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<div style="text-align: justify;">In this article we will investigate how to solve nonhomogen degenerate Cauchy problem via theory of semigroup of linear operator. The problem is formulated in Hilbert space which can be written as direct sum of subset Ker M and Ran M*. By certain assumptions the problem can be reduced to nondegenerate Cauchy problem. And then by composition between invers of operator M and the nondegenerate problem we can transform it to canonic problem, which is easier to solve than the original problem. By taking assumption that the operator A is infinitesimal generator of semigroup, the canonic problem has a unique solution. This allow to define special operator which map the solution of canonic problem to original problem. ©2016 JNSMR UIN Walisongo. All rights reserved.</div>

Fuzzy time series (FTS) is one of the forecasting methods that has been developed until now. The fuzzy time series is a forecasting method that uses the concept of fuzzy logic, which Song and Chissom first introduced. The fuzzy time series (FTS) Markov chain uses the Markov chain in defuzzification. The determination of the length of the interval in the fuzzy time series plays an important role in forming a fuzzy logic relationship (FLR), and this FLR will be used to determine the forecasting value. One method that can be used to determine the interval length is average-based. However, several studies use partitioning based on frequency density to obtain the optimal interval length to get better forecasting accuracy. This study combines the fuzzy time series Markov chain, Average-based fuzzy time series, and Fuzzy time series based on frequency density partitioning to become average-based fuzzy time series Markov chain based on the Frequency Density Partition which conducts redivided intervals based on frequency density in the average-based fuzzy time series Markov chain method. This method is implemented in forecasting the Indonesian Islamic stock index (ISSI) for the selected period. The calculation of the accuracy level using the mean square error (MSE) and the mean average percentage error (MAPE) shows that the fuzzy Markov chain-based fuzzy time series based on the frequency density partition has a high level of accuracy in forecasting.

An integral Dunford and an operator on Dunford integrable functional space have discussed in this article. The results were shown that the Dunford integrable functional space was a linear function. For every Dunford integrable function on a closed interval, there is an operator that is linear bounded and weak compact operator, whereas its adjoin operator is also linear bounded and weak compact. An operator is weak compact if and only if its adjoin operator is weak compact. Furthermore, the norm of this operator was equal to the norm of its adjoin operator.

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.