In this work , we study different chaotic properties of the product space on a one-step shift of a finite type, as well as other spaces. We prove that the product  is Lyapunove  â€“unstable if and only if at least one  or  is Lyapunove  â€“unstable. Also, we show that and locally everywhere onto (l.e.o)  if and only if is locally everywhere onto (l.e.o) .

            This work, introduces some concepts in bitopological spaces, which are nm-j-ω-converges to a subset, nm-j-ω-directed toward a set, nm-j-ω-closed mappings, nm-j-ω-rigid set, and nm-j-ω-continuous mappings. The mainline idea in this paper is nm-j-ω-perfect mappings in bitopological spaces such that n = 1,2  and m =1,2 n ≠ m. Characterizations concerning these concepts and several theorems are studied, where j = q , δ, a , pre, b, b.

This paper is concerned with introducing and studying the M-space by using the mixed degree systems which are the core concept in this paper. The necessary and sufficient condition for the equivalence of two reflexive M-spaces is super imposed. In addition, the m-derived graphs, m-open graphs, m-closed graphs, m-interior operators, m-closure operators and M-subspace are introduced. From an M-space, a unique supratopological space is introduced. Furthermore, the m-continuous (m-open and m-closed) functions are defined and the fundamental theorem of the m-continuity is provided. Finally, the m-homeomorphism is defined and some of its properties are investigated.

Spelling correction is considered a challenging task for resource-scarce languages. The Arabic language is one of these resource-scarce languages, which suffers from the absence of a large spelling correction dataset, thus datasets injected with artificial errors are used to overcome this problem. In this paper, we trained the Text-to-Text Transfer Transformer (T5) model using artificial errors to correct Arabic soft spelling mistakes. Our T5 model can correct 97.8% of the artificial errors that were injected into the test set. Additionally, our T5 model achieves a character error rate (CER) of 0.77% on a set that contains real soft spelling mistakes. We achieved these results using a 4-layer T5 model trained with a 90% error inject

The chemical properties of chemical compounds and their molecular structures are intimately connected. Topological indices are numerical values associated with chemical molecular graphs that help in understanding the physicochemical properties, chemical reactivity and biological activity of a chemical compound. This study obtains some topological properties of second and third dominating David derived (DDD) networks and computes several K Banhatti polynomial of second and third type of DDD.