Note to Rubin's Estimation of the Upper Bound to the Mean‐Square Length of a Polymer Chain

1969 ◽  
Vol 51 (5) ◽  
pp. 2281-2282
Author(s):  
Antonín Kyselka
Keyword(s):  
Polymer Chain ◽  
Upper Bound ◽  
Mean Square ◽  
The Mean

Effect on Polymer Chain Structure Caused by the Molar Fraction of 4-Hydroxybutyrate in Poly(3-hydroxybutyrate- co -4-hydroxybutyrate)

2012 ◽  
Vol 602-604 ◽  
pp. 776-780
Author(s):  
Zhi Qiang Li ◽  
Mei Li ◽  
Wei Jia Fan
Keyword(s):  
Molecular Weight ◽  
Intrinsic Viscosity ◽  
Polymer Chain ◽  
Molar Fraction ◽  
Chain Structure ◽  
The Other ◽  
Mean Square ◽  
Polarizing Microscope ◽  
Other Hand ◽  
The Mean

Poly(3-hydroxybutyrate-co-4-hydroxybutyrate)copolymer [P(3HB-co-4HB)] is a kind of biodegradable high molecular polymer produced by bioaccumulation. Because of the good biodegradability and biocompatibility, P(3HB-co-4HB)s have attracted wide attention . At first, the intrinsic viscosity[η] in good solvent of P(3HB-co-4HB) s with varying contents of 4HB was investigated in different temperature. Second, observed the changes of crystallization gathered state caused by the varying contents of 4HB by polarizing microscope. The results show that to the P(3HB-co-4HB)s in same molecular weight, the intrinsic viscosity[η] in good solvent barely changes when the mole fractions of 4HB increase. On the other hand, the mean square end to end distances[0] of macromolecular flexible chains increase with the mole fractions of 4HB. At the same time, the states of aggregation change from spherulites to dendrites. In this investigation, we discuss the reasons of the differences in depth.

Computer Simulation on Conformation Properties of Polymer Chain

2013 ◽  
Vol 341-342 ◽  
pp. 195-198
Author(s):  
Lin Lin Cui ◽  
Hua Nan Guan
Keyword(s):  
Polymer Chain ◽  
Volume Fraction ◽  
Solvent Molecule ◽  
Radius Of Gyration ◽  
Mean Square ◽  
Chain Segment ◽  
End Distance ◽  
The Mean ◽  
Multiple Chain ◽  
Linear Polymer Chain

The author adopts Monte Carlo compute method to simulate the linear polymer chain lattice model in multiple chain systems of chain lengthn=20, 50, 100 while the volume fraction Φ=0.125, and makes a research on the variational situation of the size (measured with the mean-square end-to-end distance <R2> and the mean-square radius of gyration <S2>), shape (measured with the mean asphericity factor <A>) with changing of the interaction energy between solvent molecule and polymer chain segment molecule εPS. Results indicate <R2>, <S2> and <A> have the changing rules that they become small with the increase of the εPS

Monte Carlo Simulation on Conformation Properties of Polymer Multiple Chain System

2013 ◽  
Vol 734-737 ◽  
pp. 3141-3144
Author(s):  
Lin Lin Cui ◽  
Hua Nan Guan
Keyword(s):  
Monte Carlo ◽  
Polymer Chain ◽  
Volume Fraction ◽  
Solvent Molecule ◽  
Radius Of Gyration ◽  
Mean Square ◽  
End Distance ◽  
The Mean ◽  
Multiple Chain ◽  
Linear Polymer Chain

The author adopts Monte Carlo compute method to simulate the linear polymer chain lattice model in multiple chain systems of different volume fraction Φ while chain lengthn=50, and makes a research on the variational situation of the size (measured with the mean-square end-to-end distance <R2> and the mean-square radius of gyration <S2>), shape (measured with the mean asphericity factor ) with changing of the interaction energy between solvent molecule and polymer chain segment moleculeεPS. Results indicate <R2>, <S2> and have the changing rules that they become small with the increase of theεPS.

Statistical properties of a polymer chain in the two-parametor approximation

1981 ◽  
Vol 376 (1766) ◽  
pp. 361-375 ◽  
Keyword(s):  
Polymer Chain ◽  
Numerical Data ◽  
Ring Closure ◽  
Radius Of Gyration ◽  
Mean Square ◽  
Virial Series ◽  
The Mean ◽  
Diagrammatic Method ◽  
Mean Square Distance ◽  
Self Avoiding Walks

The diagrammatic method developed in a previous paper is used to derive two terms of the virial expansion for the mean square distance from the origin, the mean square radius of gyration, and the probability of ring closure for a lattice model of a simple polymer chain. It is found that the logarithmic terms in the partition function cancel in the virial series for these universal quantities. The universality hypothesis is tested with the numerical data for self-avoiding walks (s.a.w.) for different lattices. The virial series is combined with the s.a.w. results to provide formulae for the expansion factor as a function of the excluded volume.

On Riesz means of the coefficients of the Rankin–Selberg series

1999 ◽  
Vol 127 (1) ◽  
pp. 117-131 ◽  
Author(s):  
ALEKSANDAR IVIĆ ◽  
KOHJI MATSUMOTO ◽  
YOSHIO TANIGAWA
Keyword(s):  
Asymptotic Formula ◽  
Upper Bound ◽  
Error Term ◽  
Weighted Sum ◽  
Mean Square ◽  
Riesz Means ◽  
The Mean ◽  
Voronoi Formula

We study Δ(x; ϕ), the error term in the asymptotic formula for [sum ]n[les ]xcn, where the cns are generated by the Rankin–Selberg series. Our main tools are Voronoï-type formulae. First we reduce the evaluation of Δ(x; ϕ) to that of Δ1(x; ϕ), the error term of the weighted sum [sum ]n[les ]x(x−n)cn. Then we prove an upper bound and a sharp mean square formula for Δ1(x; ϕ), by applying the Voronoï formula of Meurman's type. We also prove that an improvement of the error term in the mean square formula would imply an improvement of the upper bound of Δ(x; ϕ). Some other related topics are also discussed.

scholarly journals Mean-square upper bound of Hecke $L$-functions on the critical line.

2003 ◽  
Vol Volume 26 ◽  
Author(s):  
A Sankaranarayanan
Keyword(s):  
Cusp Form ◽  
Upper Bound ◽  
Critical Line ◽  
Congruence Subgroup ◽  
Mean Square ◽  
Absolute Value ◽  
International Audience ◽  
The Mean ◽  
The Absolute

International audience We prove the upper bound for the mean-square of the absolute value of the Hecke $L$-functions (attached to a holomorphic cusp form) defined for the congruence subgroup $\Gamma_0 (N)$ on the critical line uniformly with respect to its conductor $N$.

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