With the help of the classical Abel’s lemma on summation by parts and algorithm of q-hypergeometric summations, we deal with the summation, which can be written as multiplication of a q-hypergeometric term and q-harm...With the help of the classical Abel’s lemma on summation by parts and algorithm of q-hypergeometric summations, we deal with the summation, which can be written as multiplication of a q-hypergeometric term and q-harmonic numbers. This enables us to construct and prove identities on q-harmonic numbers. Several examples are also given.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
In this paper, we define the generalized r-Whitney numbers of the first and second kind. Moreover, we drive the generalized Whitney numbers of the first and second kind. The recurrence relations and the generating fun...In this paper, we define the generalized r-Whitney numbers of the first and second kind. Moreover, we drive the generalized Whitney numbers of the first and second kind. The recurrence relations and the generating functions of these numbers are derived. The relations between these numbers and generalized Stirling numbers of the first and second kind are deduced. Furthermore, some special cases are given. Finally, matrix representation of the relations between Whitney and Stirling numbers is given.展开更多
BACKGROUND Copy number variation(CNV)has become widely recognized in recent years due to the extensive use of gene screening in developmental disorders and epilepsy research.1q21.1 microduplication syndrome is a rare ...BACKGROUND Copy number variation(CNV)has become widely recognized in recent years due to the extensive use of gene screening in developmental disorders and epilepsy research.1q21.1 microduplication syndrome is a rare CNV disease that can manifest as multiple congenital developmental disorders,autism spectrum disorders,congenital malformations,and congenital heart defects with genetic heterogeneity.CASE SUMMARY We reported a pediatric patient with 1q21.1 microduplication syndrome,and carried out a literature review to determine the correlation between 1q21.1microduplication and its phenotypes.We summarized the patient’s medical history and clinical symptoms,and extracted genomic DNA from the patient,her parents,elder brother,and sister.The patient was an 8-mo-old girl who was hospitalized for recurrent convulsions over a 2-mo period.Whole exon sequencing and whole genome low-depth sequencing(CNV-seq)were then performed.Whole exon sequencing detected a 1.58-Mb duplication in the CHR1:145883867-147465312 region,which was located in the 1q21.1 region.Family analysis showed that the pathogenetic duplication fragment,which was also detected in her elder brother’s DNA originated from the mother.CONCLUSION Whole exon sequencing combined with quantitative polymerase chain reaction can provide an accurate molecular diagnosis in children with 1q21.1 microduplication syndrome,which is of great significance for genetic counseling and early intervention.展开更多
文摘With the help of the classical Abel’s lemma on summation by parts and algorithm of q-hypergeometric summations, we deal with the summation, which can be written as multiplication of a q-hypergeometric term and q-harmonic numbers. This enables us to construct and prove identities on q-harmonic numbers. Several examples are also given.
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In this paper, we define the generalized r-Whitney numbers of the first and second kind. Moreover, we drive the generalized Whitney numbers of the first and second kind. The recurrence relations and the generating functions of these numbers are derived. The relations between these numbers and generalized Stirling numbers of the first and second kind are deduced. Furthermore, some special cases are given. Finally, matrix representation of the relations between Whitney and Stirling numbers is given.
文摘BACKGROUND Copy number variation(CNV)has become widely recognized in recent years due to the extensive use of gene screening in developmental disorders and epilepsy research.1q21.1 microduplication syndrome is a rare CNV disease that can manifest as multiple congenital developmental disorders,autism spectrum disorders,congenital malformations,and congenital heart defects with genetic heterogeneity.CASE SUMMARY We reported a pediatric patient with 1q21.1 microduplication syndrome,and carried out a literature review to determine the correlation between 1q21.1microduplication and its phenotypes.We summarized the patient’s medical history and clinical symptoms,and extracted genomic DNA from the patient,her parents,elder brother,and sister.The patient was an 8-mo-old girl who was hospitalized for recurrent convulsions over a 2-mo period.Whole exon sequencing and whole genome low-depth sequencing(CNV-seq)were then performed.Whole exon sequencing detected a 1.58-Mb duplication in the CHR1:145883867-147465312 region,which was located in the 1q21.1 region.Family analysis showed that the pathogenetic duplication fragment,which was also detected in her elder brother’s DNA originated from the mother.CONCLUSION Whole exon sequencing combined with quantitative polymerase chain reaction can provide an accurate molecular diagnosis in children with 1q21.1 microduplication syndrome,which is of great significance for genetic counseling and early intervention.
