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The purpose of this paper is to introduce the idea of how to construct transfinite numbers and study transfinite arithmetic. The research method used is a literature review, which involves collecting various sources such as scientific papers and books related to Cantorian set theory, infinity, ordinal or transfinite arithmetic, as well as the connection between infinity and theology. The study also involves constructing the objects of study, namely ordinal numbers such as finite ordinals and transfinite ordinals, and examining their arithmetic properties. The results of this research include the methods of constructing both finite and transfinite ordinal numbers using two generation principles. Both finite and transfinite ordinal numbers are defined as well-orderings that are also transitive sets. Arithmetic of finite ordinal numbers is well-known, but the arithmetic of transfinite ordinal numbers will be introduced in this paper, including addition, multiplication, and exponentiation.

Bonds are known as one of low-risk investments and worth to be considered as a part of an investor's portfolio, however there are still underlying risks that could affect its price. This paper focuses on the effect of early redemption risk and default risk to a bond’s value. Using binomial interest rate tree method and its adjusted for default risk version, this paper wants to analyse how these risks affect Indonesian bonds’ values through simulations, while showing how these bonds can be used to construct the binomial interest rate trees. In the default risk simulation, more assumptions are made because of data limitations, which causes the first period recovery fraction to soar higher than the other periods. The analysis shows that, compared to present value of standard bonds, early redemption risk tends to cause the bond's present value to drop, while on the contrary, default risk tends to cause the bond's present value to rise. The cause of higher present value of bonds with default risk is explained by the high first period recovery fraction.

Graphs that have the properties of odd harmonious labeling are odd harmonious graphs. The research objective of this paper is to obtain odd harmonious labeling on layered graph C(x,y) and layered graph D(x,y). The research used in this paper is a qualitative method. The research flow consists of data collection, processing, and analysis. The data collection stage consists of constructing the definition of the new class graph, the data processing stage consists of constructing the vertex labeling and edge labeling, and the data analysis stage consists of constructing the theorem and proving it. The research results show that the layered graph C(x,y) and layered graph D(x,y) fulfill odd harmonious labeling. Such that the layered graph C(x,y) and layered graph D(x,y) are odd harmonious graphs. The benefit of this research is to add new properties of odd harmonious graphs. In addition, it does not rule out the possibility that this research can be developed again both in theory and application.

ABSTRAK Pendidikan 4.0 secara keseluruhan akan berkontribusi dalam membangun generasi saat ini, yang disebut “Z Generation”. Penelitian ini bertujuan untuk mengetahui pengaruh media Instagram #physics in my live pada model blended learning terhadap keterampilan berpikir Z Generation yang meliputi keterampilan berpikir kritis, berpikir kreatif & pemecahan masalah pada siswa SMA. Masalah yang diteliti adalah (1) apakah ada peningkatan keterampilan berpikir kritis, berpikir kreatif & pemecahan masalah Z Generation siswa SMA Kediri Lombok Barat; (2) apakah ada pengaruh penggunaan media instagram #physics in my live model blended learning terhadap keterampilan berpikir kritis, berpikir kreatif & pemecahan masalah Z Generation siswa SMA Kediri Lombok Barat; (3) apakah ada hubungan antara aspek keterampilan berpikir kritis, berpikir kreatif Z Generation siswa SMAN 1 Kediri Lombok Barat. Jenis penelitian yang akan digunakan adalah penelitian kuantitatif dengan metode kuasi eksperimen dengan desain one group pre-test and post-test. Perlakuan dilaksanakan pada satu kelompok ekperimen tanpa kelompok pembanding. Teknik Analisa data kuantitatif menggunakan uji Gain & uji regresi linear ganda dengan satu variabel terikat & tiga variabel bebas. Instrumen penelitian ini adalah Tes tulis, Angket & Lembar Observasi. Tes paper pencil untuk mengetahui keterampilan berpikir kritis, berpikir kreatif & pemecahan masalah Z Generation siswa SMA. Angket penelitian untuk mengetahui respon siswa terhadap media Instagram #physics in my live model blended learning, sedang lembar observasi untuk mengetahui pencapaian pembelajaran blended learning. Berdasarkan hasil penelitian dapat disimpulkan bahwa: (1) media instagram pada pembelajaran blended learning memberikan pengaruh terhadap peningkatan keterampilan berpikir kritis, berpikir kreatif dan keterampilan pemecahan masalah siswa SMA, (2) pembelajaran berbasis proyek mampu meningkatkan keterampilan pemecahan masalah siswa SMA, (3) setiap aspek keterampilan berpikir kritis mempunyai hubungan dengan aspek keterampilan berpikir kreatif. Kata kunci: instagram; keterampilan berpikir; Z generation; blended learning. ABSTRACT The purpose of this study is to determine the effect of Instagram #physics in my live blended learning model on critical thinking skills, creative thinking & problem-solving Z Generation of Kediri High School students, West Lombok. The problems to be researched are (1) whether there is an increase in critical thinking skills, creative thinking & problem solving for Z Generation students at SMA Kediri, West Lombok; (2) is there any effect of using Instagram media #physics in my live blended learning model on critical thinking skills, creative thinking & problem-solving Z Generation students of SMA Kediri, West Lombok; (3) whether there is a relationship between aspects of critical thinking skills, creative thinking & problem-solving Z Generation students of SMA Kediri, West Lombok. The type of research that will be used is mixed research that emphasizes the collection of quantitative and qualitative data. The quantitative research method used is a quasi-experimental design with one group pre-test and post-test design. The treatment was carried out in one experimental group without a comparison group. Quantitative data analysis technique using Gain test & multiple linear regression test with one dependent variable & three independent variables. The instruments of this research are Tests, Questionnaires & Observation Sheets. Paper pencil test to determine critical thinking skills, creative thinking & problem-solving Z Generation high school students. The research questionnaire was to determine student responses to the Instagram #physics in my live blended learning model, while the observation sheet was to determine the achievement of blended learning. Based on the results of the study, it can be concluded that: (1) Instagram media in blended learning has an effect on increasing critical thinking skills and creative thinking skills of high school students, (2) project-based learning is able to improve problem solving skills of high school students, (3) every aspect of skills critical thinking has a relationship with aspects of creative thinking skills. Keywords: instagram; thinking skills; Z generation; blended learning.

In this paper, we consider the boundedness of the operator families in whole space for Navier-Lame model problem in bounded domain of N dimensional Euclidean space (N≥2). To find the boundedness of the operator families, first of all we construct model problem in the form of the resolvent problem by using Laplace transform. Then, using Fourier transform, we get the solution formula of the model problem. In this paper, we use the qualitative methods to construct solution formula of velocity (u). This step is fundamental stage to find the well-posedness of the model problem. As we known that fluid motion can be described in partial differential equation (PDE). Essential point in PDE are finding existence and uniqueness of the model problem. One methods of investigating the well-posedness is R-boundedness of the solution operator families of the model problem. We can find the R-boundedness of the solution operator families not only in whole-space, half-space, bent-half space and in general domain. In this paper we investigate the R-boundedness of the solution operator families only in whole space. By using this R-boundedness, we can find that the multipliers which form of the operator families are bounded with some positive constant.

The research in graph theory has been widened by combining it with ring. In this paper, we introduce the definition of a non-braid graph of a ring. The non-braid graph of a ring R, denoted by YR, is a simple graph with a vertex set R\B(R), where B(R) is the set of x in R such that xyx=yxy for all y in R. Two distinct vertices x and y are adjacent if and only if xyx not equal to yxy. The method that we use to observe the non-braid graphs of Zn is by seeing the adjacency of the vertices and its braider. The main objective of this paper is to prove the completeness and connectedness of the non-braid graph of ring Zn. We prove that if n is a prime number, the non-braid graph of Zn is a complete graph. For all n greater than equal to 3, the non-braid graph of Zn is a connected graph.

Some conjectures related to the radical theory of rings are still open. Hence, the research on the radical theory of rings is still being investigated by some prominent authors. On the other hand, some results on the radical theory of rings can be implemented in another branch or structure. In radical theory, it is interesting to bring some radical classes into graded versions. In this chance, we implement a qualitative method to conduct the research to bring the Brown-McCoy radical class to the restricted graded Brown-McCoy radical class as research objective. We start from some known facts on the Brown-McCoy radical class and furthermore, let G be a group, we explain the Brown-McCoy radical restricted with respect to the group G. The result of this paper, we describe the Brown-McCoy radical in restricted graded version and it is denoted by G^G. Furthermore, we also give the fact by explaining G^G (A)=(〖G(A))〗_G, for any ring A, as the final outcome of this paper.

ABSTRAK Instrumen isomorfik merupakan butir soal yang memiliki representasi berbeda namun memiliki penyelesaian dan konsep-konsep fisika yang sama. Penelitian ini bertujuan mengkaji metode pengembangan instrumen, jenis instrumen isomorfik, media aplikasi instrumen, kemampuan pemahaman konsep siswa, dan materi fisika. Penelitian ini merupakan penelitian studi pustaka menggunakan metode Bibliometrik. Metode pengumpulan data sekunder hasil penelitian, artikel dari berbagai jurnal, dan sumber relevan lainnya pada database Google Scholar dan Scopus dalam kurun waktu 2015-2020, menggunakan aplikasi Publish or Perish (PoP). Langkah penelitian diawali dengan pengumpulan artikel dengan menyeleksi artikel jurnal menggunakan aplikasi PoP, melengkapi atribut artikel melalui software Mendeley, memvisualisasikan pemetaan data menggunakan software VOSviewer berdasarkan judul dan abstrak, dan mendeskripsikan topik kajian penelitian. Hasil kajian dapat disimpulkan bahwa pengembangan instrumen isomorfik digunakan sebagai asesmen sumatif maupun formatif pada pembelajaran fisika, terdiri dari dua jenis isomorfik yaitu isomorphic multiple choice dan isomorphic problem berbasis paper test, dengan media aplikasi instrumen dalam format online (web dan aplikasi), digunakan untuk menganalisis pemahaman konsep, konsistensi, miskonsepsi, mental model, pemecahan masalah fisika, dan hasil belajar siswa. Materi fisika yang sering diujikan menggunakan instrumen isomorfik adalah hukum Newton Kata kunci: instrumen isomorfik; fisika; kemampuan pemahaman konsep ABSTRACT The isomorphic instrument was several questions with different representations but have the same physics concept completion. The studied was purpose to examined development methods instrument, isomorphic types, instrument application media, ability understanding of the concept, and physics theory. This research was a literature study that used bibliometric methods. Methods of collected secondary data from research results, articles journals, and other relevant sources on the Google Scholar and Scopus databases in the 2015-2020 period used Publish or Perish (PoP). The research began with collected and selected journal articles used PoP, completed the article attributes via Mendeley, visualized data mapping used VOSviewer based on title and abstract, and described research study topic. The results of the research can be concluded that isomorphic instrument development is used as a summative and formative assessment on the physics learn consists of two types are isomorphic multiple-choice and isomorphic problem based of paper test with instrument application media in online format (web and application), used for analyzing the understanding of the concept, consistency, misconceptions, mental model, problem-solving, and study result of students by the frequently examined theory are Newton Laws. Keywords: isomorphic instrument; physic; ability understanding of the concept

In this paper, we studied a mathematical model of tuberculosis with vaccination for the treatment of tuberculosis. We considered an in-host tuberculosis model that described the interaction between Macrophages and Mycobacterium tuberculosis and investigated the effect of vaccination treatments on uninfected macrophages. Optimal control is applied to show the optimal vaccination and effective strategies to control the disease. The optimal control formula is obtained using the Hamiltonian function and Pontryagin's maximum principle. Finally, we perform numerical simulations to support the analytical results. The results suggest that control or vaccination is required if the maximal transmission of infection rate at which macrophages became infected is large. In this paper, we studied a mathematical model of tuberculosis with vaccination for the treatment of tuberculosis.We considered an in-host tuberculosis model that described the interaction between macrophages Macrophages and Mycobacterium tuberculosis and investigated the effect of vaccination treatments on uninfected macrophages. Optimal controlis applied to show the optimal vaccination and effective strategies to control the disease. The optimal control formula isobtained using the Hamiltonian function and Pontryagin's maximum principle. Finally, we perform numerical simulations to support the analytical results.The results suggest thatcontrol or vaccination is required if the maximal transmission of infection rate at which macrophages became infected is large.

In the present paper, we study some properties of the Heisenberg Lie algebra of dimension . The main purpose of this research is to construct a real Frobenius Lie algebra from the Heisenberg Lie algebra of dimension . To achieve this, we exhibit how to compute the derivation of the Heisenberg Lie algebra by following Oom’s result. In this research, we use a literature review method to some related papers corresponding to a derivation of a Lie algebra, Frobenius Lie algebras, and Plancherel measure. Determining a conjecture of a real Frobenius Lie algebra is obtained. As the main result, we prove that conjecture. Namely, for the given the Heisenberg Lie algebra, there exists a commutative subalgebra of dimension one such that its semi direct sum is a real Frobenius Lie algebra of dimension . Futhermore, in the notion of the Lie group of the Heisenberg Lie algebra which is called the Heisenberg Lie group, we compute the generalized character of its group and we determine the Plancherel measure of the unitary dual of the Heisenberg Lie group. As our contributions, we complete some examples of Frobenius Lie algebras obtained from a nilpotent Lie algebra and we also give alternative computations to find the Plancherel measure of the Heisenberg Lie group.

The Power dominating set is a minimum point of determination in a graph that can dominate the connected dots around it, with a minimum domination point. The smallest cardinality of a power dominating set is called a power domination number with the notation . The purpose of this study is to determine the Shackle operations graph value from several special graphs with a point as a link. The result operation graphs are: Shackle operation graph from Path graph , Shackle operation graph from Sikel graph , Shackle operation graph from Star graph . The method used in this paper is axiomatic deductive method in solving problems. Understanding the axiomatic method itself is a method of deductive proof principles that applies in mathematical logic by using theorems that already exist in solving a problem. In this paper begins by determining the paper object that is the Shackle point operations result graph. Next, determine the cardinality of these graphs. After that, determine the point that has the maximum degree on the graph as the dominator point of power domination. Then, check whether the nearest neighbor has two or more degrees and analyze its optimization by using a ceiling function comparison between zero forching with the greatest degree of graph. Thus it can be determined ϒp minimal and dominated. The results of the power domination number study on Shackle operation graph result with points as connectors are , for and ; , for and ; , for and .

An important aspect in the provision of a two-dimensional warranty is the expected number of failures of a component during the two-dimensional warranty period. The purpose of this paper is to present a new method to obtain the expected number of failures of a nonrepairable compo­nent from the two-dimensional renewal functions as the so­lution of two-dimensional renewal integral equations through the Mean Value Theorem for Integrals (MeVTI) method. The two-dimensional renewal integral equation involves Lu-Bhattacharyya’s bivariate Weibull model as a two-dimensional failure model. It turns out that the estimation of the expected number of failures using the MeVTI method is close to that of the other method, Riemann-Stieljies method. The bivariate data behaviour of the failures of an automobile component is also studied in this paper.

Challenges in the 21st century are increasingly complex, technology is developing rapidly and competition is getting tougher. Therefore we need quality human resources that can keep up with and anticipate the times. The use of technology involves computational thinking (CT) skills which are closely related to the problem-solving process. The stages in computational thinking are part of mathematical thinking, meaning that learning mathematics can support students' CT skills. Through the development of digital pedagogical models in CT integrated mathematics learning, it can improve problem-solving skills. This research uses design based implementation research with 4 phases including; preliminary research, prototyping, results, and design principle. The participants were 28 grade 8 junior high school students who took part in two rounds of experiment in direct CT activities and digital CT activities. In this paper, we present an iterative mathematical problem-solving process in the digital pedagogy model. The computational task, environment, tool and practices were iteratively improved over two rounds to incorporate CT effectively in mathematics. The results from CT environment demonstrated that direct CT activities are more effective than digital CT activities in mathematical problem-solving. Based on empirical research, we summarize the characteristic of the digital pedagogy model from computational tasks, computational environment and tools, and computational practices in mathematical problem solving.

Banyumasan batik is one of Indonesia's cultural heritages from Banyumas Regency, which has unique and interesting patterns and motifs. This research paper describes the mathematical modelling of geometric transformations in Banyumasan batik. Geometric transformation is a mathematical technique for changing the shape or position of an object on a plane. This is qualitative research with an ethnographic approach. Data is collected through observation, interviews, and documentations. The data obtained were analysed using content analysis to identify various Banyumasan batik motifs that can be transformed into geometric forms. This study describes the basic geometric transformations, namely translations in Sekar Tirta batik motif, reflections in Bawor Kembar batik motif, dilations in Kawung Jenggot batik motif, and rotations in Manggar Bawor batik motif which applied to Banyumasan batik motifs. The results of this study include geometric transformations of Banyumasan batik motifs which are expected to positively contribute to the development of Indonesian art and culture, especially in the field of batik, enrich insight and knowledge about the application of mathematics in everyday life, and can be used for mathematical modelling and mathematics education based on local culture.

Hyperstructures find numerous applications across various disciplines. One notable application is in chemistry, particularly in the context of chemical reactions. In 2014, Davvaz introduced the concept of bi-hyperstructures, but their application specifically in chemical reactions, has yet to be thoroughly explored in previous studies. Thus, the primary aim of this paper is to examine and analyze the different types of bi-hyperstructures present within chemical hyperstructures. The scope of this study focuses on two types of chemical hyperstructures: redox reactions and reactions in electrochemical cells. Within these chemical hyperstructures, we investigate the possibility of bi-hyperstructures among bi-semihypergroups, bi-hypergroups, bi-H_v-semigroups, and bi-H_v-groups. Next, some properties of bi-hyperstructures related to hyperstructures are also investigate.

In this paper, steady infiltration problems into a homogeneous soil from a single triangular and trapezoidal irrigation channel are considered. The governing equation is Richard's equation that represents the movement of water in unsaturated soil. It is a non-linear equation and can be solved by linearizing to become a modified Helmholtz equation. Dual Reciprocity Boundary Element Methods (DRBEM) are used in this study to numerically solve the modified Helmholtz equation. Therefore, by using a provided solution, the numerical Matric Flux Potential (MFP) is calculated. This method was applied to the homogeneous soil problem of stationer infiltration from triangular and trapezoidal single irrigation. Both numerical solutions were compared. The result show that the MFP value from the triangular irrigation is higher than the trapezoidal irrigation. This indicates that content water from the triangular irrigation channel is higher than the trapezoidal irrigation channel.

One and two-compartment pharmacokinetic models with drug-drug interactions are proposed. Two drugs are given orally simultaneously, so that their interaction affects the drug absorption process and subsequently the elimination process. The aim of this paper is to estimate the elimination and absorption rate constants by evaluating the data set of time and drug concentration. This data set was divided into two time phases: large-time elimination phase to estimate the elimination rate constant, and small-time absorption phase to estimate the absorption rate constant. Since the models are nonlinear, the Taylor expansion is employed to so that the Wagner-Nelson and the Loo-Riegelman methods can be used for estimation. Finally, simulations were performed using the generated arbitrary data set of time and concentration, instead of an actual data set, to derive the solution of drug concentration concerning time numerically. In these simulations we compared the original parameter values with their estimates for the one and two-compartment models, and we concluded that the two-compartment model produced better estimates than the one-compartment model. Qualitatively, the two-compartment model gives smaller drug concentration curve deviations between the original and the estimated curve compared with the one-compartment model.

Let R_ε=R∪{-∞}, with R being a set of all real numbers. The algebraic structure (R_ε,⊕,⊗) is called max-plus algebra. The task of finding the eigenvalue and eigenvector is called the eigenproblem. There are several methods developed to solve the eigenproblem of A∈R_ε^(n×n), one of them is by using the characteristic max-polynomial. There is another algebraic structure that is isomorphic with max-plus algebra, namely min-plus algebra. Min-plus algebra is a set of R_(ε^' )=R∪{+∞} that uses minimum (⊕^' ) and addition (⊗) operations. The eigenproblem in min-plus algebra is to determine λ∈R_(ε^' ) and v∈R_(ε^')^n such that A⊗v=λ⊗v. In this paper, we provide a method for determining the characteristic min-polynomial and solving the eigenproblem by using the characteristic min-polynomial. We show that the characteristic min-polynomial of A∈R_(ε^')^(n×n) is the permanent of I⊗x⊕^' A, the smallest corner of χ_A (x) is the principal eigenvalue (λ(A)), and the columns of A_λ^+ with zero diagonal elements are eigenvectors corresponding to the principal eigenvalue.

Antara, the official news agency of the state, reported a record-breaking population of 303 Bali starlings in the West Bali National Park (WBNP) in June 2020, attributing this achievement to the park's captive reproduction initiative. This paper presents a study on the dynamic equilibrium of Bali Starlings and proposes a mathematical model for analyzing this dynamic. The research also examines parameters ensuring the stability of the captive breeding model for Bali starlings in WBNP in a sustainable manner. The Bali starlings are categorized into two groups: those in the wild and those in captive breeding, with hatched eggs in captivity included in the latter. The dynamic model is analyzed for system stability around the endemic critical point using the Routh-Hurwitz stability criteria. As an illustrative example, a simulation is conducted to assess the model's suitability under real field conditions. The model analysis reveals that the existence of an endemic critical point can be maintained if the percentage of stolen Bali starlings or eggs reintroduced to the wild is less than the difference between the percentage of Bali starlings laying eggs and the population growth rate in WBNP. Furthermore, the stability of the endemic critical point is confirmed as long as the percentage of Bali starlings laying eggs exceeds the population growth rate. This dynamic model offers a valuable tool for evaluating the sustainability of Bali starling breeding programs and optimizing the benefits associated with their conservation efforts.

To help insurers determine insurance rates incorporating maternity factors, it is crucial to understand the maternity recovery rate, which was a metric used by insurance companies to understand how much of the expenses associated with maternity care and related medical services are covered by their policies. This paper employed Extreme Gradient Boosting (XGBoost), a powerful method for handling complex data relationships and preventing overfitting, on North American Group Long-Term Disability dataset obtained from the Society of Actuaries, which listed maternity as one of its categories, to predict the maternity recovery rate. In comparison, other machine learning methods such as Gradient Boosting Machine (GBM) and Bayesian Additive Regression Tree (BART) were used, with Root Mean Squared Error (RMSE) values calculated the difference between predicted and observed maternity recovery rates. Four datasets, 3 imbalanced and 1 fairly-balanced, were created out of the original dataset to test each method’s predictive prowess. The study revealed that XGBoost performed exceptionally well on the imbalanced datasets, while BART showed slight superiority in fairly-balanced data. Furthermore, the model identified the duration, exposures, and age of participants in both predicting maternity recovery rates and the underwriting process.

This study investigates the effect of learning opportunities and beliefs on the efficacy of teaching mathematics on knowledge of mathematics for teaching. Using a structured questionnaire along with paper and pencil tests adapted from the literature review, data were collected from 187 prospective primary school teachers at the Indonesian Teacher Education Institute. Data were analyzed using Smart PLS 3.0. The results of the structural equation model show that both opportunities to learn-Practicum (β = 0.395, p < 0.001) and opportunities to learn -Program (β = 0.324, p < 0.001) are positively related to mathematics. The second regression analysis is to examine the influence of teaching efficacy beliefs on mathematics. The results showed that beliefs about expectations of learning outcomes in mathematics (β = 0.322, p < 0.001) were positively related to mathematics, while personal mathematics teaching efficacy beliefs (β = 0.017, p > 0.1) were not related to mathematics. Overall, the confidence factor and opportunities to learn explain the total 54% variance in mathematics. The implications of these findings for the successful implementation of teacher education programs in Indonesia are further elaborated.

Assumed that (S,+,.) is a semiring. Semiring is a algebra structure as a generalization of a ring. A set I⊆S is called an ideal over semiring S if for any α,β∈I, we have α-β∈I and sα=αs∈I for every s in semiring S. Based on this definition, there is a special condition namely prime ideal P, when for any αβ∈P, then we could prove that α or β are elements of ideal P. Furthermore, an ideal I of S is irreducible if Ia is an intersection ideal from any ideal A and B on S, then I=A or I=B. We also know the strongly notion of the irreducible concept. The ideal I of S is a strongly irreducible ideal when I is a subset of the intersection of A and B (ideal of S), then I is a subset of A, or I is a subset of B. In this paper, we discussed the characteristics of the semiring of the non-negative integer set. We showed that pZ^+ is an ideal of semiring of the non-negative integer Z^+ over addition and multiplication. We find a characteristic that 〖pZ〗^+ is a prime ideal and also a strongly irreducible ideal of the semiring Z^+ with p is a prime number.

Quantifying dependence among variables is the core of all modelling efforts in financial models. In the recent years, copula was introduced to model the dependence structure among financial assets return, and its application developed fast. A large number of studies on copula have been performed, but the study of multivariate extremes related with copulas was quite behind in comparison with the research on copulas. The COVID-19 pandemic is an extreme event that has caused the collapse of various economic activities which resulted in the decline of stock prices. The modelling of extreme events is therefore important to mitigate huge financial losses. Extreme-value copula can be suitable to quantify dependencies among assets under an extreme event. In this paper, we study the modelling of extreme value dependence using extreme value copulas on finance data. This model was applied in the portfolio of the IDX Composite Index (IHSG), Straits Times Index (STI) and Kuala Lumpur Stock Exchange (KLSE). Each individual asset return is modelled by the ARMA-GARCH and the joint distribution is modelled using extreme value copulas. This empirical study showed that Gumbel copula is the most appropriate extreme value copulas for the three indices. The results of this study are expected to be used as a basis for investors in the formation of a portfolio consisting of 2 financial assets and a portfolio consisting of 3 financial assets.

Graph theory is a branch of algebra that is growing rapidly both in concept and application studies. This graph application can be used in chemistry, transportation, cryptographic problems, coding theory, design communication network, etc. There is currently a bridge between graphs and algebra, especially an algebraic structures, namely theory of graph algebra. One of researchs on graph algebra is a graph that formed by prime ring elements or called prime graph over ring R. The prime graph over commutative ring R (PG(R))) is a graph construction with set of vertices V(PG(R))=R and two vertices x and y are adjacent if satisfy xRy={0}, for x≠y. Girth is the shortest cycle length contains in PG(R) or can be written gr(PG(R)). Order in PG(R) denoted by |V(PG(R))| and size in PG(R) denoted by |E(PG(R))|. In this paper, we discussed prime graph over cartesian product over rings Z_m×Z_n and its complement. We focused only for m=p_1, n=p_2 and m=p_1, n=〖p_2〗^2, where p_1 and p_2 are prime numbers. Then, we obtained some properties related to order and size, degree, and girth. We also observe some examples. Moreover, we found that a correction in the statement of (Pawar & Joshi, 2019) about the complement graph over prime graph over a ring and gave a counter example for that.

Assuming that G is a finite group and H is a subgroup of G, the graph known as the relative coprime graph of G with respect to H (denoted as Γ_(G,H)) has vertices corresponding to elements of G. Two distinct vertices x and y are adjacent by an edge if and only if (|x|,|y|)=1 and x or y belongs to H. This paper will focus on finding the general formula for some topological indices of the relative coprime graph of a dihedral group. The study of topological indices in graph theory offers valuable insights into the structural properties of graphs. This study is conducted by reviewing many past literatures and then from there we infer a new result. The obtained outcomes will include measurements of distance, degree of vertex, and various topological indices such as the first Zagreb index, second Zagreb index, Wiener index, and Harary index that are associated with distance and degree of vertex.

This study proposes a SVEIL model of tuberculosis disease spread with imperfect vaccination. Susceptible individuals can receive imperfect vaccination, but over the time the vaccine efficacy will decrease. Vaccinated individuals are in vulnerable class since they still have probability to get reinfected. The proposed model includes treatment for both high-risk latent and active TB patients. In fact, after getting appropriate treatment (get recovered) the individuals still have bacteria in their body and it is classified to low-risk laten class. Dynamical behaviour of the model is analyzed to understand the local stability equilibrium. The Routh-Hurwitz criterion is used to analyze the local stability equilibrium in disease free equilibrium (DFE) point and Center Manifold theorem is used to prove the local stability of the endemic equilibrium (EE) point. The local stability equilibrium state totally depends on the effective reproduction number R_v. If R_v1 the EE point is locally asymptotically stable . The parameter used in this paper is based on the previous researches related to TB and the initial subpopulations are assumed. Numerical simulations show that the disease transmission rate affect the effective reproduction number, therefore it influences the stability of equilibrium points..

Foot and Mouth Disease (FMD) is an acute infectious disease that attacks livestock, thus threatening the availability of food and the husbandry industry. This paper discusses the formulation of a mathematical model for the spread of FMD in livestock with a saturated incidence rate. The research method used is quantitative mathematical modeling with simulation, with stages including problem identification, determining assumptions, model formulation, analysis and model simulation. The discussion results obtained two equilibrium points, namely the non-endemic equilibrium point and the endemic equilibrium point, and then analyzed for stability. Numerical simulation is presented using Runge-Kutta approximation with MATLAB. Furthermore, after a sensitivity analysis, the parameters that greatly influenced the spread of FMD were direct or indirect contact (which led to the entry of the FMD virus) and the supporting capacity of livestock. Then the most influential parameter in reducing the spread of FMD is the application of culling on exposed animals and infected animals. The FMD modeling is a form of mathematical application to simulate the spread of disease on livestock.

Complex yet hierarchical mathematical material in fact does not mean that the process of schema acquisition and accommodation in long-term memory is easy. For an instance, year seven students learn to solve problems related to the compound area of quadrilaterals and triangles. This problem may be categorized as a well-structured problem, therefore, it might be procedurally obvious to reach the solution, for those who have possessed sufficient prior knowledge of the aspects of the shapes. This study aims to discuss instructional strategy for novices when learning to solve problems like these. Based on a cognitive load theory, the strategy of worked example could be applied effectively for novices. The design of the worked example must be presented in such a way it could minimize extraneous cognitive load. The composite shapes of quadrilateral and triangles might be challenging to be described in an integrated format to avoid the split attention effect. This paper shows how this can be done by using techniques, such as (1) number sequences of the solution steps; (2) different colors for prompting attention resource; (3) less wording, more procedure application; and (4) consistency of layout within overall material. In order to apply these, the instructional designers should follow ADD steps, (1) Analyze the learning content, arrange them accordingly; (2) Design one worked example, evaluate its accuracy and niche; and (3) Develop into several worked example pairs. Two worked example with paired problems have been successfully developed and declare valid. Eventually, it might be suggested that comprehensive consideration when designing worked example that contain pictures should reflects how students use their cognitive resource during learning.

In this paper, an asymptotic distribution of the estimator for the variance function of a compound periodic Poisson process with power function trend is discussed. The periodic component of this intensity function is not assumed to have a certain parametric form, except it is a periodic function with known period. The slope of power function trend is assumed to be positive, but its value is unknown. The objectives of this research are to modify the existing variance function estimator and to determine its asymptotic distribution. This research begins by modifying the formulation of the variance function estimator. After the variance function is obtained, the research is continued by determining the asymptotic distribution of the variance function estimator of the compound periodic Poisson process with a power function trend. The first result is modification of existing estimator so that its asymptotic distribution can be determined. The main result is asymptotic normality of the estimator of variance function of a compound periodic Poisson process with power function trend.

Graph Coloring Problem (GCP) is the assignment of colors to certain elements in a graph based on certain constraints. GCP is used by assigning a color label to each node with neighboring nodes assigned a different color and the minimum number of colors used. Based on this, GCP can be drawn into an optimization problem that is to minimize the colors used. Optimization problems in graph coloring can occur due to uncertainty in the use of colors to be used, so it can be assumed that there is an uncertainty in the number of colored vertices. One of the mathematical optimization methods in the presence of uncertainty is Robust Optimization (RO). RO is a modeling methodology combined with computational tools to process optimization problems with uncertain data and only some data for which certainty is known. This paper will review research on Robust GCP with model validation to be applied to electrical circuit problems using a systematic review of the literature. A systematic literature review was carried out using the Preferred Reporting Items for Systematic reviews and Meta Analysis (PRISMA) method. The keywords used in this study were used to search for articles related to this research using a database. Based on the results of the search for articles obtained from PRISMA and Bibliometric R Software, it was found that there was a relationship between the keywords Robust Optimization and Graph Coloring, this means that at least there is at least one researcher who has studied the problem. However, the Electricity keyword has no relation to the other two keywords, so that a gap is obtained and it is possible if the research has not been studied and discussed by other researchers. Based on the results of this study, it is hoped that it can be used as a consideration and a better solution to solve optimization problems.

The World Health Organization (WHO) asserted the recently discovered severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), also known as COVID-19, a pandemic on March 11, 2020. Since the genesis and growth mechanisms of this virus are unclear and impossible to detect, there are still many uncertainties concerning it and no vaccination or effective treatment. The main goal is to halt its global spread. This paper employed a regression growth model with an extended Weibull function on the dynamics of COVID-19 to make predictions about its spread. Our findings demonstrate the viability of using this model to forecast the spread of the virus. Using a logistic growth regression model, the note tabulates the COVID-19-related final epidemic sizes for a few sites, including Ireland and Israel.

Technological advances during the COVID-19 pandemic in Indonesia gave rise to the PeduliLindungi application which is developed by the government to prevent the spread of COVID-19. The advantages and disadvantages of developing PeduliLindungi can be seen from the responses and opinions from users, one of which is through the Twitter. A person's opinion about PeduliLindungi based on the tweet can be classified into positive, negative, or neutral categories using a Machine Learning approach with the Support Vector Machine (SVM) algorithm. In this paper, multiobjective optimization modeling is used to maximize the performance metrics, which are the value of Accuracy, Precision, Recall, and F1-Score. The value of the performance metrics is considered to contain uncertainty factors. Therefore, the optimization problem is solved by using Robust Optimization to handle the uncertainty factor. The data uncertainty is assumed to be belongs to polyhedral uncertainty set thus the resulted robust is computationally tractable. Numerical experiment is presented to complete the discussion.

Multi-objective integer optimization model that contain uncertain parameter can be handled using the Adjustable Robust Counterpart (ARC) methodology with Polyhedral Uncertainty Set. The ARC method has two stages of completion, so completing the second stage can be assisted by the Benders Decomposition. This paper discusses the systematic review on this topic using the Preferred Reporting Items for Systematic Reviews and Meta-Analysis (PRISMA). PRISMA presents a database mining algorithm for previous articles and related topics sourced from Scopus, Science Direct, Dimensions, and Google Scholar. Four stages of the algorithm are used, namely Identification, Screening, Eligibility, and Included. In the Eligibility stage, 16 articles obtained and called Dataset 1, used for bibliometric mapping and evolutionary analysis. Moreover, in the Included stage, six final databases obtained and called Dataset 2, which was used to analyze research gaps and novelty. The analysis was carried out on two datasets, assisted by the output visualisation using RStudio software with the " bibliometrix" package, then we use the command 'biblioshiny()' to create a link to the “shiny web interface”. At the final stage of the article using six articles analysis, it is concluded that there is no research on the ARC multi-objective integer optimization model with Polyhedral Uncertainty Sets using the Benders Decomposition Method, which focuses on discussing the general model and its mathematical analysis. Moreover, this research topic is open and becomes the primary references for further research in connection.

Ethnomathematics studies the relationship between mathematics and culture. Indonesia has many traditional cultures. One of them is traditional cloth. The traditional cloth from East Nusa Tenggara (NTT) province is called tenun ikat. Since the motif of tenun ikat consists of symmetrical and repeated patterns, it can be generated using Frieze groups. The Frieze groups are the plane symmetry groups of patterns whose subgroups of translations are isomorphic to Z. There are seven groups in the Frieze groups, i.e., F_1, F_2, F_3, F_4, F_5, F_6, and F_7. Translation, reflection, rotation, and glide reflection are the transformation used in the Frieze groups. In this paper, Frieze groups are used to generate digital tenun ikat motifs from the basic pattern. Since one piece of original tenun ikat may consist of some basic patterns, the basic patterns are identified first, and then each of them is generated into the desired pattern, according to the suitable Frieze groups. Furthermore, a GUI Matlab program is developed to generate the Frieze groups. Three motifs of tenun ikat are presented to demonstrate the implementation of Frieze groups. With the Frieze group, users can generate other patterns from a basic pattern, so users can generate new motifs of tenun ikat without leaving the cultural characteristics of NTT province.

The unequal amount of questions that test students' lower and higher-order thinking skills is a concern in mathematics learning evaluation. The unequal amount of questions resulted in poor cognitive levels among students. The purpose of this study is to develop questions that may be used to assess the higher-order thinking skills of senior high school pupils and fulfill the requirements of being valid, practical, and effective. This study is a developmental research that employs the formative evaluation model. Validation sheets, student evaluation sheets, and examinations that measure students' higher-order thinking skills are among the data gathering procedures employed. According to the researchers' findings, the questions generated matched the valid, practical, and effective requirements. The valid criteria are based on the validators' assessments, and both validators agree that the questions created are good and possible to utilize with a few adjustments. Student answer papers to the established questions serve as the basis for practical criteria. According to the response form, 85 percent of pupils answered positively to the questions. Effective criteria are based on pupils' abilities to answer the questions that have been established. According to the test findings, 85 percent of students satisfied the minimal completeness requirements for both questions testing higher-order thinking skills, indicating that the questions were effective. The questions that have been developed by researchers can be used by teachers to evaluate students' higher-order thinking skills so that teachers do not only provide questions that measure students' lower-order thinking skills. In addition, the questions that have been developed can be used as a reference for developing questions that can measure students' higher-order thinking skills.

Wave equation is one of the second order of the linear hyperbolic equation. Telegraph equation as a special case of wave equation has interesting point to investigate in the numerical point of view. In this paper, we consider the numerical methods for one dimensional telegraph equation by using cubic B-spline collocation method. Collocation method is one method to solve the partial differential equation model problem. Cubic spline interpolation is an interpolation to a third order polynomial. This polynomial interpolate four point. B-Spline is one of spline function which related to smoothness of the partition. For every spline function with given order can be written as linear combination of those B-spline. As we known that the result of the numerical technique has difference with the exact result which we called as, so that we have an error. The numerical results are compared with the interpolating scaling function method which investigated by Lakestani and Saray in 2010. This numerical methods compared to exact solution by using RMSE (root mean square error), L2 norm error and L_∞ norm error . The error of the solution showed that with the certain function, the cubic collocation of numerical method can be used as an alternative methods to find the solution of the linear hyperbolic of the PDE. The advantages of this study, we can choose the best model of the numerical method for solving the hyperbolic type of PDE. This cubic B-spline collocation method is more efficiently if the error is relatively small and closes to zero. This accuration verified by test of example 1 and example 2 which applied to the model problem.

ABSTRAK Tujuan penulisan ini yaitu untuk mendeksripsikan efektivitas modul pembelajaran berbasis problem based learning dalam pembelajaran fisika untuk meningkatkan kemampuan pemecahan masalah siswa pada materi dinamika rotasi. Jenis penilitian yang digunakan yaitu peneliian eksperimen. Penelitian ini dilakukan terhadap siswa kelas XI MIPA SMAN Taraju dengan menggunakan desain penelitian One Group Pre test-Post test. Kemudian untuk mengukur kemampuan pemecahan masalah menggunakan desain One Group Pre test-Post test dan memperoleh nilai efek size sebesar 3.19 sehingga menunjukan bahwa terjadi peningkatan dengan taraf atau kriteria sangat besar serta dilakukan uji t-test dan mendapatkan hasil thitung > ttabel yaitu 17,77 > 2,86 Jadi, dapat disimpulkan bahwa penelitian ini dapat menghasilkan modul pembelajaran berbasis Problem Based Learning yang layak pada materi dinamika rotasi dan juga dapat membantu meningkatkan kemampuan pemecahan masalah pada siswa. Kata kunci: problem based learning; keterampilan pemecahan masalah; dinamika rotasi. ABSTRACT. The purpose of this paper is to describe the effectiveness of problem-based learning modules in physics learning to improve students' problem-solving abilities in the rotational dynamics material. The type of research used is experimental research. This research was conducted on class XI MIPA students of SMAN Taraju by using the research design One Group Pre-test-Post test. Then to measure the problem solving ability using the One Group Pre-test-Post test design and obtain an effect size value of 3.19 so that it shows that there is an increase with a very large level or criteria and a t-test is carried out and gets the results of tcount> ttable, namely 17.77> 2.86 So, it can be concluded that this study can produce a problem-based learning module that is feasible on the rotational dynamics material and can also help improve students' problem-solving abilities. Keywords: problem based learning; problem solving skills; rotational dynamics.

Under classical statistics Hotelling 〖 T〗^2 control chart is applied when the observations of quality characteristics are precise, exact, or crips data. However, in reality, under uncertain conditions, the observations are not necessarily precise, exact, or indeterminacy. As a consequence, the classical Hotelling〖 T〗^2control chart is not appropriate to monitor the process for this condition. To tackle this situation, we proposed new Hotelling 〖 T〗^2 monitoring scheme based on a fuzzy neutrosophic concept. Neutrosophic is the generalization of fuzzy. It is used to handle uncertainty using indeterminacy. The combination of statistics based on neutrosophic Hotelling 〖 T〗^2 and classical Hotelling 〖 T〗^2 control chart will be proposed to tackle indeterminacy observations. The proposed Hotelling 〖 T〗^2 statistics, its call neutrosophic Hotelling 〖 T〗^2 (T_N^2 ) control chart. This chart involves the indeterminacy of observations, its call neutrosophic data and will be expressed in the indeterminacy interval. T_N^2 control charts consist T_N^2 lower chart and T_N^2 upper chart. In this paper, the neutrosophic Hotelling T^2will be applied to individual observations of glass production and will be compared by using classical Hotelling T^2 control chart. Based on T_N^2 control charts of glass production, nine points fall outside of 〖UCL〗_N of lower control chart and 24 points outside from 〖UCL〗_N of upper control chart. Whereas using classical Hotelling T^2 control chart, just one point outside frim UCL. From the comparison, it concluded that the neutrosophic Hotelling T^2 control chart is more suitable for the indeterminacy of observations.

In this article, the notion of (α,β)-intuitionistic fuzzy ideals (briefly, (α,β)-IF ideals) of ternary semigroups is described using ”belong to” relation (ϵ) and “quasi-coincidence with” relation (q) connecting two objects, i.e., an intuitionistic fuzzy point (IFP, for short) and an intuitionistic fuzzy set (briefly, IFS). Throughout this paper, α∈{ϵ,q,ϵ∨q} and β∈{ϵ,q,ϵ∨q,ϵ∧q}. The main purposes of this research are to construct the definition of (α,β)-intuitionistic fuzzy ideals of ternary semigroups and to investigate the relations between some types of these ideals. To achieve these goals, we use literature review method to study previous researches regarding (α,β)-fuzzy ideals of ternary semigroups and (α,β)-IF ideals of semigroups. As a result, we find the conditions for an IFS and an ideal of a ternary semigroup to be classified as an (α,ϵ∨q)-IF ideal of ternary semigroup. Relations between some types of (α,β)-IF ideals of a ternary semigroup are also discussed here.

We consider Susceptible-Infected-Recovered (SIR) models of infectious disease spread without and with vital dynamics. We recall some existing analytical approximate iterative methods for solving these models. We observe that all these methods solve the models accurately only for points close to the initialisation. These methods produce inaccurate, and even, unrealistic solutions to the SIR models if the time domain is sufficiently large. In this paper, our research objective is to propose an analytical-numerical iterative method, which is able to solve the SIR models accurately on the whole domain. The research method used is quantitative mathematical modelling with simulation. By implementing this analytical-numerical iterative method into a finite number of small consecutive subintervals of the domain, our research results show that the proposed method produces accurate solutions to the SIR models on the whole domain.

The purpose of this research finds a link between adaptive reasoning, mathematical problem solving, and cognitive style. This research is a literature study. This paper showed the relationship between adaptive reasoning, mathematical problem solving, and cognitive styles. In particular, field-independent (FI) and field-dependent (FD) cognitive styles. Adaptive reasoning in mathematical problems is the legitimacy of problem strategies. Whereas the cognitive style provides opportunities for students to give reasonable arguments according to the tendency of the cognitive style. Mathematical problem-solving in the FI and the FD can vary according to students' cognitive styles. It was found that adaptive reasoning, mathematical problems, cognitive styles (FI, FD) have a strong stylistic relationship where adaptive reasoning will be able to give direction to students in solving math problems. Therefore, the basis for researching the adaptive reasoning profile of students in solving mathematical problems in terms of FI and FD cognitive styles is our understanding of the relationship between adaptive reasoning, mathematical problem solving, and cognitive style

ABSTRAK Penelitian ini dilakukan untuk mengetahui nilai muatan listrik, nilai kapasitor yang dirangkai secara seri dan paralel kemudian nilai energi yang tersimpan pada kapasitor berdasarkan jenis-jenis material dengan konstanta dielektrik yang telah ditentukan. Metode yang dilakukan adalah mengkaji masing-masing nilai tersebut berdasarkan konstanta dielektrik dengan perhitungan matematis berbantuan Microsoft excel untuk menghasilkan grafiknya. Adapun tegangan yang digunakan pada metode ini adalah 20, 40, 60 dan 100 volt dan material yang dianalisis antara lain kaca, udara, mika, kertas dan polystiren. Analisis dilakukan menggunakan analisis kuantitatif yang memanfaatkan data berupa angka yang dipresentasikan melalui grafik. Ditemukan hasil perhitungan matematis bahwa kaca memiliki jumlah muatan yang lebih besar dibandingkan dengan material yang lainnya. pada saat dirangkaiakan secara seri dan paralel diperoleh bahwa jumlah muatan kapasitor yang dirangkaiakan secara paralel jauh lebih besar dibandingkan seri, energi yang tersimpan pada kapasitor yang dirangkaikan secara paralel juga memiliki nilai yang besar dari pada kapasitor yang dirangkai secara seri. Kata kunci: kapasitor; konstata dielektrik; tegangan. ABSTRACT This research aims are to analyze the value of the electric charge, the value of the capacitor connected in series and paralel then the energy value stored in the capacitor based on the types of materials with determined a dielectric constant. The method used to examine each of these values based on the dielectric constant with mathematical calculations assisted by Microsoft Excel to produce the graph. The voltages used on this methode are 20, 40, 60, and 100. The analysis using quantitative analysis that utilizes data in the form of numbers presented through graphs. It has been studied based on the physical formulation of the value of an electric charge against the dielectric constant of glass, air, mica, paper and polystyrene. It was found that the results of mathematical calculations show that glass has a greater amount of charge compared to other materials. When assembled in series and paralel, it is found that the amount of charge on the capacitor connected in paralel is much greater than in series, the energy stored in the capacitor connected in paralel also has a greater value than the capacitor connected in series. Keywords: capasitor; dielectric constant; voltage.

In this paper, length biased Weibull distribution is considered for Bayesian analysis. The expressions for Bayes estimators of the parameter have been derived under squared error, precautionary, entropy, K-loss, and Al-Bayyati’s loss functions by using quasi and gamma priors.

ABSTRAK Penulisan makalah ini bertujuan untuk mengetahui pengaruh media pembelajaran online dalam pemahaman dan minat siswa dalam mempelajari konsep-konsep fisika dan untuk respon siswa mengenai penerapan media pembelajaran online, pemerintah memberlakukan kebijakan jarak sosial atau dikenal sebagai jarak fisik dan oleh karena itu sekolah memutuskan untuk belajar jarak jauh yang ditakuti selama pandemi menggunakan media pembelajaran online minat belajar siswa akan menurun.Dari hasil survai menggunakan kuesioner di google form terhadap siswa Sekolah Menengah Atas diperoleh 18,8% setuju menggunakan media pembelajaran online dan 52,75% tidak setuju menggnakan media pembelajran online. dari hasil data tersebut dikatakan "tidak kuat" ini menunjukkan respon siswa segatif terhadap menggunakan media pembelajran online. minat belajar siswa lebih meningkat bila pembelajaran online dilakukan di dalam kelas. selama pandemi berlangsung sebaiknya pendidik atau guru lebih keatif dalam melakukan pemebelajaran online dirumah agar siswa tidak merasa bosan dan semangat saat sedang pembelajaran jarak jauh. sehingga minat belajar siswa dapat meningkat dan memahami konsep fisik Kata kunci: covid 19; pembelajaran; fisika. ABSTRACT Writing this paper aims to determine the effect of online learning media in students' understanding and interest in learning the concepts of physics and for student responses regarding the application of online learning media, the government imposed social distancing policy or known as physical distancing and therefore schools decided to distance learning feared during the pandemic using online learning media student learning interest will decrease. From the survey results using a questionnaire on the Google form of high school students obtained 18.8% agreed to use online learning media and 52.75% did not agree to use online learning media. from the results of the data it is said "not strong" this shows the response of students to use the online learning media. Student interest in learning increases when online learning is done in the classroom. During the pandemic, educators or teachers should be more active in conducting online learning at home so that students do not feel bored and excited while studying distance learning. so that students' interest in learning can increase and understand physical concepts Keywords: covid 19; learning; physics

Abstrak: Root finding adalah salah satu topik dalam metode numerik dalam menentukan akar suatu persamaan , biasanya persamaan tersebut dalam bentuk rumit dan sulit diselesaikan secara analitik. Dalam artikel ini suatu polinomial berderajat tinggi yaitu polinomial Wilkinson akan digunakan untuk menguji perbandingan akurasi metode Halley dan Olver, yang mana metode tersebut jarang digunakan karena kalah populer dengan metode Newton-Raphson, akan tetapi kedua metode tersebut memiliki kinerja yang cukup bagus dan lebih cepat konvergen dengan iterasi lebih sedikit. Berdasarkan hasil simulasi metode Halley pada iterasi ke-4 mendapatkan persentase galat 0,0029%, Metode Olver pada iterasi ke-5 mendapatkan persentase galat 0,0004% sedangkan metode Newton-Raphson membutuhkan iterasi ke-7 untuk mendapatkan persentase galat 0,0098%. Abstract: Root finding is one of the topics in numerical methods for determaining roots of an equation , usually that equation in the form of complicated and it will be difficult to be solved analytically. In this paper, high order polynomial like Polynomial Wilkinson will be used to test comparison the accuracy on Halley and Olver methods, which is those methods are rarely used for losing popular than Newton-Raphson method, but both methods had powerful performance and faster converging with less iteration. Based on the simulation, Halley method on 4th iteration got percentage error 0,0029%, Olver method on 5th iteration got percentage error 0,0004%, while Newton-Raphson method are need 7th iteration got percentage error 0,0098%.

Abstrak: Di dalam tulisan ini disajikan analisa kestabilan, diselidiki eksistensi dan kestabilan limit cycle, dan ditentukan solusi pendekatan dengan menggunakan metode multiple scale dari persamaan Van der Pol. Penelitian ini dilakukan dalam tiga tahapan metode. Pertama, menganalisa perilaku dinamik persamaan Van der Pol di sekitar ekuilibrium, meliputi transformasi persamaan ke sistem persamaan, analisa kestabilan persamaan melalui linearisasi, dan analisa kemungkinan terjadinya bifukasi pada persamaan. Kedua, membuktikan eksistensi dan kestabilan limit cycle dari persamaan Van der Pol dengan menggunakan teorema Lienard. Ketiga, menentukan solusi pendekatan dari persamaan Van der Pol dengan menggunakan metode multiple scale. Hasil penelitian adalah, berdasarkan variasi nilai parameter kekuatan redaman, daerah kestabilan dari persamaan Van der Pol terbagi menjadi tiga. Untuk parameter kekuatan redaman bernilai positif mengakibatkan ekuilibrium tidak stabil, dan sebaliknya, untuk parameter kekuatan redaman bernilai negatif mengakibatkan ekuilibrium stabil asimtotik, serta tanpa kekuatan redaman mengakibatkan ekuilibrium stabil. Pada kondisi tanpa kekuatan redaman, persamaan Van der Pol memiliki solusi periodik dan mengalami bifurkasi hopf. Selain itu, dengan menggunakan teorema Lienard dapat dibuktikan bahwa solusi periodik dari persamaan Van der Pol berupa limit cycle yang stabil. Pada akhirnya, dengan menggunakan metode multiple scale dan memberikan variasi nilai amplitudo awal dapat ditunjukkan bahwa solusi persamaan Van der Pol konvergen ke solusi periodik dengan periode dua. Abstract: In this paper, the stability analysis is given, the existence and stability of the limit cycle are investigated, and the approach solution is determined using the multiple scale method of the Van der Pol equation. This research was conducted in three stages of method. First, analyzing the dynamic behavior of the equation around the equilibrium, including the transformation of equations into a system of equations, analysis of the stability of equations through linearization, and analysis of the possibility of bifurcation of the equations. Second, the existence and stability of the limit cycle of the equation are proved using the Lienard theorem. Third, the approach solution of the Van der Pol equation is determined using the multiple scale method. Our results, based on variations in the values of the damping strength parameters, the stability region of the Van der Pol equation is divided into three types. For the positive value, it is resulting in unstable equilibrium, and contrary, for the negative value, it is resulting in asymptotic stable equilibrium, and without the damping force, it is resulting in stable equilibrium. In conditions without damping force, the Van der Pol equation has a periodic solution and has hopf bifurcation. In addition, by using the Lienard theorem, it is proven that the periodic solution is a stable limit cycle. Finally, by using the multiple scale method with varying the initial amplitude values, it is shown that the solution of the Van der Pol equation is converge to a periodic solution with a period of two.