How to solve the polynomial ordinary differential equations

This paper discussed how to solve the polynomial ordinary differential equations. At first, we construct the theory of the linear equations about the unknown one variable functions with constant coefficients. Secondly, we use this theory to convert the polynomial ordinary differential equations into the simultaneous first order linear ordinary differential equations with constant coefficients and quadratic equations. Thirdly, we work out the general solution of the polynomial ordinary differential equations which is no longer concerned with the differential. Finally, we discuss the necessary and sufficient condition of the existence of the solution.     

A bijective enumeration of labeled trees with given indegree sequence

For a labeled tree on the vertex set {1,2,…,n}, the local direction of each edge (ij) is from i to j if i<j. For a rooted tree, there is also a natural global direction of edges towards the root. The number of edges pointing to a vertex is called its indegree. Thus the local (resp. global) indegree sequence λ=e₁¹e₂²… of a tree on the vertex set {1,2,…,n} is a partition of n−1. We construct a bijection from (unrooted) trees to rooted trees such that the local indegree sequence of a (unrooted) tree equals the global indegree sequence of the corresponding rooted tree. Combining with a Prüfer-like code for rooted labeled trees, we obtain a bijective proof of a recent conjecture by Cotterill and also solve two open problems proposed by Du and Yin. We also prove a q-multisum binomial coefficient identity which confirms another conjecture of Cotterill in a very special case.     

Theoretical Study on Structural Stability and Nonlinear Optical Properties of Non-conjugated Carbon and Nitrogen Diradical Molecules

The geometrical structures and stability of non-conjugated C5H10 and C3H8N2 singlet and triplet diradical molecules have been investigated at the UCCSD/6-311g** level. The effects of molecular structure, radical position, amount of Hartree Hork (HF) exchange and spin multiplicity on the nonlinear optical (NLO) coefficients have been also investigated. The reliable UCCSD results show that the triplets of all diradical molecules are more stable compared to their singlet analogues. In addition, the α s and β tot values of C5H10 and C3H8N2 triplet diradical mo-lecules have been investigated by the UBHandHLYP, UB3LYP, UBLYP, UHF and UCCSD methods. The investigation shows that the variations in α s and β tot values are closely connected to the amount of HF exchange. The increasing amount of HF exchange results in monotonic decreases in α s and β tot values, while the α s and β totvalues of singlet diradical molecules and the γ s of C5H10 and C3H8N2 singlet and triplet diradical molecules have been studied by the UBHandHLYP method. The results illustrate that the NLO coefficients for our studied non-conjugated carbon and nitrogen diradical species can be tuned by molecular structure, radical position and spin multiplicity, which are very significant for designing NLO materials.     

Landscape of wave focusing and localization at low frequencies

High-contrast scattering problems are special among classical wave systems as they allow for strong wave focusing and localization at low frequencies. We use an asymptotic framework to develop a landscape theory for high-contrast systems that resonate in a subwavelength regime. Our from-first-principles asymptotic analysis yields a characterization in terms of the generalized capacitance matrix, giving a discrete approximation of the three-dimensional scattering problem. We develop landscape theory for the generalized capacitance matrix and use it to predict the positions of three-dimensional wave focusing and localization in random and non-periodic systems of subwavelength resonators.     

自适应斜Z字形采样Hadamard单像素成像

单像素成像技术在成像速度和质量上已经取得一定的进展,然而,如何在保证成像质量的前提下提升成像速度仍然是一个待解决的问题。对Hadamard系数矩阵中每条斜列上的系数绝对值求取平均值后,发现其具有迅速减小的特征,基于此发现提出了一种自适应斜Z字形采样方法,以实现预期的成像质量和速度提升。实验结果表明,该方法可以有效减少采样次数与数据采集量,兼顾成像质量的同时提升了成像速度。在相同的采样率下,其成像质量优于顺序采样的单像素成像。     

Perturbation Theory of Polymer Chains Revisited, 2—Intermolecular Excluded Volume of Random and Self‐Avoiding Walks

The intermolecular compatibility of 10⁷–10⁸ independent pair configurations of linear chains (chain length 64 ≤ N ≤ 8192) with and without intramolecular interaction on various lattices is checked and the intermolecular excluded volume u evaluated. From the (average) number of possible i‐tuples of overlaps in the pair configurations the parameters C_k of the perturbation theory are calculated. Differences between the various lattice types at finite N vanish completely for N → ∞. While the theoretical values of C₁ and C₂ were recovered within very narrow limits for chains without intramolecular interaction, markedly higher figures apply to self‐avoiding walks. The excluded volume u itself and higher C_k parameters are called upon to test several classical and renormalized perturbation theories.

     

Abraham model correlations describing the solubilising ability of peanut oil

Experimental data have been compiled from the published literature for the logarithm of the gas-to-peanut oil partition coefficient, log K, and for the logarithm of the water-to-peanut oil partition coefficient, log P, for a series of 93 solutes having a wide range of solute polarities and hydrogen-bonding character. The two sets of partition coefficients were correlated with the Abraham solvation parameter model. The derived Abraham model correlations described the experimental log K and log P values within standard deviations of 0.13 and 0.14 log units, respectively. Principal component analysis was used to compare the solvent properties of peanut oil to four other natural oils (olive oil, soybean oil, triolein and oleyl alcohol). Peanut oil was found to be close to olive oil and soybean oil, and quite far away from triolein and oleyl alcohol.