We describe the relationships among several notions of isoperimetry on a class of pre-fractal Sierpinski Carpets that satisfy a set of geometric assumptions as well as on the graphs corresponding to these carpets. We compute the L1-isoperimetric profile at scale h, and we use this computation to determine both the classical isoperimetric profile and a profile computed using Jordan curve boundaries.
In the context of a heat kernel diffusion which admits a Gaussian type estimate with parameter β on a local Dirichlet space, we consider the log asymptotic behavior of the negative exponential moments of the Wiener sausage. We show that the log asymptotic behavior up to time t β V (x,t) is V (x,t), which is analogous to the Euclidean result.Here V (x,t) represents the mass of the ball of radius t about a point x of the local Dirichlet space. The proof uses a known coarse graining technique to obtain the upper asymptotic, but must be adapted for use without translation invariance in this setting. This result provides the first such asymptotics for several other contexts, including diffusions on complete Riemannian manifolds with non-negative Ricci curvature.
Globally, thousands of institutions house nearly three billion scientific collections offering unparallelled resources that contribute to both science and society. For herbaria alone - facilities housing dried plant collections - there are over 3,000 herbaria worldwide with an estimated 350 million specimens that have been collected over the past four centuries. Digitisation has greatly enhanced the use of herbarium data in scientific research, impacting diverse research areas, including biodiversity informatics, global climate change, analyses using next-generation sequencing technologies and many others. Despite the entrance of herbaria into a new era with enhanced scientific, educational and societal relevance, museum specimens remain underused. Natural history museums can enhance learning and engagement in science, particularly for school-age and undergraduate students. Here, we outline a novel approach of a natural history museum using touchscreen technology that formed part of an interactive kiosk in a temporary museum exhibit on biological specimens. We provide some preliminary analysis investigating the efficacy of the tool, based on the Zooniverse platform, in an exhibit environment to engage patrons in the collection of biological data. We conclude there is great potential in using crowd‐sourced science, coupled with online technology to unlock data and information from digital images of natural history specimens themselves. Sixty percent of the records generated by community scientists (citizen scientists) were of high enough quality to be utilised by researchers. All age groups produced valid, high quality data that could be used by researchers, including children (10 and under), teens and adults. Significantly, the paper outlines the implementation of experiential learning through an undergraduate mathematics course that focuses on projects with actual data to gain a deep, practical knowledge of the subject, including observations, the collection of data, analysis and problem solving. We here promote an intergenerational model including children, high school students, undergraduate students, early career scientists and senior scientists, combining experiential learning, museum patrons, researchers and data derived from natural history collections. Natural history museums with their dual remit of education and collections-based research can play a significant role in the field of community engagement and people-powered research. There also remains much to investigate on the use of interactive displays to help learners interpret and appreciate authentic research. We conclude with a brief insight into the next phase of our ongoing people-powered research activities developed and designed by high school students using the Zooniverse platform.
Let X be a complete local Dirichlet space with a local Poincaré inequality, local volume doubling, and volumes of balls of a fixed radius bounded away from both 0 and ∞. When X is a co-compact covering of a finitely generated group, the large time behavior of their heat kernels are comparable. This is an extension of work by Pittet and Saloff-Coste (J Geom Anal 10: 2000).
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