Using a result by Chazal, Crawley-Boevey and de Silva concerning radicals of persistence modules, we show that every lower semi-continuous q-tame module can be decomposed as direct sum interval modules upper product modules.

In 1965, Njastad [2] introduced the notion of α-sets in topological space. In 1983, Mashhour et al. [1] introduced, with the help of α-sets, a weak form of continuity which they termed as α-continuity. Noiri [3] introduced the same concept, but under the name strong semicontinuity. Noiri [4] defined with the aid of α-sets a new weakened form of continuous mapping called weakly α-continuous mapp...

In [6], a sufficient condition for space to be ?-stratifiable was presented and it asked whether the is necessary. Jin et al [4] gave negative answer question by showing that with zero-dimensional. this paper, we show precisely an almost discrete space. Moreover, introduce notions of strongly lower (upper) semi-continuous functions, which characterizations spaces are presented.

We introduce the minimal maximally predictive models ( -machines) of processes generated by certain hidden semi-Markov models. Their causal states are either hybrid discrete-continuous or continuous random variables and causal-state transitions are described by partial differential equations. Closed-form expressions are given for statistical complexities, excess entropies, and differential info...

Several classes of hereditarily normal spaces are characterized in terms of extending upper semi-continuous compact-valued mappings. The case of controlled extensions is considered as well. Applications are obtained for real-valued semi-continuous functions.