Design, analysis, and implementation of a multiprecision polynomial rootfinder

We present the design, analysis, and implementation of an algorithm for the computation of any number of digits of the roots of a polynomial with complex coefficients. The real and the imaginary parts of the coefficients may be integer, rational, or floating point numbers represented with an arbitrary number of digits. The algorithm has been designed to deal also with numerically hard polynomials like those arising from the symbolic preprocessing of systems of polynomial equations, where the degree and the size of the coefficients are typically huge.The algorithm is based on an adaptive strategy which automatically exploits any specific feature of the input polynomial, like its sparsity or the conditioning of its roots, in order to speed up the computation. We introduce different concepts and tools suitably designed to arrive at an adaptive implementation, such as the concepts of root neighborhood, inclusion discs and some inclusion and conditioning theorems for their determination. The main engine for shrinking the inclusion discs is the simultaneous iteration method of Ehrlich–Aberth, complemented with a suitable technique for cluster analysis that is used for getting rid of the slow convergence in case of clustered or multiple roots.The algorithm, implemented in C, relies on the GNU multiprecision package GMP and allows many options. Counting, isolating and approximating all roots in a given set are the main goals that the algorithm provides. Automatic determination of multiplicities and the detection of real or imaginary roots can be selected as well. Polynomials having coefficients with a bounded precision may be processed too. Comparisons with the polynomial rootfinders of the packages Mathematica, Mapleand Pari, performed on a wide class of test polynomials show that our algorithm is generally much faster: in most cases the speedup factor is greater than 10 and, for certain polynomials, it is greater than 1000. The MPSolve package can be downloaded from the numeralgo library of netlib.     

Identification of p-Coumaric Acid and Ethyl p-Coumarate as the Main Phenolic Components of Hemp (Cannabis sativa L.) Roots

Hemp (Cannabis sativa L.) contains a variety of secondary metabolites, including cannabinoids, such as psychoactive (−)-trans-Δ⁹-tetrahydrocannabinol. The present study was conducted to identify the major phenolic components contained in hemp root, which has been relatively under-researched compared to other parts of hemp. The aqueous ethanol extract of hemp roots was fractionated into methylene chloride (MC), ethyl acetate (EA), and water (WT) fractions, and high-performance liquid chromatography with photodiode array detection (HPLC-DAD) analysis was performed. The main ultraviolet (UV)-absorbing phenolic compound contained in the EA fraction was identified as p-coumaric acid by comparing the retention time and UV absorption spectrum with a standard. Silica gel column chromatography was performed to isolate a hydrophobic derivative of p-coumaric acid contained in the MC fraction. Nuclear magnetic resonance (NMR) analysis identified the isolated compound as ethyl p-coumarate. For comparative purposes, ethyl p-coumarate was also chemically synthesized by the esterification reaction of p-coumaric acid. The content of p-coumaric acid and ethyl p-coumarate in the total extract of hemp root was estimated to be 2.61 mg g⁻¹ and 6.47 mg g⁻¹, respectively, by HPLC-DAD analysis. These values correspond to 84 mg Kg⁻¹ dry root and 216 mg Kg⁻¹ dry root, respectively. In conclusion, this study identified p-coumaric acid and ethyl p-coumarate as the main phenolic compounds contained in the hemp roots.     

用标准化变量方法简化介绍线性模型

用标准化变量方法,对线性回归模型的描述进行了简化,使其统计模型变得直观和容易理解.     

相伴于平方根算子的Hp空间

本文利用椭圆算子的平方根所生成的热核给出了Hp空间的极大函数刻画.     

(2,3)-Generation of SL(n,q).I. cases n=5,6,7

In this paper we prove that the groupws SL(n,q), q=p^m, are factors of the modular groups PSL(2,Z) When n=5,6,7 and P≠2, q≠9     

Crystal structures of mollugin and lucidin

Mollugin was isolated from the extract of Rubia tinctorum roots. Lucidin was obtained by semisynthesis from xanthopurpurin. Mollugin crystallises in space group P-1 (No. 2), a = 8.5857(7), b = 11.473(4), c = 15.024(1) Å, α = 77.64(2)^∘, β = 89.36(1)^∘, γ = 89.71(2)^∘, and V = 1445.5(5) ų, lucidin crystallises in space group P2₁/c (No.14), a = 16.800(6), b = 9.637(2), c = 7.073(7) Å, β = 98.01(5)^∘, and V = 1134(1) ų.     

Derivation Algebras of Toric Varieties

We study the Lie algebra of derivations of the coordinate ring of affine toric varieties defined by simplicial affine semigroups and prove the following results:Such toric varieties are uniquely determined by their Lie algebra if they are supposed to be Cohen–Macaulay of dimension 2 or Gorenstein of dimension =1.In the Cohen–Macaulay case, every automorphism of the Lie algebra is induced from a unique automorphism of the variety.Every derivation of the Lie algebra is inner.     

Dn Basic Hypergeometric Series

We study multiple series extensions of basic hypergeometric series related to the root system D_n. We make a small change in the notation used for C_n and D_n series to bring them closer to A_n series. This allows us to combine the three types of series, and get D_n extensions of the following classical summation and transformation theorems: The q-Pfaff-Saalschütz summation, Rogers' _{6 5} sum, the q-Gauss summation, q-Chu-Vandermonde summations, Watson's q-analogue of Whipple's transformation, and the q-Dougall summation theorem. We also define A_n and C_n extensions of the Rogers-Selberg function, and prove a reduction formula for both of them. This generalizes some work of Andrews. We use some techniques originally developed to study multiple basic hypergeometric series related to the root system A_n (U(n + 1) basic hypergeometric series).     

带强迫项的多时滞差分方程解的渐近性态

研究了一个带有强迫项的多时滞差分方程△x_n+sum from j=1 to m p_jx_(n-k_j)=r_n解的某种渐近性态.当这个方程的系数以及强迫项满足一定的约束条,其中p_j,r_n是实数,n,r_j是非负整数,λ_0是上述差分方程的特征方程的唯一实特征根,则表达式■λ_0~(-n)·x_n存在,并且给出了它的极限值.     

简单函数生成Weyl-Heisenberg框架的充要条件

设φ(x)=∑Nn=0anxE(x-n).当E是z -tile且a是一正有理数时,我们证明了{e2ximxφ(x-na):m,n∈z}成为框架的充要条件是多项式p(z)=∑Nn=0anzn无单位根.此结果推广了Casazza和Kalton的结果[3],并且给出了Weyl-Heisenberg框架与多项式根的相互关系.