geometric structure
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2021 ◽  
pp. 1-53
Author(s):  
Yu Nie ◽  
Yang Zhang

Abstract Large meridional excursions of a jet stream are conducive to blocking and related midlatitude weather extremes, yet the physical mechanism of jet meandering is not well understood. This paper examines the mechanisms of jet meandering in boreal winter through the lens of a potential vorticity (PV)-like tracer advected by reanalysis winds in an advection-diffusion model. As the geometric structure of the tracer displays a compact relationship with PV in observations and permits a linear mapping from tracer to PV at each latitude, jet meandering can be understood by the geometric structure of tracer field that is only a function of prescribed advecting velocities. This one-way dependence of tracer field on advecting velocities provides a new modeling framework to quantify the effects of time mean flow versus transient eddies on the spatiotemporal variability of jet meandering. It is shown that the mapped tracer wave activity resembles the observed spatial pattern and magnitude of PV wave activity for the winter climatology, interannual variability, and blocking-like wave events. The anomalous increase in tracer wave activity for the composite over interannual variability or blocking-like wave events is attributed to weakened composite mean winds, indicating that the low-frequency winds are the leading factor for the overall distributions of wave activity. It is also found that the tracer model underestimates extreme wave activity, likely due to the lack of feedback mechanisms. The implications for the mechanisms of jet meandering in a changing climate are also discussed.


Author(s):  
V. A. Babkin ◽  
D. S. Andreev ◽  
E. S. Titova ◽  
S. V. Chepurnov ◽  
R. O. Boldyrev ◽  
...  

For the first time, the geometric and electronic structure of graphene oxide was calculated within the framework of the Hoffman and Ress models by the quantum-chemical method MNDO. Optimized structures of this compound are obtained. The acidic strength (pKa = 28 and pKa = 14) was theoretically estimated in the Hoffman and Ress model, respectively.


2021 ◽  
Vol 28 (4) ◽  
pp. 167-178
Author(s):  
Ewa Piątkowska

Abstract This paper reports on a study of the influence of solid particle contamination on the wear process in water-lubricated slide bearings (steel-acrylonitrile-butadiene rubber (NBR) and steel-polytetrafluoroethylene (PTFE)). To compare the wear of the shaft journal and bushes (NBR and PTFE) when lubricated with fresh water and contaminated water, an experiment was carried out to identify key factors that influence the state of wear of slide bearing. The amount of wear was checked by means of geometric structure measurements on the journals, namely, roughness profile measurements using both a contact profilometer and an optical microscope. The obtained results enabled correlations between the material comprising the sliding sleeve, roughness of the journals and contamination inside the water-lubricated slide bearings.


Nature ◽  
2021 ◽  
Vol 600 (7887) ◽  
pp. 70-74
Author(s):  
Alex Davies ◽  
Petar Veličković ◽  
Lars Buesing ◽  
Sam Blackwell ◽  
Daniel Zheng ◽  
...  

AbstractThe practice of mathematics involves discovering patterns and using these to formulate and prove conjectures, resulting in theorems. Since the 1960s, mathematicians have used computers to assist in the discovery of patterns and formulation of conjectures1, most famously in the Birch and Swinnerton-Dyer conjecture2, a Millennium Prize Problem3. Here we provide examples of new fundamental results in pure mathematics that have been discovered with the assistance of machine learning—demonstrating a method by which machine learning can aid mathematicians in discovering new conjectures and theorems. We propose a process of using machine learning to discover potential patterns and relations between mathematical objects, understanding them with attribution techniques and using these observations to guide intuition and propose conjectures. We outline this machine-learning-guided framework and demonstrate its successful application to current research questions in distinct areas of pure mathematics, in each case showing how it led to meaningful mathematical contributions on important open problems: a new connection between the algebraic and geometric structure of knots, and a candidate algorithm predicted by the combinatorial invariance conjecture for symmetric groups4. Our work may serve as a model for collaboration between the fields of mathematics and artificial intelligence (AI) that can achieve surprising results by leveraging the respective strengths of mathematicians and machine learning.


2021 ◽  
Author(s):  
Christian Rosendal

This book provides a general framework for doing geometric group theory for many non-locally-compact topological transformation groups that arise in mathematical practice, including homeomorphism and diffeomorphism groups of manifolds, isometry groups of separable metric spaces and automorphism groups of countable structures. Using Roe's framework of coarse structures and spaces, the author defines a natural coarse geometric structure on all topological groups. This structure is accessible to investigation, especially in the case of Polish groups, and often has an explicit description, generalising well-known structures in familiar cases including finitely generated discrete groups, compactly generated locally compact groups and Banach spaces. In most cases, the coarse geometric structure is metrisable and may even be refined to a canonical quasimetric structure on the group. The book contains many worked examples and sufficient introductory material to be accessible to beginning graduate students. An appendix outlines several open problems in this young and rich theory.


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