Micron-Sized Zeolite Beta Single Crystals Featuring Intracrystal Interconnected Ordered Macro-Meso-Microporosity Displaying Superior Catalytic Performance

Zeolite Beta single crystals with intracrystalline hierarchical porosity at macro-, meso-, and micro-length scales can effectively overcome the diffusion limitations in the conversion of bulky molecules. However, the construction of large zeolite Beta single crystals with such porosity is a challenge. We report herein the synthesis of hierarchically ordered macro-mesoporous single-crystalline zeolite Beta (OMMS-Beta) with a rare micron-scale crystal size by an in situ bottom-up confined zeolite crystallization strategy. The fully interconnected intracrystalline macro-meso-microporous hierarchy and the micron-sized single-crystalline nature of OMMS-Beta lead to improved accessibility to active sites and outstanding (hydro)thermal stability. Higher catalytic performances in gas-phase and liquid-phase acid-catalyzed reactions involving bulky molecules are obtained compared to commercial Beta and nanosized Beta zeolites. The strategy has been extended to the synthesis of other zeolitic materials, including ZSM-5, TS-1, and SAPO-34.     

Certain fractional calculus and integral transform results of incomplete ℵ-functions with applications

In this paper, we study certain interesting and useful properties of incomplete $\aleph$-functions. The incomplete $\aleph$-function is an extension of the $\aleph$-function. We find several useful classical integral transforms of these functions. Further, we examine the fractional calculus with the incomplete $\aleph$-functions and point out several special cases. Finally, we give the applications of incomplete $\aleph$-functions in detecting glucose supply in human blood.