Ramanujan's Theta Functions and Parity of Parts and Cranks of Partitions

In this paper, we explore intricate connections between Ramanujan's theta functions and a class of partition functions defined by the nature of the parity of their parts. This consequently leads us to the parity analysis of the crank of a partition and its correlation with the number of partitions with odd number of parts, self-conjugate partitions, and also with Durfee squares and Frobenius symbols.

Comprehensive partitions and optimisation strategies based on tourism urbanisation and resources environment carrying capacity in the Yellow River Basin, China

A better understanding of the spatial coordination relationship between tourism urbanisation and resources environment carrying capacity (RECC) is vital for the regional selection of key eco-liveable tourist cities in the Yellow River Basin. This study addressed this research issue by evaluating and partitioning tourism urbanisation level and RECC of the Yellow River Basin in 2005, 2011, and 2018 using a geographic information system (GIS) technology, spatial autocorrelation model, and partition method. Empirical results suggest that the tourism urbanisation level of Shaanxi Province maintains its leading position, while Shanxi Province has great development potential. The high-value areas of RECC are concentrated in Gansu, Inner Mongolia, and Shandong provinces. The degree of spatial agglomeration of the tourism urbanisation level and RECC has been improved. The RECC exhibits a greater restrictive effect on the current high-level areas of tourism urbanisation, and the spatial correspondence between them is weak. Based on the findings of this study, a series of optimisation strategies have been proposed to promote the sustainable development of tourism urbanisation in the Yellow River Basin.

Can we migrate COVID-19 spreading risk?

It is well recognized that Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) virus could be spread through touch and large droplets. However, we may have under-estimated the disease transmission by small droplets or aerosols that contain SARS-CoV-2 virus. Social distancing in public transport vehicles, such as airplanes, is not feasible. It is also not possible to wear masks in restaurant. This paper recommended wearing masks in airplanes and use partition screens in the middle of a table in a restaurant to reduce the infectioncausedbySARS-CoV-2virus. Advanced ventilation systems, such as personalized ventilation and displacement ventilation, are strongly recommended for transport vehicles and buildings.

An extensive analysis of the parity of broken 3-diamond partitions

In 2007, Andrews and Paule introduced the family of functions [Formula: see text] which enumerate the number of broken k-diamond partitions for a fixed positive integer k. Since then, numerous mathematicians have considered partitions congruences satisfied by [Formula: see text] for small values of k. In this work, we provide an extensive analysis of the parity of the function [Formula: see text], including a number of Ramanujan-like congruences modulo 2. This will be accomplished by completely characterizing the values of [Formula: see text] modulo 2 for [Formula: see text] and any value of [Formula: see text]. In contrast, we conjecture that, for any integers [Formula: see text], [Formula: see text] and [Formula: see text] is infinitely often even and infinitely often odd. In this sense, we generalize Subbarao's Conjecture for this function [Formula: see text]. To the best of our knowledge, this is the first generalization of Subbarao's Conjecture in the literature.