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D. Bokal proved that the crossing number is additive for the zip product under the condition of having two coherent bundles in the zipped graphs. This property is very effective when dealing with the crossing numbers of (capped) Cartesian product of trees with graphs containing a dominating vertex. In this paper, we first prove that the crossing number is still additive for the zip product under a weaker condition. Based on the new condition, we then establish some general expressions for bounding the crossing numbers of (capped) Cartesian product of trees with graphs (possibly without dominating vertex). Exact values of the crossing numbers of Cartesian product of trees with most graphs of order at most five are obtained by applying these expressions, which extend some previous results due to M. Klešč. In fact, our results can also be applied to deal with Cartesian product of trees with graphs of order more than five. [ABSTRACT FROM AUTHOR]

Mild cognitive impairment affects 36% of people aged 65 years and over in China, and around 50% transition from mild cognitive impairment to dementia within 3 years. Early intervention can slow down disease progression and thus delay dementia onset. The purpose of this article is to outline the protocol of an ongoing randomized controlled trial in mainland China that will evaluate the effects and feasibility of a 6‐month multicomponent integrative intervention on the speed of progression of mild cognitive impairment to dementia. Ninety‐six community‐dwelling older adults, aged 65 years and older, will be recruited (recruitment will be completed in May 2020), using strict inclusion/exclusion criteria, from two community health service centers in Guangzhou, Guangdong province. Participants will be allocated to receive either the multicomponent integrative intervention or usual care. The core components of the intervention are cognitive training, dietary instruction, physical activity, and management of vascular risk factors. Data are collected at the beginning of the study, then at 1, 3, and 6 months. The primary outcome is cognitive function. The main secondary outcomes are exercise capacity, comprehensive physical capacity, depression, and quality of life. An intention‐to‐treat analysis will be conducted. The study will be completed in 2021. The multicomponent integrative intervention detailed in this protocol could be incorporated into dementia prevention programs in community health service centers, or other similar settings, to delay the onset of dementia. [ABSTRACT FROM AUTHOR]