四次函数实零点的完全判据和正定条件

本文给出了四次函数实零点的完全判据和正定条件,这一结果在研究多项式系统的奇点分析和分支问题时给出了可以实用的判据．     

New light on Hensel''s lemma

The historical development of Hensel's lemma is briefly discussed (Section 1). Using Newton polygons, a simple proof of a general Hensel's lemma for separable polynomials over Henselian fields is given (Section 3). For polynomials over algebraically closed, valued fields, best possible results on continuity of roots (Section 4) and continuity of factors (Section 6) are demonstrated. Using this and a general Krasner's lemma (Section 7), we give a short proof of a general Hensel's lemma and show that it is, in a certain sense, best possible (Section 8). All valuations here are non-Archimedean and of arbitrary rank. The article is practically self-contained.     

Studies on mass production of transformed Panax ginseng hairy roots in bioreactor

The growth properties of Panax ginseng hairy roots transformed by Agrobacterium rhizogenes were compared between flask and aerated column or stirred bioreactor. In flask cultures, sucrose, initially 30 g/L, was nearly exhausted after 45 d of culture. The pH of the medium dropped from 5.5 to 4.96 after 10 d, but afterward it gradually increased to 6.4. After 45 d, hairy roots grew about 16-folds. The growth rate of hairy roots in air-bubble column or stirred bioreactor cultures was 1.13 (1.11) to 1.23 (1.20) g fresh wt (dry wt)/(g of cells·d), respectively. For both bioreactors, growth was about three times as high as in the flask cultivation.     

16-Demethoxymethyllycaconitine,a new norditerpenoid alkaloid from Delphinium cuneatum

A new norditerpenoid alkaloid has been isolated from the roots of Delphinium cuneatum. On the basis of ¹H, ¹³C NMR, IR, and mass spectra, the structure of the alkaloid was established as 1,6,14-trimethoxy-7,8-dihydroxy-18-(2-methyl)succinylanthranoyloxyaconane (16-demethoxymethyllycaconitine). The roots of Delphinium cuneatum also contain the known alkaloid methyllycaconitine and N,N"-di(methoxycarbonyl)-3,4-diaminotoluene.     

Datura stramonium hairy roots tropane alkaloid content as a response to changes in Gamborg’s B5 medium

Hairy roots are less susceptible to manipulation by changes in medium composition than callus and cell suspension cultures. However, when the individual components of the Gamborg’s B₅ medium were modified one by one, several of them modified the yield of root cultures. Nitrogen (N) source changes, the absence of phosphorus (P), calcium (Ca), and cobalt (Co), and the increase in the sucrose content of the medium increased the level of alkaloids inDatura stramonium hairy root cultures. The use of ammoniun as N source provoked the alkaloid excretion into the medium. The addition of different auxins inhibited the alkaloid accumulation, but the addition of abscisic acid (ABA) did not change the alkaloid content of the cultures. The use of MeJa increased the content of hyoscyamine to 100%.     

A Note on twin primes and the modular ring Z 6

The modular ring ?₆ defines integers via (6r _i + (i ? 3)) where i is the class and r _i the row when tabulated in an array. Since only Classes 2₆ and 4₆ contain odd primes, this modular ring is ideally suited to the analysis of twin primes. The calculations are facilitated by the use of the right-end digit (RED) technique.     

Analysis of the multiple roots of the Lundberg fundamental equation in the PH (n) risk model

In the study of the Sparre Andersen risk model with phase‐type (n) inter‐claim times (PH (n) risk model), the distinct roots of the Lundberg fundamental equation in the right half of the complex plane and the linear independence of the eigenvectors related to the Lundberg matrix L_δ(s) play important roles. In this paper, we study the case where the Lundberg fundamental equation has multiple roots or the corresponding eigenvectors are linearly dependent in the PH (n) risk model. We show that the multiple roots of the Lundberg fundamental equation det[L_δ(s)] = 0 can be approximated by the distinct roots of the generalized Lundberg equation introduced in this paper and that the linearly dependent eigenvectors can be approximated by the corresponding linearly independent ones as well. Using this result we derive the expressions for the Gerber–Shiu penalty function. Two special cases of the generalized Erlang(n) risk model and a Coxian(3) risk model are discussed in detail, which illustrate the applicability of main results. Finally, we consider the PH(2) risk model and conclude that the roots of the Lundberg fundamental equation in the right half of the complex plane are distinct and that the corresponding eigenvectors are linearly independent. Copyright © 2011 John Wiley & Sons, Ltd.     

On the structure of split Leibniz triple systems

We study the structure of arbitrary split Leibniz triple systems with a coherent 0-root space. By developing techniques of connections of roots for this kind of triple systems, under certain conditions, in the case of T being of maximal length, the simplicity of the Leibniz triple systems is characterized.