This paper presents a complete method to prove geometric theorem by decomposing the corresponding polynomial system. into strong regular sets, by which one can compute some components for which the geometry theorem is...This paper presents a complete method to prove geometric theorem by decomposing the corresponding polynomial system. into strong regular sets, by which one can compute some components for which the geometry theorem is true and exclude other components for which the geometry theorem is false. Two examples are given to show that the geometry theorems are conditionally true for some components which are excluded by other methods.展开更多
Using Galois rings and Galois fields, we construct several infinite classes of partial geometric difference sets, and partial geometric difference families, with new parameters. Furthermore, these partial geometric di...Using Galois rings and Galois fields, we construct several infinite classes of partial geometric difference sets, and partial geometric difference families, with new parameters. Furthermore, these partial geometric difference sets(and partial geometric difference families) correspond to new infinite families of directed strongly regular graphs. We also discuss some of the links between partially balanced designs, 2-adesigns(which were recently coined by Cunsheng Ding in "Codes from Difference Sets"(2015)), and partial geometric designs, and make an investigation into when a 2-adesign is a partial geometric design.展开更多
In the study of(partial)difference sets and their generalizations in groups G,the most widely used method is to translate their definition into an equation over group ring Z[G]and to investigate this equation by apply...In the study of(partial)difference sets and their generalizations in groups G,the most widely used method is to translate their definition into an equation over group ring Z[G]and to investigate this equation by applying complex representations of G.In this paper,we investigate the existence of(partial)difference sets in a different way.We project the group ring equations in Z[G]to Z[N]where N is a quotient group of G isomorphic to the additive group of a finite field,and then use polynomials over this finite field to derive some existence conditions.展开更多
This note characterizes the set of Fréchet-differentiable points of the projection operator on a polyhedral set and the B-subdifferential of this projection operator at any point.
文摘This paper presents a complete method to prove geometric theorem by decomposing the corresponding polynomial system. into strong regular sets, by which one can compute some components for which the geometry theorem is true and exclude other components for which the geometry theorem is false. Two examples are given to show that the geometry theorems are conditionally true for some components which are excluded by other methods.
文摘Using Galois rings and Galois fields, we construct several infinite classes of partial geometric difference sets, and partial geometric difference families, with new parameters. Furthermore, these partial geometric difference sets(and partial geometric difference families) correspond to new infinite families of directed strongly regular graphs. We also discuss some of the links between partially balanced designs, 2-adesigns(which were recently coined by Cunsheng Ding in "Codes from Difference Sets"(2015)), and partial geometric designs, and make an investigation into when a 2-adesign is a partial geometric design.
基金This work is partially supported by Natural Science Foundation of Hunan Province(No.2019JJ30030)Training Program for Excellent Young Innovators of Changsha(No.kql905052).
文摘In the study of(partial)difference sets and their generalizations in groups G,the most widely used method is to translate their definition into an equation over group ring Z[G]and to investigate this equation by applying complex representations of G.In this paper,we investigate the existence of(partial)difference sets in a different way.We project the group ring equations in Z[G]to Z[N]where N is a quotient group of G isomorphic to the additive group of a finite field,and then use polynomials over this finite field to derive some existence conditions.
基金supported by the National Natural Science Foundation of China(Nos.12071055,11971089 and 11731013).
文摘This note characterizes the set of Fréchet-differentiable points of the projection operator on a polyhedral set and the B-subdifferential of this projection operator at any point.
