On the Geometry of Horizontally Homotheitic Maps and Harmonic Morphisms

ＯｎｔｈｅＧｅｏｍｅｔｒｙｏｆＨｏｆｉｚｏｎｔａｌｌｙＨｏｍｏｔｈｅｔｉｃＭａｐｓａｎｄＨａｒｍｏｎｉｃＭｏｒｐｈｉｓｍｓＭｏＸｉａｏｈｕａｎ（莫小欢）（ＰｅｋｉｎｇＵｎｉυｅｒｓｉｔｙ，Ｂｅｊｉｎｇ，１００８７１）ＣｏｍｍｕｎｉｃａｔｅｄｂｙＺｈａ...     

Landau-Lifshitz Equations of the Ferromagnetic Spin Chain and Harmonic Maps

Let M be a Rimannian manifold without boundary and N be S~2. Our aim is to solvethe following Landau-Lifshitz type equations of M into S~2 _tu = -α_u×(u×△_Mu) +α_2u×△u (1)where α_1>0 and α_2 are constants. The heat flow for Larmonic map of M into S~2 is given by     

Spectra of Tensor Triangulated Categories over a Category Algebra—A New Approach

Applied Categorical Structures - We provide a new method to compute the Balmer spectra of the bounded derived category and the singularity category of the category algebra of a finite EI category...     

An Adjoint Equivalence of the Category IS of Inverse-sheaves

We have obtained the category IS of r(r≥0)-dimensional inverse-sheaves of open setsof R~n and the followingTheorem1 Let U:C~r→IS be the forgetful functor.Then there is an adjoint equivalence:IS—C~r.Corollary1 C~r-manifolds(X,(?))and(Y,(?))are diffeomorphic iff U(X,(?))and U(Y,(?))are isomorphic.     

On the Continuity of Julia Sets and Hausdorff Dimension of N CP and Parabolic Maps

Denote by HD(J(f))the Hansdorff dimension of the Julia set J(f)of a rational function f.Our first result asserts that if f is an NCP map,and f_n→f horocyclically, preserving sub-critical relations,then f_n is an NCP map for all n(?)0 and J(f_n)→J(f)in the Hausdorff topology.We also prove that if f is a parabolic map and f_n is an NCP map for all n(?)0 such that f_n→f horocyclically,then J(f_n)→J(f)in the Hansdorff topology, and HD(J(f_n))→HD(J(f)).     

Spaces over a Category and Assembly Maps in Isomorphism Conjectures in K‐ and L‐Theory

We give a unified approach to the Isomorphism Conjecture of Farrell and Jones on the algebraicKandLtheory of integral group rings and to the Baum–Connes Conjecture on the topologicalKtheory of reducedC*algebras of groups. The approach is through spectra over the orbit category of a discrete groupG.We give several points of view on the assembly map for a family of subgroups and characterize such assembly maps by a universal property generalizing the results of Weiss and Williams to the equivariant setting. The main tools are spaces and spectra over a category and their associated generalized homology and cohomology theories, and homotopy limits.     

On the Structure of ∞-Harmonic Maps

Let H ∈ C ²(? ^N×n ), H ≥ 0. The PDE system arises as the Euler-Lagrange PDE of vectorial variational problems for the functional E _∞(u, Ω) = ‖H(Du)‖ _{L ^∞(Ω)} defined on maps u: Ω ? ? ⁿ  → ? ^N . (1) first appeared in the author's recent work. The scalar case though has a long history initiated by Aronsson. Herein we study the solutions of (1) with emphasis on the case of n = 2 ≤ N with H the Euclidean norm on ? ^N×n , which we call the “∞-Laplacian”. By establishing a rigidity theorem for rank-one maps of independent interest, we analyse a phenomenon of separation of the solutions to phases with qualitatively different behaviour. As a corollary, we extend to N ≥ 2 the Aronsson-Evans-Yu theorem regarding non existence of zeros of |Du| and prove a maximum principle. We further characterise all H for which (1) is elliptic and also study the initial value problem for the ODE system arising for n = 1 but with H(·, u, u′) depending on all the arguments.     

Horizontal Isometric Maps and Gauss Maps

In 1970,Rub and Vilms [1]proved a famous result that the Gauss map ofan lsometric immersion into an Euclidean space is harmonic if and only if theimmersion has parallel mean curvature.Here we generalized the resutlt from“iso-     

The Cartesian Closedness of the Category CDL and Function Spaces on Topological CDL''''s

ＴｈｅＣａｒｔｅｓｉａｎＣｌｏｓｅｄｎｅｓｓｏｆｔｈｅＣａｔｅｇｏｒｙＣＤＬａｎｄＦｕｎｃｔｉｏｎＳｐａｃｅｓｏｎＴｏｐｏｌｏｇｉｃａｌＣＤＬ＇ｓＹａｎｇＺｈｏｎｇｑｉａｎｇ（杨忠强）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＳｈａａｎｘｉＮ...     

Some Properties of Fuzzy Filter and Its Category

Ｉｎｏｒｄｅｒｔｏｓｔｕｄｙｍａｎｙｖａｌｕｅｄｌｏｇｉｃｓｙｓｔｅｍａｎｄｅｖｅｎｔｈｅｌｏｇｉｃｓｙｓｔｅｍｂａｓｅｄｏｎｌａｔｔｉｃｅ，ａｎａｌｇｅｂｒａｉｃｓｔｒｕｃｔｕｒｅ－ｌａｔｔｉｃｅｉｍｐｌｉｃａｔｉｏｎａｌｇｅｂｒａｗａｓｉｎｔｒｏｄｕｃｅｄｉｎｒｅｆｅｒｅｎｃｅ［１］．Ｔｈｅｓｔｕｄｙｏｆｌａｔｔｉｃｅｉｍｐｌｉｃａｔｉｏｎａｌｇｅｂｒａａｓａｌｇｅｂｒａｉｃｓｙｓｔｅｍｉｓｏｆｉｎｄｅｐｅｎｄｅｎｔｉｎｔｅｒｅｓｔ．Ｉｎｒｅｆｅｒｅｎｃｅ［２，３］ｓｏｍｅｐｒｏｐｅｒｔｉｅ…