SPECTRUM-PRESERVING ELEMENTARY OPERATORS ON B（X）

1.IntroductionLetXbeaninfinitedimensionalcomplexBanachspaceandB(X)theBanachalgebraofallboundedlinearoperatorsonX.ForTEB(X),a(T),asusual,willdenotethespectrumofT.Let4bealinearmapfromB(X)intoitself.4isspectrum-preservingifa(di(T))=a(T)forallTEB(X);4isspectrum-compressingifa(4(T))ga(T)forallTEB(X).Itisclearthatif4isunital(i.e.,ac(I)=I),thenacisspectrum-preserving(spectrum-compressing)ifandonlyif4preservesinvertibilityinbothdirections(preservesinvertibility),i.e.,4(T)isinvertibleifando…     

Metastability in the Potts model on the Cayley tree

Metastability in the ferromagneticp-state Potts model defined on the Cayley tree is discussed. It is shown that the sign of the boundary fieldH _s determines the order of the transition as well as the stability of the low-temperature phase. Lowering the temperature withH _s >0, a system withp<2 (p>2) will display a second (first)-order transition to a metastable (stable) phase. ForH _s >0 a second (first)-order transition to a metastable (stable) phase occurs ifp>2 (p<2). In this case the system also has a residual entropy which is negative forp<2.     

Timelike Minimal Surfaces via Loop Groups

This work consists of two parts. In Part I, we shall give a systematic study of Lorentz conformal structure from structural viewpoints. We study manifolds with split-complex structure. We apply general results on split-complex structure for the study of Lorentz surfaces.In Part II, we study the conformal realization of Lorentz surfaces in the Minkowski 3-space via conformal minimal immersions. We apply loop group theoretic Weierstrass-type representation of timelike constant mean curvature for timelike minimal surfaces. Classical integral representation formula for timelike minimal surfaces will be recovered from loop group theoretic viewpoint.     

Degree-regular triangulations of torus and Klein bottle

A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In [8], Lutz has classified all the weakly regular triangulations on at most 15 vertices. In [5], Datta and Nilakantan have classified all the degree-regular triangulations of closed surfaces on at most 11 vertices. In this article, we have proved that any degree-regular triangulation of the torus is weakly regular. We have shown that there exists ann-vertex degree-regular triangulation of the Klein bottle if and only if n is a composite number ≥ 9. We have constructed two distinctn-vertex weakly regular triangulations of the torus for eachn ≥ 12 and a (4m + 2)-vertex weakly regular triangulation of the Klein bottle for eachm ≥ 2. For 12 ≤n ≤ 15, we have classified all then-vertex degree-regular triangulations of the torus and the Klein bottle. There are exactly 19 such triangulations, 12 of which are triangulations of the torus and remaining 7 are triangulations of the Klein bottle. Among the last 7, only one is weakly regular.     

Finite Degree Holomorphic Covers of Compact Riemann Surfaces

A conjecture of Ehrenpreis states that any two compact Riemann surfaces of genus at least two have finite degree unbranched holomorphic covers that are arbitrarily close to each other in moduli space. Here we prove a weaker result where certain branched covers associated with arithmetic Riemann surfaces are allowed, and investigate the connection of our result with the original conjecture.     

The π-π-theorem for manifold pairs with boundaries

The surgery obstruction of a normal map to a simple Poincaré pair (X, Y) lies in the relative surgery obstruction group L _*(π ₁(Y) → π ₁(X)). A well-known result of Wall, the so-called π-π-theorem, states that in higher dimensions a normal map of a manifold with boundary to a simple Poincaré pair with π ₁(X) ? π ₁(Y) is normally bordant to a simple homotopy equivalence of pairs. In order to study normal maps to a manifold with a submanifold, Wall introduced the surgery obstruction groups LP _* for manifold pairs and splitting obstruction groups LS _*. In the present paper, we formulate and prove for manifold pairs with boundary results similar to the π-π-theorem. We give direct geometric proofs, which are based on the original statements of Wall’s results and apply obtained results to investigate surgery on filtered manifolds.     

Periodic solutions of nonlinear impulsive differential inclusions with constraints

In this paper we obtain the existence of periodic solutions for nonlinear ``invariance' problems monitored by impulsive differential inclusions subject to impulse effects.

     

Arithmetic Hodge Structure and Nonvanishing of the Cycle Class of 0-Cycles

We construct 0-cycles on a self-product of a curve whose cycle classes are linearly independent in the extension group of arithmetic Hodge structure. This is an extended version of Theorem 4.5 of Asakura's contribution to CMR Lecture Notes Ser. 24 ((2000), pp. 133–154).     

Lagrangian Duality in Set-Valued Optimization

In this paper, we study optimization problems where the objective function and the binding constraints are set-valued maps and the solutions are defined by means of set-relations among all the images sets (Kuroiwa, D. in Takahashi, W., Tanaka, T. (eds.) Nonlinear analysis and convex analysis, pp. 221–228, 1999). We introduce a new dual problem, establish some duality theorems and obtain a Lagrangian multiplier rule of nonlinear type under convexity assumptions. A necessary condition and a sufficient condition for the existence of saddle points are given. The authors thank the two referees for valuable comments and suggestions on early versions of the paper. The research of the first author was partially supported by Ministerio de Educación y Ciencia (Spain) Project MTM2006-02629 and by Junta de Castilla y León (Spain) Project VA027B06.     

Territories,corridors, and networks: A biological model for the premodern state

When depicted on maps as homogenous territorial wholes, ancient states are visually summarized as static entities in a way that conceals the highly fluid dynamics of polity formation, maintenance, and growth. Models derived from studies of animal behavior show that “territory” does not consist of an undifferentiated use of the landscape. Instead, the concept of territory can be parsed into a series of resource‐rich nodes linked by corridors of access, surrounded by unutilized regions and boundaries marked at points of competition. Ancient human groups also can be analyzed as having perceived and occupied landscapes through strategies of flexible networks in which nodes and corridors were surrounded by unutilized spaces around which boundaries were selectively identified and defended. This strategy is identifiable in human social groups at different levels of complexity ranging from hunter‐gatherers through ancient chiefdoms and states. © 2007 Wiley Periodicals, Inc. Complexity 12: 28–35, 2007