Tuberculosis is an infectious disease; it caused by Mycobacterium tuberculosis. In this paper, we discuss how to use the Routh-Hurwitz stability criterion to analyze the stability of disease free of the tuberculosis transmission model. From this method, can be found the number of roots of the characteristic polynomial (eigenvalues) with positive real parts is equal to the number of changes in sign of the first column of the Routh array. If all of the eigenvalues are negative, then the model is stable. While the bifurcation method is used to analyze the stability of the endemic equilibrium point of the tuberculosis transmission, the endemic equilibrium point is locally asymptotically stable if reproduction number greater than one and additional parameters requirement that bifurcation met. Finally, numerical simulations are demonstrated to verify the used method.
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.
This article discusses the convergence theorem of the Dunford integrals. We examine the sufficient conditions so that limit of the sequence of integral value whose Dunford integrable is same as limit of functions sequence. We have obtained that to guarantee a function to be Dunford integrable and its limit of functions sequence are same as value of the functions, then a sequence of Dunford integrable function is uniform convergent or weakly convergent, weakly monoton, and its limit exist. Furthermore, its weakly convergent and bounded.
This study discussed the integral of Dunford and compact linear operator on space of Dunford integral function. For each f which is Dunford integral on [a,b] is defined as an operator DL by DL (x *) = x * f, for each x * ∈ X * This study resulted that the operator DL is both a continuous linear operator and weakly compact operators. Then, it was defined as the adjoint of the operator D L * by D L * ( h ) ( x * ) = ∫ a b h D L ( x * ) each h ∈ (L 1)* The adjoint operator D L * is continuous and weakly compact linear operators.
We are discussed operator norms on spce of Dunford integral function. We show that for a function which Dunford integral, operator from dual space into space of Lebesgue integral is a bounded linear operator. Furthermore, sets of all bounded linear operator is a linear space and it is a normed space by norm certain. Finally, the distance function generated by the norm is metrix space.
In this paper, discuss the relationship between approachable function with bounded variations, measurement, and absolute continuity. Futhermore, If f is approachable function of interval [a, b] then f is a bounded variation function and f is a measurable function of interval [a, b]. In relation with absolute continuity, if f is an absolute continuous of interval [a, b] then f is approachable function of [a, b].
In this paper, we have defined an approachable function using a simple function on a compact sets. Furthermore the simple properties of the function was examined and it was obtained that measurable function, continuous function, and bounded function are approachable function along the function space is a linier space.
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