Adjustment Method of S-λ Curves and Bezier Curves with 2 Shape Parameters

Fan和Zeng在[1]中提出了S-λ基函数和S-λ曲线的概念，并指出可以通过扰动生成函 数系数来对S-λ曲线形状进行调整。本文在此基础上进一步给出了当对生成函数单一系 数进行扰动时，扰动后的曲线与原曲线之间的相应关系以及两者到控制顶点的距离之 间的关系。对两类特殊控制多边形结构的三次Bezier曲线受到扰动后曲线变化情况做了 相应的分析，并给出了曲线变化的趋势。 另一方面，针对经典的Bezier曲线在控制顶点给定，曲线就唯一固定，缺乏进一步 调整曲线形状能力的这一状况，本文给出了一种带两个形状参数的Bezier曲线。这种带 形状参数的Bezier曲线保留了经典Bezier曲线的...In [1], Fan and Zeng introduced S-λ bases and S-λ curves, they also pointed out that we can adjust the shape of S-λ curves through disturbing the coefficients of generating function. In this paper, we do a further research on this method and give some conclusions about the relationships between the after-disturbing curve and the normal curve and the function of distance between the two curve and a...学位：理学硕士院系专业：数学科学学院_概率论与数理统计学号：1902010115250

Adjustment Method of S-λ Curves and Bezier Curves with 2 Shape Parameters

Fan和Zeng在[1]中提出了S-λ基函数和S-λ曲线的概念，并指出可以通过扰动生成函 数系数来对S-λ曲线形状进行调整。本文在此基础上进一步给出了当对生成函数单一系 数进行扰动时，扰动后的曲线与原曲线之间的相应关系以及两者到控制顶点的距离之 间的关系。对两类特殊控制多边形结构的三次Bezier曲线受到扰动后曲线变化情况做了 相应的分析，并给出了曲线变化的趋势。 另一方面，针对经典的Bezier曲线在控制顶点给定，曲线就唯一固定，缺乏进一步 调整曲线形状能力的这一状况，本文给出了一种带两个形状参数的Bezier曲线。这种带 形状参数的Bezier曲线保留了经典Bezier曲线的...In [1], Fan and Zeng introduced S-λ bases and S-λ curves, they also pointed out that we can adjust the shape of S-λ curves through disturbing the coefficients of generating function. In this paper, we do a further research on this method and give some conclusions about the relationships between the after-disturbing curve and the normal curve and the function of distance between the two curve and a...学位：理学硕士院系专业：数学科学学院_概率论与数理统计学号：1902010115250

Theoretical investigation of finite size effects at DNA melting

We investigated how the finiteness of the length of the sequence affects the phase transition that takes place at DNA melting temperature. For this purpose, we modified the Transfer Integral method to adapt it to the calculation of both extensive (partition function, entropy, specific heat, etc) and non-extensive (order parameter and correlation length) thermodynamic quantities of finite sequences with open boundary conditions, and applied the modified procedure to two different dynamical models. We showed that rounding of the transition clearly takes place when the length of the sequence is decreased. We also performed a finite-size scaling analysis of the two models and showed that the singular part of the free energy can indeed be expressed in terms of an homogeneous function. However, both the correlation length and the average separation between paired bases diverge at the melting transition, so that it is no longer clear to which of these two quantities the length of the system should be compared. Moreover, Josephson's identity is satisfied for none of the investigated models, so that the derivation of the characteristic exponents which appear, for example, in the expression of the specific heat, requires some care