We study zeros of the Jones polynomial and their distributions for torus knots and 2-bridge knots. We prove that e(2m+1)πi/2and e(2m+1)πi/4(m is a positive integer)can not be the zeros of Jones polynomial for torus ...We study zeros of the Jones polynomial and their distributions for torus knots and 2-bridge knots. We prove that e(2m+1)πi/2and e(2m+1)πi/4(m is a positive integer)can not be the zeros of Jones polynomial for torus knots T p,q by the knowledge of the trigonometric function. We elicit the normal form of Jones polynomials of the 2-bridge knot C(-2, 2, ···,(-1)r2) by the recursive form and discuss the distribution of their zeros.展开更多
An H（2）-move is a local move of a knot which is performed by adding a half-twisted band. It is known an H（2）-move is an unknotting operation. We define the H（2）-unknotting number of a knot K to be the minimum nu...An H（2）-move is a local move of a knot which is performed by adding a half-twisted band. It is known an H（2）-move is an unknotting operation. We define the H（2）-unknotting number of a knot K to be the minimum number of H（2）-moves needed to transform K into a trivial knot. We give several methods to estimate the H（2）-unknotting number of a knot. Then we give tables of H（2）-unknotting numbers of knots with up to 9 crossings.展开更多
Extension of a B-spline curve or surface is a useful function in a CAD system. This paper presents an algorithm for extending cubic B-spline curves or surfaces to one or more target points. To keep the extension curve...Extension of a B-spline curve or surface is a useful function in a CAD system. This paper presents an algorithm for extending cubic B-spline curves or surfaces to one or more target points. To keep the extension curve segment GC2-continuous with the original one, a family of cubic polynomial interpolation curves can be constructed. One curve is chosen as the solution from a sub-class of such a family by setting one GC2 parameter to be zero and determining the second GC2 parameter by mini- mizing the strain energy. To simplify the final curve representation, the extension segment is reparameterized to achieve C2-continuity with the given B-spline curve, and then knot removal from the curve is done. As a result, a sub-optimized solution subject to the given constraints and criteria is obtained. Additionally, new control points of the extension B-spline segment can be determined by solving lower triangular linear equations. Some computing examples for comparing our method and other methods are given.展开更多
The bushy root-2(brt-2)tomato mutant has twisting roots,and slower plant development.Here we used whole genome resequencing and genetic mapping to show that brt-2 is caused by a serine to cysteine(S75C)substitution in...The bushy root-2(brt-2)tomato mutant has twisting roots,and slower plant development.Here we used whole genome resequencing and genetic mapping to show that brt-2 is caused by a serine to cysteine(S75C)substitution in the DNA binding domain(DBD)of a heat shock factor class B(HsfB)encoded by SolycHsfB4a.This gene is orthologous to the Arabidopsis SCHIZORIZA gene,also known as AtHsfB4.The brt-2 phenotype is very similar to Arabidopsis lines in which the function of AtHsfB4 is altered:a proliferation of lateral root cap and root meristematic tissues,and a tendency for lateral root cap cells to easily separate.The brt-2 S75C mutation is unusual because all other reported amino acid substitutions in the highly conserved DBD of eukaryotic heat shock factors are dominant negative mutations,but brt-2 is recessive.We further show through reciprocal grafting that brt-2 exerts its effects predominantly through the root genotype even through BRT-2 is expressed at similar levels in both root and shoot meristems.Since AtHsfB4 is induced by root knot nematodes(RKN),and loss-of-function mutants of this gene are resistant to RKNs,BRT-2 could be a target gene for RKN resistance,an important trait in tomato rootstock breeding.Gene&accession numbers SolycHsfB4a-Solyc04g078770.展开更多
基金Supported by the National Science Foundation of China（11471151） Supported by Program for Liaoning Excellent Talents in University（LR2011031）
Acknowledgment The authors would like to thank the referees for kind suggestions and many useful comments
文摘We study zeros of the Jones polynomial and their distributions for torus knots and 2-bridge knots. We prove that e(2m+1)πi/2and e(2m+1)πi/4(m is a positive integer)can not be the zeros of Jones polynomial for torus knots T p,q by the knowledge of the trigonometric function. We elicit the normal form of Jones polynomials of the 2-bridge knot C(-2, 2, ···,(-1)r2) by the recursive form and discuss the distribution of their zeros.
基金supported by Grant-in-Aid (20540079) for Scientific Research (C),Japan Society for the Promotion of Science
文摘An H（2）-move is a local move of a knot which is performed by adding a half-twisted band. It is known an H（2）-move is an unknotting operation. We define the H（2）-unknotting number of a knot K to be the minimum number of H（2）-moves needed to transform K into a trivial knot. We give several methods to estimate the H（2）-unknotting number of a knot. Then we give tables of H（2）-unknotting numbers of knots with up to 9 crossings.
文摘Extension of a B-spline curve or surface is a useful function in a CAD system. This paper presents an algorithm for extending cubic B-spline curves or surfaces to one or more target points. To keep the extension curve segment GC2-continuous with the original one, a family of cubic polynomial interpolation curves can be constructed. One curve is chosen as the solution from a sub-class of such a family by setting one GC2 parameter to be zero and determining the second GC2 parameter by mini- mizing the strain energy. To simplify the final curve representation, the extension segment is reparameterized to achieve C2-continuity with the given B-spline curve, and then knot removal from the curve is done. As a result, a sub-optimized solution subject to the given constraints and criteria is obtained. Additionally, new control points of the extension B-spline segment can be determined by solving lower triangular linear equations. Some computing examples for comparing our method and other methods are given.
基金The research was supported by BBSRC-UKRI fundingthe RootLINK(BB/L01954X/1)project focused on the“Understanding the Genetic Basis of Traits for Rootstock Improvement in Vegetable Crops”.
文摘The bushy root-2(brt-2)tomato mutant has twisting roots,and slower plant development.Here we used whole genome resequencing and genetic mapping to show that brt-2 is caused by a serine to cysteine(S75C)substitution in the DNA binding domain(DBD)of a heat shock factor class B(HsfB)encoded by SolycHsfB4a.This gene is orthologous to the Arabidopsis SCHIZORIZA gene,also known as AtHsfB4.The brt-2 phenotype is very similar to Arabidopsis lines in which the function of AtHsfB4 is altered:a proliferation of lateral root cap and root meristematic tissues,and a tendency for lateral root cap cells to easily separate.The brt-2 S75C mutation is unusual because all other reported amino acid substitutions in the highly conserved DBD of eukaryotic heat shock factors are dominant negative mutations,but brt-2 is recessive.We further show through reciprocal grafting that brt-2 exerts its effects predominantly through the root genotype even through BRT-2 is expressed at similar levels in both root and shoot meristems.Since AtHsfB4 is induced by root knot nematodes(RKN),and loss-of-function mutants of this gene are resistant to RKNs,BRT-2 could be a target gene for RKN resistance,an important trait in tomato rootstock breeding.Gene&accession numbers SolycHsfB4a-Solyc04g078770.
