On the abundance of chaotic behavior for generic one-parameter families of maps

In this paper, we want to show the abundance of chaotic systems with absolutely continuous probability measures in the generic regular family with perturbable points. More precisely, we prove that iff _a:I I, a P is a regular family satisfying some conditions described in the next section, then there exists a Borel set P of positive Lebesgue measure such that for everya ,f _a admits an absolutely continuous invariant probability measure w.r.t. the Lebesgue measure. The idea of proof in this paper, as compared with that shown in [1] and [7], follows a similar line.Supported by the NSFC and the National 863 Project.     

Some results on P-harmonic maps and exponentially harmonic maps between Finsler manifolds

This paper studies the stability of P-harmonic maps and exponentially harmonic maps from Finsler manifolds to Riemannian manifolds by an extrinsic average variational method in the calculus of variations. It generalizes Li＇s results in [2] and [3].     

A note on stability spectrum of generic structures

We show that if a class K of finite relational structures is closed under quasi‐substructures, then there is no saturated K ‐generic structure that is superstable but not ω ‐stable (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Remarks on generic stability in independent theories

In NIP theories, generically stable Keisler measures can be characterized in several ways. We analyze these various forms of “generic stability” in arbitrary theories. Among other things, we show that the standard definition of generic stability for types coincides with the notion of a frequency interpretation measure. We also give combinatorial examples of types in NSOP theories that are finitely approximated but not generically stable, as well as ϕ-types in simple theories that are definable and finitely satisfiable in a small model, but not finitely approximated. Our proofs demonstrate interesting connections to classical results from Ramsey theory for finite graphs and hypergraphs.     

Completely positive module maps and completely positive extreme maps

Let be unital -algebras and be the set of all completely positive linear maps of into . In this article we characterize the extreme elements in , for all , and pure elements in in terms of a self-dual Hilbert module structure induced by each in . Let be the subset of consisting of -module maps for a von Neumann algebra . We characterize normal elements in to be extreme. Results here generalize various earlier results by Choi, Paschke and Lin.

     

Map operations and k-orbit maps

A k-orbit map is a map with k flag-orbits under the action of its automorphism group. We give a basic theory of k-orbit maps and classify them up to k?4. “Hurwitz-like” upper bounds for the cardinality of the automorphism groups of 2-orbit and 3-orbit maps on surfaces are given. Furthermore, we consider effects of operations like medial and truncation on k-orbit maps and use them in classifying 2-orbit and 3-orbit maps on surfaces of small genus.     

Characterizations of relatively generalized monotone maps

New concepts of relative monotonicity were introduced in Konnov (Oper Res Lett 28:21–26, 2001a) which extend the usual ones. These concepts enable us to establish new existence and uniqueness results for variational inequality problems over product sets. This paper presents first-order characterizations of new (generalized) monotonicity concepts. Specialized results are obtained for the affine case.      

Two maps on one surface

Two embeddings of a graph in a surface S are said to be “equivalent” if they are identical under an homeomorphism of S that is orientation‐preserving for orientable S. Two graphs cellularly embedded simultaneously in S are said to be “jointly embedded” if the only points of intersection involve an edge of one graph transversally crossing an edge of the other. The problem is to find equivalent embeddings of the two graphs that minimize the number of these edge‐crossings; this minimum we call the “joint crossing number” of the two graphs. In this paper, we calculate the exact value for the joint crossing number for two graphs simultaneously embedded in the projective plane. Furthermore, we give upper and lower bounds when the surface is the torus, which in many cases give an exact answer. In particular, we give a construction for re‐embedding (equivalently) the graphs in the torus so that the number of crossings is best possible up to a constant factor. Finally, we show that if one of the embeddings is replaced by its “mirror image,” then the joint crossing number can decrease, but not by more than 6.066%. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 198–216, 2001     

Periodic points of nonexpansive maps and nonlinear generalizations of the Perron-Frobenius theory

Let and suppose that f : K ⁿ →K ⁿ is nonexpansive with respect to the l ₁-norm, , and satisfies f (0) = 0. Let P ₃(n) denote the (finite) set of positive integers p such that there exists f as above and a periodic point of f of minimal period p. For each n≥ 1 we use the concept of 'admissible arrays on n symbols' to define a set of positive integers Q(n) which is determined solely by number theoretical and combinatorial constraints and whose computation reduces to a finite problem. In a separate paper the sets Q(n) have been explicitly determined for 1 ≤n≤ 50, and we provide this information in an appendix. In our main theorem (Theorem 3.1) we prove that P ₃(n) = Q(n) for all n≥ 1. We also prove that the set Q(n) and the concept of admissible arrays are intimately connected to the set of periodic points of other classes of nonlinear maps, in particular to periodic points of maps g : D _g→D _g, where is a lattice (or lower semilattice) and g is a lattice (or lower semilattice) homomorphism.     

Geometrical properties of nonlinear maps and their application, Part I: The topological degree of quasiruled Fredholm maps on quasicylindrical domains

In this paper we introduce the notion of quasicylindrical domains in Banach spaces and develop a concept of a degree for quasiruled Fredholm mappings on quasicylindrical domains. Note that this quasicylindrical structure appears in a rather natural way, whenever “analytically” given nonlinear pseudodifferential operators are investigated in spaces of sufficiently smooth functions. Moreover, the class of quasiruled Fredholm mappings on quasicylindrical domains is sufficiently large, so that within this framework one can study a quite large class of nonlinear boundary value problems which are related to pseudodifferential operators.     

Irreducible positive linear maps on operator algebras

Motivated by the classical results of G. Frobenius and O. Perron on the spectral theory of square matrices with nonnegative real entries, D. Evans and R. Høegh-Krohn have studied the spectra of positive linear maps on general (noncommutative) matrix algebras. The notion of irreducibility for positive maps is required for the Frobenius theory of positive maps. In the present article, irreducible positive linear maps on von Neumann algebras are explicitly constructed, and a criterion for the irreducibility of decomposable positive maps on full matrix algebras is given.

     

Universally meager sets and principles of generic continuity and selection in Banach spaces

In this paper we use large cardinals to address some problems about generic continuity and generic selection that occur in the study of fragmentability and differentiability in the context of Banach spaces.     

Gauss-Prym maps on Enriques surfaces

We prove that the kth Gaussian map ${\gamma }_H^k$ is surjective on a polarized unnodal Enriques surface $(S,H)$ with $\phi (H)>2k+4$. In particular, as a consequence, when $\phi (H)>4(k+2)$, we obtain the surjectivity of the kth Gauss-Prym map ${\gamma }_{{\omega }_C\otimes \alpha }^k$, with $\alpha :={\omega }_{{S|}_C}$, on smooth hyperplane sections $C\in |H|$. In case $k=1$, it is sufficient to ask $\phi (H)>6$.     

Characterizations of generalized monotone maps

This paper is a sequel to Ref. 1 in which several kinds of generalized monotonicity were introduced for maps. They were related to generalized convexity properties of functions in the case of gradient maps. In the present paper, we derive first-order characterizations of generalized monotone maps based on a geometrical analysis of generalized monotonicity. These conditions are both necessary and sufficient for generalized monotonicity. Specialized results are obtained for the affine case.     

The manifold structure of maps between open manifolds

We establish in a canonical manner a manifold structure for the completed space of bounded maps between open manifoldsM andN, assuming thatM andN are endowed with Riemannian metrics of bounded geometry up to a certain order. The identity component of the corresponding diffeomorphisms is a Banach manifold and metrizable topological group.