We set forth a time-fractional logistic model and give an implicit finite difference scheme for solving of the model. The L^2 stability convergence are proved with aids discrete Gronwall inequality, numerical examples presented to support theoretical analysis.

In this paper, the connection between Menger probabilistic norms and H"{o}hle probabilistic norms is discussed. In addition, the correspondence between probabilistic norms and Wu-Fang fuzzy (semi-) norms is established. It is shown that a probabilistic norm (with triangular norm $min$) can generate a Wu-Fang fuzzy semi-norm and conversely, a Wu-Fang fuzzy norm can generate a probabilistic norm.

We prove several new results regarding the relationship between probabilistic time, BPTime (t), and alternating time, ΣO(1)Time (t). Our main results are the following: 1. We prove that BPTime (t) ⊆ Σ3Time (t · poly log t). Previous results show that BPTime (t) ⊆ Σ2Time ( t2 · log t ) (Sipser and Gács, STOC ’83; Lautemann, IPL ’83) and BPTime (t) ⊆ ΣcTime (t) for a large constant c > 3 (Ajtai, ...

Probabilistic and soft computing methods are increasingly adopted in various fields of engineering geology. In the past decades, extensive original and innovative studies of various engineering geology problems have been conducted using probabilistic and soft computing methods, and substantial new theories, methods, insights, experiences, and data have been attained. This special issue is aimed...