The Sixth Painlevé Equation as Similarity Reduction of $${\widehat{\mathfrak{gl}}_3}$$ Generalized Drinfel’d–Sokolov Hierarchy

Scaling symmetry of -type Drinfel’d–Sokolov hierarchy is investigated. Applying similarity reduction to the hierarchy, one can obtain the Schlesinger equation with (n + 1) regular singularities. Especially in the case of n = 3, the hierarchy contains the three-wave resonant system and the similarity reduction gives the generic case of the Painlevé VI equation. We also discuss Weyl group symmetry of the hierarchy.      

Conditions of synthesis for PpIX

Photodynamic therapy (PDT) is an effective treat-ment for cancer[1-3].In PDT treatment ,a tumor-locali-zing photosensitizer is administered,and sti mulated bylight with special wavelengthtoinitiate a chemical reac-tionleadingto the death of tumor cells .I…     

New Double Wronskian Solutions of the Whitham-Broer-Kaup System: Asymptotic Analysis and Resonant Soliton Interactions

In this paper, by the Darboux transformation together with the Wronskian technique, we construct new double Wronskian solutions for the Whitham-Broer-Kaup (WBK) system. Some new determinant identities are developed in the verification of the solutions. Based on analyzing the asymptotic behavior of new double Wronskian functions as t → ±∞, we make a complete characterization of asymptotic solitons for the non-singular, non-trivial and irreducible soliton solutions. It turns out that the solutions are the linear superposition of two fully-resonant multi-soliton configurations, in each of which the amplitudes, velocities and numbers of asymptotic solitons are in general not equal as t → ±∞. To illustrate, we present the figures for several examples of soliton interactions occurring in the WBK system.     

Isogroups of differential ideals of vector-valued differential forms: Application to partial differential equations

An older geometric technique for the study of invariance groups of partial differential equations, originally proposed by one of the authors and F. B. Estabrook, is generalized and extended to problems involving exterior equations for vector-valued or Lie algebra-valued exterior differential forms. Use of the method is demonstrated in the study of the symmetry groups of the two-dimensional Dirac equation and the full Yang-Mills free-field equations in Minkowski spacetime.     

一个基本的Hilbert型不等式

本文研究了一个新的、基本的不含参量的Hilbert型不等式.应用改进的Euler-Maclaurin求和公式以估算权系数,证明了其常数系数为最佳值以及此不等式的等价式.     

Effective computation of periodic orbits and bifurcation diagrams in delay equations

Summary By employing a numerical method which uses only rather classical tools of Numerical Analysis such as Newton's method and routines for ordinary differential equations, unstable periodic solutions of differential-difference equations can be computed. The method is applied to determine bifurcation diagrams with backward bifurcation.Dedicated to Professor Lothar Collatz on the occasion of his 70th birthdayThis paper has been read at the Conference of Numerical Mathematicians, Zeist, The Netherlands, October 12–15, 1979     

On a class of martingale inequalities

Let Z = {Z₀, Z₁, Z₂,…} be a martingale, with difference sequence X₀ = Z₀, X_i = Z_i ? Z_{i ? 1}, i ≥ 1. The principal purpose of this paper is to prove that the best constant in the inequality λP(sup_i |X_i| ≥ λ) ≤ C sup_iE |Z_i|, for λ > 0, is C = (log 2)^?1. If Z is finite of length n, it is proved that the best constant is $\text{C}{}_{\mathrm{n}}\text{= [n(2}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}\text{? 1)]}{}^{\mathrm{?1}}$. The analogous best constant C_n(z) when Z₀ ≡ z is also determined. For these finite cases, examples of martingales attaining equality are constructed. The results follow from an explicit determination of the quantity G_n(z, E) = sup_zP(max_i=1,…,n |X_i| ≥ 1), the supremum being taken over all martingales Z with Z₀ ≡ z and E|Z_n| = E. The expression for G_n(z,E) is derived by induction, using methods from the theory of moments.     

Chemical Synthesis and Immunological Evaluation of a Pentasaccharide Bearing Multiple Rare Sugars as a Potential Anti-pertussis Vaccine

With the infection rate of Bordetella pertussis at a 60-year high, there is an urgent need for new anti-pertussis vaccines. The lipopolysaccharide (LPS) of B. pertussis is an attractive antigen for vaccine development. With the presence of multiple rare sugars and unusual glycosyl linkages, the B. pertussis LPS is a highly challenging synthetic target. In this work, aided by molecular dynamics simulation and modeling, a pertussis-LPS-like pentasaccharide was chemically synthesized for the first time. The pentasaccharide was conjugated with a powerful carrier, bacteriophage Qβ, as a vaccine candidate. Immunization of mice with the conjugate induced robust anti-glycan IgG responses with IgG titers reaching several million enzyme-linked immunosorbent assay (ELISA) units. The antibodies generated were long lasting and boostable and could recognize multiple clinical strains of B. pertussis, highlighting the potential of Qβ-glycan as a new anti-pertussis vaccine.