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Abstract We consider time fractional parabolic equations in divergence and non-divergence form when the leading coefficients $$a^{ij}$$ are measurable functions of $$(t,x_1)$$ except for $$a^{11}$$, which is a measurable function of either $$t$$ or $$x_1$$. We obtain the solvability in Sobolev spaces of the equations in the whole space, on a half space, and on a partially bounded domain. The proofs use a level set argument, a scaling argument, and embeddings in fractional parabolic Sobolev spaces for which we give a direct and elementary proof. 

All solid materials are created via nucleation. In this evolutionary process, nuclei form in solution or at interfaces, expand by monomeric growth and oriented attachment, and undergo phase transformation. Nucleation determines the location and size of nuclei, whereas growth controls the size, shape, and aggregation of newly formed nanoparticles. These physical properties of nanoparticles can affect their functionalities, reactivities, and porosities, as well as their fate and transport. Recent advances in nanoscale analytical technologies allow in situ real-time observations, enabling us to uncover the molecular nature of nuclei and the critical controlling factors for nucleation and growth. Although a single theory cannot yet fully explain such evolving processes, we have started to better understand how both classical andnonclassical theories can work together, and we have begun to recognize the importance of connecting these theories. This review discusses the recent convergence of knowledge about the nucleation and growth of nanoparticles. 

Abstract This review article highlights recent advances in designing biomaterials to be interfaced with food and plants, with the goal of enhancing the resilience of the AgroFood infrastructure by boosting crop production, mitigating environmental impact, and reducing losses along the supply chain. Special attention is given to innovations in biomaterial‐based approaches and platforms for 1) seed enhancement through encapsulation, preservation, and controlled release of payloads (e.g., plant growth‐promoting microbes) to the seeds and their rhizosphere; 2) precision delivery of multi‐scale payloads to targeted plant tissues, organelles, and vasculature; 3) edible food coatings that regulate gas exchanges and provide antimicrobial properties to extend the shelf life of perishable food; and 4) food spoilage detection based on different sensor/reporter systems. Within each domain, biomaterials design principles, emerging micro‐/nanofabrication strategies, and the advantages and disadvantages of different delivery/preservation/sensing platforms are introduced and critically discussed. Views of future requirements, aims, and trends are also given based on the opportunities and challenges of applying biomaterials in the AgroFood system. 

Abstract Food quality monitoring, particularly, the detection of bacterial pathogens and spoilage throughout the food supply chain, is critical to ensure global food safety and minimize food loss. Incorporating sensors into packaging is promising, but it is challenging to achieve the required sampling volume while using food‐safe sensor materials. Here, by leveraging water‐based processing of silk fibroin, a platform for the detection of pathogenic bacteria in food is realized using a porous silk microneedle array; the microneedle array samples fluid from the interior of the food by capillary action, presenting the fluid to polydiacetylene‐based bioinks printed on the backside of the array. Through the colorimetric response of bioink patterns,Escherichia colicontamination in fish fillets is identified within 16 h of needle injection. This response is distinct from spoilage measured via the increase in sample pH. It is also shown that the microneedles can pierce commercial food packaging, and subsequently sample fluid and present it to the sensor, enabling the adaptation of the technology downstream in food supply chains such as in stores or at home. This study highlights that regenerated structural biopolymers can serve as safe materials for food contact and sensing with robust mechanical properties and tailored chemistry.