Exact solutions for KdV system equations hierarchy

Exact solutions for KdV system equations hierarchy are obtained by using the inverse scattering transform. Exact solutions of isospectral KdV hierarchy, nonisospectral KdV hierarchies and τ$\tau$-equations related to the KdV spectral problem are obtained by reduction. The interaction of two solitons is investigated.     

Factorizing τ-spectra of mappings

In this paper by a spectrum of mappings we mean a morphism of spectra of spaces. However, using the notion of a mapping of mappings, we give the definition of a spectrum of mappings similar to that of a spectrum of spaces. In this case, the formulations of the given results are also similar to the formulations of the corresponding results concerning the spectra of spaces.For the spectra of mappings we define the notion of a τ-spectrum of mappings factorizing in a special sense and prove a version of the Spectral Theorem for such spectra. Furthermore, to a given indexed collection F of mapping we associate a τ-spectrum factorizing in the above special sense whose mappings are Containing Mappings for F constructed in Iliadis (2005) [4]. These associated τ-spectra and the corresponding version of the Spectral Theorem imply that for a given indexed collection F of mappings any so-called “natural” τ-spectrum for F factorizing in the special sense contains a cofinal and τ-closed subspectrum whose mappings are Containing Mapping for F. Thus, Containing Mappigs for F appear here without any concrete construction. The associated τ-spectra are used also in order to define and characterize the so-called second-type saturated classes of mappings (which are “saturated” by universal elements).     

ν-Stable τ-Tilting Modules

Inspired by τ-tilting theory [3 Adachi , T. , Iyama , O. , Reiten , I. ( 2014 ). τ-tilting theory . Compos. Math. 150 ( 3 ): 415 – 452 .[Crossref], [Web of Science ®] , [Google Scholar]], we introduce the notion of ν-stable support τ-tilting modules. For any finite dimensional selfinjective algebra Λ, we give bijections between two-term tilting complexes in K ^b (proj Λ), ν-stable support τ-tilting Λ-modules, and ν-stable functorially finite torsion classes in modΛ. Moreover, these objects correspond bijectively to selfinjective cluster tilting objects in 𝒞 if Λ is a 2-CY tilted algebra associated with a Hom-finite 2-CY triangulated category 𝒞. We also study some properties of support τ-tilting modules over 2-CY tilted algebras, and we give a necessary condition such that algebras are 2-CY tilted in terms of support τ-tilting modules.     

Relative Flatness,Mittag–Leffler Modules,and Endocoherence

Let M _R be a right R-module over a ring R with S = End(M _R ). We study the coherence of the left S-module _S M relative to a hereditary torsion theory for the category of right R-modules. Various results are developed, many extending known results.     

S-系的扭类

假设S是有0,1的半群,τ是S-系的遗传扭论,对任意的右S-系M,Tτ（M）是M的τ-扭根。x∈Tτ（M）当且仅当存在某个τ-稠密右同余ρ,使得对任意的（S1,S2）∈ρ均有xs1=xs2,同时,当右S-系M是τ-扭自由时,M的τ-稠密同余是M的本质同余,特别,对忠实的遗传扭论τ,S的τ-稠密右同余是S     

Delay-dependent stability of symmetric schemes in Boundary Value Methods for DDEs

We consider the symmetric schemes in Boundary Value Methods (BVMs) applied to delay differential equations y^′(t)=ay(t)+by(t-τ) with real coefficients a and b. If the numerical solution tends to zero whenever the exact solution does, the symmetric scheme with (k₁+m,k₂)-boundary conditions is called τ_k1,k2(0)-stable. Three families of symmetric schemes, namely the Extended Trapezoidal Rules of first (ETRs) and second (ETR₂s) kind, and the Top Order Methods (TOMs), are considered in this paper.By using the boundary locus technology, the delay-dependent stability region of the symmetric schemes are analyzed and their boundaries are found. Then by using a necessary and sufficient condition, the considered symmetric schemes are proved to be τ_ν,ν-1(0)-stable.     

非线性互补问题的组合同伦算法

针对拟P_*-映射和P(τ,α,β)-映射所对应的非线性互补问题,本文对其解的存在性及有效求解算法进行了研究.文中利用组合同伦方法给出了这两类非线性互补问题存在有界解的构造性证明,并利用预估校正方法对同伦路径进行跟踪,得到了互补问题的解.通过数值算例验证了该算法的有效性.     

非交换的Kolmogorov不等式和Zygmund不等式

In this paper,we discuss some properties of distribution functions of τ-measurable operators and obtain non-commutative analogue of Kolmogorov inequality and Zygmund inequality.     

On Downing-Kirk's theorem

In this paper, using the notion of τ-distance, we prove a generalization of Downing-Kirk's theorem.