The main results are as follows:( i ) For the number of chord diagrams of order n, an exact formula is given.( ii ) For the number of spine diagrams of order n, the upper and lower bounds are obtained. These bounds sh...The main results are as follows:( i ) For the number of chord diagrams of order n, an exact formula is given.( ii ) For the number of spine diagrams of order n, the upper and lower bounds are obtained. These bounds show that the estimation is asymptotically the best.As a byproduct, an upper bound is obtained, for the dimension of Vassiliev knot invariants of order n, that is, 1/2 ( n -1)! for any n≥3, and 1/2( n - 1)! - 1/2( n - 2)! for bigger n . Our upper bound is based on the work of Chmutov and Duzhin and is an improvement of their bound ( n - 1)! . For n = 3, and 4,1/2( n - 1)! is already the best.展开更多
Large and growing data resources on the diversity, distribution, and properties of minerals are ushering in a new era of data-driven discovery in mineralogy. The most comprehensive international mineral database is th...Large and growing data resources on the diversity, distribution, and properties of minerals are ushering in a new era of data-driven discovery in mineralogy. The most comprehensive international mineral database is the IMA database, which includes information on more than 5400 approved mineral species and their properties, and the mindat.org data source, which contains more than 1 million species/locality data on minerals found at more than 300 000 localities. Analysis and visualization of these data with diverse techniques—including chord diagrams, cluster diagrams, Klee diagrams, skyline diagrams, and varied methods of network analysis—are leading to a greater understanding of the co-evolving geosphere and biosphere. New data-driven approaches include mineral evolution, mineral ecology, and mineral network analysis—methods that collectively consider the distribution and diversity of minerals through space and time. These strategies are fostering a deeper understanding of mineral co-occurrences and, for the first time, facilitating predictions of mineral species that occur on Earth but have yet to be discovered and described.展开更多
文摘The main results are as follows:( i ) For the number of chord diagrams of order n, an exact formula is given.( ii ) For the number of spine diagrams of order n, the upper and lower bounds are obtained. These bounds show that the estimation is asymptotically the best.As a byproduct, an upper bound is obtained, for the dimension of Vassiliev knot invariants of order n, that is, 1/2 ( n -1)! for any n≥3, and 1/2( n - 1)! - 1/2( n - 2)! for bigger n . Our upper bound is based on the work of Chmutov and Duzhin and is an improvement of their bound ( n - 1)! . For n = 3, and 4,1/2( n - 1)! is already the best.
基金grants from the Alfred P. Sloan Foundation (G-2016-7065)the W. M. Keck Foundation (grant entitled ‘‘Co-Evolution of the Geosphere and Biosphere”), the John Templeton Foundation (60645)the NASA Astrobiology Institute (1-NAI8_2-0007), a private foundation, and the Carnegie Institution for Science. Sergey V. Krivovichev acknowledges support from the Russian Science Foundation (19-17-00038).
文摘Large and growing data resources on the diversity, distribution, and properties of minerals are ushering in a new era of data-driven discovery in mineralogy. The most comprehensive international mineral database is the IMA database, which includes information on more than 5400 approved mineral species and their properties, and the mindat.org data source, which contains more than 1 million species/locality data on minerals found at more than 300 000 localities. Analysis and visualization of these data with diverse techniques—including chord diagrams, cluster diagrams, Klee diagrams, skyline diagrams, and varied methods of network analysis—are leading to a greater understanding of the co-evolving geosphere and biosphere. New data-driven approaches include mineral evolution, mineral ecology, and mineral network analysis—methods that collectively consider the distribution and diversity of minerals through space and time. These strategies are fostering a deeper understanding of mineral co-occurrences and, for the first time, facilitating predictions of mineral species that occur on Earth but have yet to be discovered and described.
