The development of technology is growing very rapidly, then it should be used for improving many important aspects of our daily life, such as learning innovation. This study aims to develop an android-based interactive e-module on trigonometries’ topic to enhance the learning motivation of students. The topic of trigonometry is chosen because it lacks to be discussed. Many Android-based interactive e-modules are generally limited on numbers. On the other hand, online learning policies during pandemics make many students feel bored, one of which is because the learning applied by the learning media or methods chosen by teachers is monotonous and less attractive. The contents of this e-module consist of learning outcomes, material descriptions, learning videos, and quizzes. The method in this study is Research and Development with the ADDIE model. The ADDIE model has five stages: Analyze, Design, Development, Implementation, and Evaluation. The main key in this method is the iteration process. Before the trial, the e-module is validated by substantial experts and learning media’s experts. The substantial experts gave a score of 4.32 out of 5 which indicates the modules are valid substantially. The learning media’s expert gave a score of 4.18 out of 5 which indicates the modules are very valid. Then, these e-modules are implemented into small classes and large classes. The practicality and the effectiveness of these e-modules are measured. The practicality of these e-modules in the small class has a score 4.18 out of 5, while it has a score of 4.28 out of 5 in the large class. The effectiveness of this e-modules in small class have score 4.28 out of 5, while it has score 4.31 out of 5 in a large class. These results indicate that android-based interactive e-modules are effective and recommended to be used in the teaching-learning process on trigonometry.The development of technology is growing very rapidly, then it should be used for improving many important aspects of our daily life, such as learning innovation. This study aims to develop an android-based interactive e-module on trigonometries’ topic to enhance the learning motivation of students. The topic of trigonometry is chosen because it lacks to be discussed. Many android-based interactive e-modules are generally limited on numbers. On the other hand, online learning policies during pandemic make many students feel bored, one of which is because the learning applied by the learning media or methods chosen by teachers is monotonous and less attractive. The contents of this e-module consist of learning outcomes, material descriptions, learning videos, and quizzes. The method in this study is Research and Development with the ADDIE model. The ADDIE model has five stages: Analyze, Design, Development, Implementation, and Evaluation. The main key in this method is the iteration process. Before the trial, the e-module is validated by substancial experts and learning media’s experts. The substancial experts gave a score 4.32 out of 5 which indicates the modules are valid substancially. The learning media’s expert gave a score 4.18 out of 5 which indicates the modules are very valid. Then, this e-modules is implemented into small class and large class. The practicality and the effectiveness of this e-modules are measured. The practicality of this e-modues in the small class have score 4.18 out of 5, while it have score 4.28 out of 5 in the large class. The effectiveness of this e-modules in small class have score 4.28 out of 5, while it have score 4.31 out of 5 in large class. These results indicate that android-based interactive e-modules are effective and recommended to be used in teaching-learning process on trigonometry.

The mathematical model of COVID-19 considered in this study is the SEIR model which is defined by four ordinary differential equations that describe the number of susceptible, infected, infected and cured individuals by applying optimal control theory in the form of treatment and quarantine. To characterize the optimal control in the COVID-19 seir mathematical model, the Pontryagin maximum principle is used. The purpose of this study was to reduce the number of susceptible, infected and infected individuals and increase the number of recovered individual populations. The covid-19 mathematical model with optimal control is solved using the Runge-kutta order 4 method and the results are represented graphically. The results obtained from the simulations carried out show that optimal control can work well on the Covid-19 mathematical model that has been formed with the data used being actual data on Covid-19 cases in Indonesia.

In crisp graphs, if a graph G is given then we can define line graph L(G) of it. In a similar way, for a given fuzzy graph, we can determine its fuzzy line graph. A number of important results about fuzzy line graphs have been obtained. However, study of anti fuzzy line graphs is still open. Motivated by this, we are interested in studying anti fuzzy line graphs. In this paper, we define the concept of anti fuzzy line graphs. We derive a necessary and sufficient condition for an anti fuzzy graph to be isomorphic to its corresponding anti fuzzy line graph. We investigate when an isomorphism between two anti fuzzy graphs follows from an isomorphism of their corresponding anti fuzzy line graphs. We conjecture a necessary and sufficient condition for an anti fuzzy graph to be the anti fuzzy line graph of some anti fuzzy graph. In particular, we give some result related to regular anti fuzzy line graphs.

In this article, we investigate the dynamical analysis of predator prey model. Interaction among preys and predators use Holling type II functional response, and assuming prey refuge as well as harvesting in both populations. This study aims to study the predator prey model and to determine the effect of overharvesting which consequently will affect the ecosystem. In the model found three equilibrium points, i.e., E 1(0,0) is the extinction of predator and prey equilibrium, E 2(x 2, 0) is the equilibrium with predatory populations extinct and the last equilibrium points E 3 (x 3, y 3) is the coexist equilibrium. All equilibrium points are asymptotically stable (locally) under certain conditions. These analytical findings were confirmed by several numerical simulations.

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