In this paper, we establish the global existence and uniqueness of solutions to stochastic heat equations with logarithmic nonlinearity driven by Brownian motion on a bounded domain D in setting L2(D) space. The result is valid for all initial values contrast existing literature. Sobolev inequality plays an important role.

We introduce the notion of log R-maps, and develop a proper moduli stack stable R-maps in case hybrid gauged linear sigma model. Two virtual cycles (canonical reduced) are constructed for these stacks. The main results two comparison theorems relating reduced cycle to cosection localized cycle, as well canonical cycle. This sets foundation new technique computing higher genus Gromov-Witten inva...

in this paper, we study rst order linear fuzzy dierential equations with fuzzy coecient and initial value. we use the generalized dierentiability concept and apply the exponent matrix to present the general form of their solutions. finally, one example is given to illustrate our results.