UNDERSTANDING SCHUBERT’S BOOK(Ⅱ)

In this paper,we give rigorous justification of the ideas put forward in§20,Chapter 4 of Schubert’s book;a section that deals with the enumeration of conics in space.In that section,Schubert introduced two degenerate conditions about conics,i.e.,the double line and the two intersection lines.Using these two degenerate conditions,he obtained all relations regarding the following three conditions:conics whose planes pass through a given point,conics intersecting with a given line,and conics which are tangent to a given plane.We use the language of blow-ups to rigorously treat the two degenerate conditions and prove all formulas about degenerate conditions stemming from Schubert’s idea.

UNDERSTANDING SCHUBERT’S BOOK(Ⅰ)

Hilbert Problem 15 required an understanding of Schubert’s book[1],both its methods and its results.In this paper,following his idea,we prove that the formulas in§6,§7,§10,about the incidence of points,lines and planes,are all correct.As an application,we prove formulas 8 and 9 in§12,which are frequently used in his book.

UNDERSTANDING SCHUBERT'S BOOK(Ⅲ)

In§13 of Schubert’s famous book on enumerative geometry,he provided a few formulas called coincidence formulas,which deal with coincidence points where a pair of points coincide.These formulas play an important role in his method.As an application,Schubert utilized these formulas to give a second method for calculating the number of planar curves in a one dimensional system that are tangent to a given planar curve.In this paper,we give proofs for these formulas and justify his application to planar curves in the language of modern algebraic geometry.We also prove that curves that are tangent to a given planar curve is actually a condition in the space of planar curves and other relevant issues.

Preface

Editors-in-Chief： Gui-Qiang Chen, Banghe Li and Xiping Zhu This special issue of Acta Mathematica Scientia is dedicated to the memory of Professor Xiaqi Ding, a former Editor-in-Chief of this journal, on the occasion of his ninetieth anniversary.

A much better replacement of the Michaelis—Menten equation and its application

Michaelis-Menten equation is a basic equation of enzyme kinetics and gives acceptable approximations of real chemical reaction processes.Analyzing the derivation of this equation yields the fact that its good performance of approximating real reaction processes is due to Michaelis-Menten curve(8).This curve is derived from Quasi-Steady-State Assumption(QSSA),which has been proved always true and called Quasi-Steady-State Law by Banghe Li et al.[Quasi-steady state laws in enzyme kinetics,J.Phys.Chem.A 112(11)(2008)2311-2321].Here,we found a polynomial equation with total degree of four A(S,E)= 0(14),which gives more accurate approximation of the reaction process in two aspects:during the quasi-steady-state of the reaction,Michaelis-Menten curve approximates the reaction well,while our equation A(S,E)= 0 gives better approximation;near the end of the reaction,our equation approaches the end of the reaction with a tangent line the same to that of the reaction process trajectory simulated by mass action,while MicheielisMenten curve does not.In addition,our equation A(S,E)= 0 differs to Michaelis-Menten curve less than the order of 1/S^3 as S approaches +∞.By considering the above merits of A(S,E)= 0,we suggest it as a replacement of Michaelis-Menten curve.Intuitively,this new equation is more complex and harder to understand.But,just because of its complexity,it provides more information about the rate constants than Michaelis-Menten curve does.Finally,we get a better replacement of the Michaelis-Menten equation by combing 4(S,E)= 0 and the equation dP/dt = k2C(t).

MINIMUM HAUSDORFF DISTANCE UNDER RIGID MOTIONS AND COMPARISON OF PROTEIN STRUCTURES

Hausdorff distance between two compact sets, defined as the maximum distance from a point of one set to another set, has many application in computer science. It is a good measure for the similarity of two sets. This paper proves that the shape distance between two compact sets in R^n defined by nfinimum Hausdorff distance under rigid motions is a distance. The authors introduce similarity comparison problems in protein science, and propose that this measure may have good application to comparison of protein structure as well. For calculation of this distance, the authors give one dimensional formulas for problems （2, n）, （3, 3）, and （3, 4）. These formulas can reduce time needed for solving these problems. The authors did some data, this formula can reduce time needed to one As n increases, it would save more time. numerical experiments for （2, n）. On these sets of fifteenth of the best algorithms known on average.