Let f（q） = arq^r ＋…＋asqs, with ar ≠ 0 and as ≠ 0, be a real polynomial. It is a palindromic polynomial of darga n if r＋s= n and ar＋i =as-i for all i. Polynomials of darga n form a linear subspace Pn（q） of R...Let f（q） = arq^r ＋…＋asqs, with ar ≠ 0 and as ≠ 0, be a real polynomial. It is a palindromic polynomial of darga n if r＋s= n and ar＋i =as-i for all i. Polynomials of darga n form a linear subspace Pn（q） of R（q）n＋l of dimension [n/2] ＋ 1. We give transition matrices between two bases {q^j（1 ＋ q ＋… q^n-2j）}, {q^j（1 ＋ q）^n-2j } and the standard basis {q^j（1 ＋ q^n-2j）} of Pn （q）. We present some characterizations and sufficient conditions for palindromic polynomials that can be expressed in terms of these two bases with nonnegative coefficients. We also point out the link between such polynomials and rank-generating functions of posets.展开更多
Double-crossover-like(DXL)molecules are a series of DNA motifs containing two strands with identical or different sequences.These homo-or hetero-dimers can further polymerize into bulk structures through specific hydr...Double-crossover-like(DXL)molecules are a series of DNA motifs containing two strands with identical or different sequences.These homo-or hetero-dimers can further polymerize into bulk structures through specific hydrogen bonding between sticky ends.DXL molecules have high designability,predictivity and sequence robustness;and their supramolecular polymerization products would easily achieve controllable morphology.In addition,among all available DNA nanomotifs,DXL molecules are small in size so that the cost of DXL-based nanostructures is low.These properties together make DXL-based nanostructures good candidates for patterning,templating,information and matter storage,etc.Herein,we will discuss DXL motifs in terms of the detailed molecular design,and their supramolecular polymerization in various dimensions,and related applications.展开更多
基金Supported by National Natural Science Foundation of China(Grant Nos.11071030,11371078)the Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.20110041110039)
文摘Let f（q） = arq^r ＋…＋asqs, with ar ≠ 0 and as ≠ 0, be a real polynomial. It is a palindromic polynomial of darga n if r＋s= n and ar＋i =as-i for all i. Polynomials of darga n form a linear subspace Pn（q） of R（q）n＋l of dimension [n/2] ＋ 1. We give transition matrices between two bases {q^j（1 ＋ q ＋… q^n-2j）}, {q^j（1 ＋ q）^n-2j } and the standard basis {q^j（1 ＋ q^n-2j）} of Pn （q）. We present some characterizations and sufficient conditions for palindromic polynomials that can be expressed in terms of these two bases with nonnegative coefficients. We also point out the link between such polynomials and rank-generating functions of posets.
基金financially supported by NSF(Nos.CCF-2107393 and CCMI-2025187 to C.M.).
文摘Double-crossover-like(DXL)molecules are a series of DNA motifs containing two strands with identical or different sequences.These homo-or hetero-dimers can further polymerize into bulk structures through specific hydrogen bonding between sticky ends.DXL molecules have high designability,predictivity and sequence robustness;and their supramolecular polymerization products would easily achieve controllable morphology.In addition,among all available DNA nanomotifs,DXL molecules are small in size so that the cost of DXL-based nanostructures is low.These properties together make DXL-based nanostructures good candidates for patterning,templating,information and matter storage,etc.Herein,we will discuss DXL motifs in terms of the detailed molecular design,and their supramolecular polymerization in various dimensions,and related applications.
