COVID-19 is still a pandemic in Indonesia, and Central Java is no exception. New positive cases of COVID-19 in Central Java are being discovered every day. Therefore, researchers try to predict new positive cases in Central Java. Many forecasting methods are currently developing, one of which is fuzzy time series (FTS). FTS has been also developed until now, one of which is a development of the FTS by combining the Markov chain as a defuzzification process. In FTS there is no definite formula to determine the length of the interval, so the researcher uses an average based to determine the length of the interval in the FTS Markov chain. Next, the researcher repartitioned based on the modified frequency density. The results of this study are that forecasting new positive cases of COVID-19 in Central Java using the average based-FTS Markov chain based on a modified frequency density partitioning method has a good level of accuracy, this can be seen from the MAPE value of the method.

In a ring with involution we already know that the Moore Penrose inverse can be used to construct the properties of normal elements. By using the fact that the group inverse of an element in a ring will be commutative with element which is commutative with that element, this paper explains that the generalization of Moore Penrose inverse can be also to establish some properties of normal elements in a ring with an involution. Some elements in the ring such as symmetric, EP, partial isometries, etc are also can be expressed in group inverse. So the results of this paper much be required for built the properties of those elements.

No abstract

Invers Moore Penrose Elemen Normal pada Ring dengan involusi telah dibahas oleh beberapa peneliti. Dengan memperumum konsep invers Moore Penrose menjadi invers Moore Penrose diperumum, sifat elemen Invers Moore Penrose Diperumum dari Elemen Normal juga telah diperoleh. Selain memperumum konsep invers Moore Penrose, definisi elemen normal juga telah diperumum dengan memperumum pangkat 1 menjadi pangkat 𝑛 ∈ ℕ. Diperoleh bahwa irisan antara himpunan elemen Normal diperumum dan himpunan elemen yang mempunyai Invers Moore Penrose Diperumum tidak kosong. Hal ini mengindikasikan ada sifat bersama yang dimiliki oleh keduanya, sehingga tulisan ini bertujuan membangun syarat perlu dan cukup suatu elemen normal diperumum mempunyai invers Moore Penrose diperumum dengan menggunakan sifat2 tersebut. Metode yang dilakukan adalah dengan mencari kesamaan sifat yang dimiliki oleh elemen normal diperumum dan elemen yang mempunyai invers Moore Penrose diperumum. Langkah selanjutnya adalah menggunakan sifat involusi untuk memperoleh hasil akhir. Pendekatan yang dilakukan selain melalui invers Moore Penrose Diperumum, juga melalui invers grup Kata Kunci:

Moore-Penrose inverse is one of development of generalized inverse. In this paper, we defined and studied a relation between the Moore-Penrose inverse and the Drazin inverse in the setting of rings with involution. The results of this paper are new characterizations of Moore-Penrose inverse by applying Drazin inverse with an algebraic proof.