◽  
Vol 5 (Supplement_1) ◽  
Author(s):  
Qurat Ul Ain Amjad ◽  
Spencer Ellis

Abstract Case report - Introduction Rheumatic disease occurring as a paraneoplastic manifestation such as dermatomyositis is well recognised, whilst the symptoms of cancer may frequently mimic the presentations of common rheumatic disorders. It is essential to recognise these differential diagnoses to avoid delay in the detection of malignancy. We present an unusual case of rheumatoid arthritis where, conversely, the diagnosis of inflammatory arthritis was delayed due to the attribution of musculoskeletal symptoms to existing malignancy undergoing treatment. This case highlights the importance of careful history and examination. Cancer and inflammatory disorders may co-exist and delayed diagnosis of either condition can result in increased morbidity. Case report - Case description A 65-year-old male presented to Ear, Nose and Throat (ENT) clinic with a facial lump in association with swelling in his throat, tongue and neck associated with a decrease in appetite and weight loss. He reported an active lifestyle, including mountain climbing until a month prior. He was a non-smoker with alcohol intake of less than 10units/week. Other medical history included osteoarthritis of his left knee which did not limit function. Imaging of the head and neck confirmed a mass in the oropharynx, tonsils, and tongue muscles. Subsequent biopsy revealed moderate to poorly differentiated squamous cell carcinoma (SCC) of the left tonsil. 11 months prior to this presentation, he had developed pain and swelling involving knees, shoulders, and small joints of hands. His mobility became progressively impaired eventually requiring a wheelchair for any outdoor excursion. The systemic and musculoskeletal symptoms were attributed to a paraneoplastic manifestation of his malignant process. A rheumatology opinion was eventually requested in view of the persistence of joint symptoms despite effective cancer treatment with radical radiotherapy. He described early morning stiffness lasting more than 3 hours. Synovitis was evident at his hands, wrists and knees, whilst shoulder movement was restricted, resulting in a high disease activity score (DAS) of 7.44. Blood workup showed normal rheumatoid factor and cyclic citrullinated peptide (CCP). Acute phase markers were elevated with ESR 99 mm/hr, CRP 79 mg/l and ferritin 1194 ng/ml. Ultrasound (US) hands confirmed widespread active synovitis involving bilateral wrist and metacarpophalangeal joints. X-rays showed only degenerative changes. A diagnosis of seronegative inflammatory arthritis was established, and he was initiated on hydroxychloroquine with reducing regime of prednisolone. Methotrexate (MTX) was briefly deferred pending oncology advice. Following initiation of MTX he improved dramatically, returning to normal mobility, and steroids were successfully tapered off. Case report - Discussion Our patient had symptoms of inflammatory arthritis prior to identification of a SCC with local spread; however, his systemic and musculoskeletal features were initially regarded as paraneoplastic. Paraneoplastic arthritides encompass the musculoskeletal manifestations of malignancy. This includes polyarthritis, inflammatory myopathies, hypertrophic osteoarthropathy and palmar fasciitis. Symptoms may precede or occur concurrently with cancer presentations and often respond to treatment of the underlying cancer. Persistence of musculoskeletal symptoms may be a clue to co-existence of a rheumatic diagnosis. Reassessment of history and examination can guide physicians to a concurrent second diagnosis when clinical progress is not as expected. Our case also highlights the question of safe use of immunosuppressive drugs in rheumatic patients with cancer and whether they may promote or induce malignant disease. Literature is challenging to assess as many systemic inflammatory disorders, including RA, have an established increased risk of certain cancers, including lymphoproliferative disease. He was managed by prednisolone and hydroxychloroquine, followed rapidly by MTX, leading to dramatic symptomatic improvement. Low-dose MTX (25—30mg weekly) is considered first-line treatment in RA. Of the commonly used anti-rheumatic therapies it has the least evidence to suggest a potential increased malignancy risk. Other anti -rheumatic drugs such as Tumour Necrosis Factor (TNF) inhibitors also have a favourable risk profile regarding cancer development. Most observational studies evaluating the use of biologic therapy in RA patients with previous solid tumours do not show an increased risk of recurrence. Coordination of care with the patient and their oncologist is essential in the management of cases of individuals with rheumatic disease and concomitant cancer. Case report - Key learning points


Author(s):  
Lara Du ◽  
Jeffrey C. Lagarias

Let [Formula: see text] the product of the elements of the [Formula: see text]th row of Pascal’s triangle. This paper studies the partial factorizations of [Formula: see text] given by the product [Formula: see text] of all prime factors [Formula: see text] of [Formula: see text] having [Formula: see text], counted with multiplicity. It shows [Formula: see text] as [Formula: see text] for a limit function [Formula: see text] defined for [Formula: see text]. The main results are deduced from study of functions [Formula: see text] that encode statistics of the base [Formula: see text] radix expansions of the integer [Formula: see text] (and smaller integers), where the base [Formula: see text] ranges over primes [Formula: see text]. Asymptotics of [Formula: see text] and [Formula: see text] are derived using the prime number theorem with remainder term or conditionally on the Riemann hypothesis.


Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1548
Author(s):  
Xianyong Huang ◽  
Shanhe Wu ◽  
Bicheng Yang

In this paper, by virtue of the symmetry principle, we construct proper weight coefficients and use them to establish a more accurate half-discrete Hilbert-type inequality involving one upper limit function and one partial sum. Then, we prove the new inequality with the help of the Euler–Maclaurin summation formula and Abel’s partial summation formula. Finally, we illustrate how the obtained results can generate some new half-discrete Hilbert-type inequalities.


Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1320
Author(s):  
Pedro Ortiz ◽  
Juan Carlos Trillo

In this paper, we introduce and analyze the behavior of a nonlinear subdivision operator called PPH, which comes from its associated PPH nonlinear reconstruction operator on nonuniform grids. The acronym PPH stands for Piecewise Polynomial Harmonic, since the reconstruction is built by using piecewise polynomials defined by means of an adaption based on the use of the weighted Harmonic mean. The novelty of this work lies in the generalization of the already existing PPH subdivision scheme to the nonuniform case. We define the corresponding subdivision scheme and study some important issues related to subdivision schemes such as convergence, smoothness of the limit function, and preservation of convexity. In order to obtain general results, we consider σ quasi-uniform grids. We also perform some numerical experiments to reinforce the theoretical results.


Author(s):  
. Kado

In the study of calculus, the concept of limit of function occupies a central role as it is important instruments used in the study of the theory of rate of change, continuity, integral calculus, and differential calculus. Despite its significance, the secondary students hold the inadequate understanding of the limit concepts, more over their concept image of the limit function deviated from the concept definition resulting in the misconception. This study aims to identify the misconception in the limit of function and possible causes of misconceptions. This study was done in two phases, a concept test based on limit of function was administered to all 25 students of Samtse Higher Secondary School. Subsequently, based on the errors and misconception demonstrated by students from the concept test, five students were purposively selected and interviewed to corroborate the finding from concept test to confirm the existence of misconception and its causes. Data from the transcripts, capturing essential and relevant bits of student's responses to each question, was collected.  The data were analyzed and result of the study can be described as follows; it was found that learners only think of the manipulative aspect when solving problems on limits and not of the limit concept, confusion over the concept of the limit and value of function, and ambiguity regarding the formal definition of the limit of function. The possible cause of the misconceptions can be attributed to instrumental learning and lack of the sound knowledge in algebra which is cornerstone to understand the limit concept.


2021 ◽  
Vol 5 (1) ◽  
pp. 1-11
Author(s):  
Aris Budiyanto ◽  
Pinta Deniyanti Sampoerno ◽  
Makmuri Makmuri

High-level mathematical thinking skills are the main goal in learning mathematics. So far, the practice of learning in schools has not facilitated students to develop these abilities. PMRI (the Indonesian version of realistic mathematics education) is a learning approach that is oriented towards the students` ability to understand the realistic situations that can be imagined and integrated with the ability to understand concepts, use progressive mathematical models, interactivity and take advantage of student contributions are expected to improve students' abilities in learning mathematics. This study aims to determine the implementation of PMRI approach in improving high-order thinking skills of mathematics in class XI SMA Mutiara Bangsa 2 Tangerang on limit function through e-learning. This research was developed based on a research design that includes three, those are preliminary design, experiment, and retrospective analysis. This study was designed with five meetings using six research subjects. The subjects of this study consisted of six students who were selected based on their evaluation test and their activeness during learning process. The research instruments analyzed were formative and summative test results based on higher-order thinking skills, findings note, student questionnaires, and the hypothetical learning trajectory (HLT). The results obtained from the retrospective analysis show that the implementation of PMRI in learning and instruction of limit function is able to develop higher-order thinking skills of class XI students of SMA Mutiara Bangsa 2 Tangerang.


2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Juhong Kuang ◽  
Weiyi Chen ◽  
Zhiming Guo

<p style='text-indent:20px;'>In this paper, we develop a new method to study Rabinowitz's conjecture on the existence of periodic solutions with prescribed minimal period for second order even Hamiltonian system without any convexity assumptions. Specifically, we first study the associated homogenous Dirichlet boundary value problems for the discretization of the Hamiltonian system with given step length and obtain a sequence of nonnegative solutions corresponding to different step lengths by using discrete variational methods. Then, using the sequence of nonnegative solutions, we construct a sequence of continuous functions which can be shown to be precompact. Finally, by utilizing the limit function of convergent subsequence and the symmetry of the potential, we will obtain the desired periodic solution. In particular, we prove Rabinowitz's conjecture in the case when the potential satisfies a certain symmetric assumption. Moreover, our main result greatly improves the related results in the literature in the case where <inline-formula><tex-math id="M1">\begin{document}$ N = 1 $\end{document}</tex-math></inline-formula>.</p>


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