Localizing sets for invariant compact sets of discrete dynamical systems with perturbation and control

We consider the localization problem for the invariant compact sets of a discrete dynamical system with perturbation and control, that is, the problem of constructing domains in the system state space that contain all invariant compact sets of the system. The problem is solved on the basis of a functional method used earlier in localization problems for time-invariant continuous and discrete systems and also for control systems. The properties of the corresponding localizing sets are described.     

Localizing sets for invariant compact sets of continuous dynamical systems with a perturbation

A functional method of localization of invariant compact sets, which was earlier developed for autonomous continuous and discrete systems, is generalized to continuous dynamical systems with perturbations. We describe properties of the corresponding localizing sets. By using that method, we construct localizing sets for positively invariant compact sets of the Lorenz system with a perturbation.     

轨道逼近时间集的密度

任意给定0pq1,证明了在符号系统中(进而在帐篷映射中)存在Mycielski集C,使得C中任意两个互异的点的轨道按照下密度p,上密度q的"速率"逼近.构造了线段上的连续映射,使其具有一个满Lebesgue测度的Mycielski集S,使得S中任意两个互异的点的轨道按照下密度p,上密度q的"速率"逼近.     

Decomposition behavior in aggregated data sets

One way to aggregate data is to combine several sets with the same structure, but no overlap in their ranges of values — for instance, aggregating prices before and after a period of hyperinflation. Looking at nonparametric tests on three ‘items’, we compute the relation of the decomposition of the underlying voting profiles of such aggregated sets to those for the original data. We focus on the Basic components, including examples of ‘pure Basic’ sets, computed using Sage. This yields several interesting results about consistency of nonparametric tests with respect to this kind of aggregation, and suggests types of non-uniformity which are not detected by standard tests.     

Sesquitransitive and Localizing Operator Algebras

An algebra of operators on a Banach space X is said to be transitive if X has no nontrivial closed subspaces invariant under every member of the algebra. In this paper we investigate a number of conditions which guarantee that a transitive algebra of operators is “large” in various senses. Among these are the conditions of algebras being localizing or sesquitransitive. An algebra is localizing if there exists a closed ball B ∌ 0 such that for every sequence (x _n ) in B there exists a subsequence and a bounded sequence (A _k ) in the algebra such that converges to a non-zero vector. An algebra is sesquitransitive if for every non-zero z ∈ X there exists C > 0 such that for every x linearly independent of z, for every non-zero y ∈ X, and every there exists A in the algebra such that and ||Az|| ≤ C||z||. We give an algebraic version of this definition as well, and extend Jacobson’s density theorem to algebraically sesquitransitive rings. The second and the third authors were supported by NSERC.     

Localizing the axioms

We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by Loc(ZFC), says that every set belongs to a transitive model of ZFC. LZFC consists of Loc(ZFC) plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All ${\Pi_2}We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by Loc(ZFC), says that every set belongs to a transitive model of ZFC. LZFC consists of Loc(ZFC) plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All P₂{\Pi_2} consequences of ZFC are provable in LZFC. LZFC strongly extends Kripke-Platek (KP) set theory minus Δ₀-Collection and minus ?{\in} -induction scheme. ZFC+ “there is an inaccessible cardinal” proves the consistency of LZFC. In LZFC we focus on models rather than cardinals, a transitive model being considered as the analogue of an inaccessible cardinal. Pushing this analogy further we define α-Mahlo models and P₁¹{\Pi_1^1} -indescribable models, the latter being the analogues of weakly compact cardinals. Also localization axioms of the form Loc(ZFC+f){Loc({\rm ZFC}+\phi)} are considered and their global consequences are examined. Finally we introduce the concept of standard compact cardinal (in ZFC) and some standard compactness results are proved.     

Limit behavior of ellipsoids bounding attainable sets

We study the asymptotic behavior of some ellipsoidal bounds for attainable sets of stable linear control dynamic systems. We show that, in a variety of situations, there is exactly one ellipsoid which can serve as a limit of the above bounds ast. This ellipsoid is shown to be in many cases locally attractive.This work was supported by Grant 93-013-16286 of the Russian Fundamental Research Foundation and by Grant NMAF000 of the International Science Foundation.The author thanks his colleagues F. L. Chernousko and Y. N. Reshetnyak for useful discussions.     

Dynamical behavior of endomorphisms on certain invariant sets

We study the Hausdorff dimension of the intersection between local stable manifolds and the respective basic sets of a class of hyperbolic polynomial endomorphisms on the complex projective space ?². We consider the perturbation (z ² +?z +b?w ², w ²) of (z ², w ²) and we prove that, for b sufficiently small, it is injective on its basic set Λ_? close to Λ:= {0} × S ¹. Moreover we give very precise upper and lower estimates for the Hausdorff dimension of the intersection between local stable manifolds and Λ _? , in the case of these maps.     

Localizing combinatorial properties of partitions

The purpose of this paper is to develop a framework for the analysis of combinatorial properties of partitions. Our focus is on the relation between global properties of partitions and their localization to subpartitions. First, we study properties that are characterized by their local behavior. Second, we determine sufficient conditions for classes of partitions to have a member that has a given property. These conditions entail the possibility of being able to move from an arbitrary partition in the class to one that satisfies the given property by sequentially satisfying local variants of the property. We apply our approach to several properties of partitions that include consecutiveness, nestedness, order-consecutiveness, full nestedness and balancedness, and we demonstrate its usefulness in determining the existence of optimal partitions that satisfy such properties.     

Lower bounds on Ricci curvature and quantitative behavior of singular sets

Let Y ⁿ denote the Gromov-Hausdorff limit $M^{n}_{i}\stackrel{d_{\mathrm{GH}}}{\longrightarrow} Y^{n}$ of v-noncollapsed Riemannian manifolds with ${\mathrm{Ric}}_{M^{n}_{i}}\geq-(n-1)$ . The singular set $\mathcal {S}\subset Y$ has a stratification $\mathcal {S}^{0}\subset \mathcal {S}^{1}\subset\cdots\subset \mathcal {S}$ , where $y\in \mathcal {S}^{k}$ if no tangent cone at y splits off a factor ? ^k+1 isometrically. Here, we define for all η>0, 0<r≤1, the k-th effective singular stratum $\mathcal {S}^{k}_{\eta,r}$ satisfying $\bigcup_{\eta}\bigcap_{r} \,\mathcal {S}^{k}_{\eta,r}= \mathcal {S}^{k}$ . Sharpening the known Hausdorff dimension bound $\dim\, \mathcal {S}^{k}\leq k$ , we prove that for all y, the volume of the r-tubular neighborhood of $\mathcal {S}^{k}_{\eta,r}$ satisfies ${\mathrm {Vol}}(T_{r}(\mathcal {S}^{k}_{\eta,r})\cap B_{\frac{1}{2}}(y))\leq c(n,{\mathrm {v}},\eta)r^{n-k-\eta}$ . The proof involves a quantitative differentiation argument. This result has applications to Einstein manifolds. Let $\mathcal {B}_{r}$ denote the set of points at which the C ²-harmonic radius is ≤r. If also the $M^{n}_{i}$ are Kähler-Einstein with L ₂ curvature bound, $\| Rm\|_{L_{2}}\leq C$ , then ${\mathrm {Vol}}( \mathcal {B}_{r}\cap B_{\frac{1}{2}}(y))\leq c(n,{\mathrm {v}},C)r^{4}$ for all y. In the Kähler-Einstein case, without assuming any integral curvature bound on the $M^{n}_{i}$ , we obtain a slightly weaker volume bound on $\mathcal {B}_{r}$ which yields an a priori L _p curvature bound for all p<2. The methodology developed in this paper is new and is applicable in many other contexts. These include harmonic maps, minimal hypersurfaces, mean curvature flow and critical sets of solutions to elliptic equations.     

Emergent behavior in two complex cellular automata rule sets

Cellular automata systems often produce complex behavior from simple rule sets. The behaviors and results of two complex combinations of cellular automata rules are analyzed. Both two‐dimensional rule sets add complexities to typical cellular automata systems by attaching attributes and rules to each cell. One of the rule sets produces gliders that reproduce upon collision, whereas the other grows into an intricate shape. Projection and entropy analysis classify the rule sets as complex for the intricate shape, but measurements indicate that the self‐reproducing gliders fall between ordered and complex classification, despite their complex appearance. © 2005 Wiley Periodicals, Inc. Complexity 10: 45–55, 2005     

Vandermonde sets and super-Vandermonde sets

Given a set TGF(q), |T|=t, w_T is defined as the smallest positive integer k for which ∑_yTy^k≠0. It can be shown that w_Tt always and w_Tt−1 if the characteristic p divides t. T is called a Vandermonde set if w_Tt−1 and a super-Vandermonde set if w_T=t. This (extremal) algebraic property is interesting for its own right, but the original motivation comes from finite geometries. In this paper we classify small and large super-Vandermonde sets.     

Wiggly sets and limit sets

We show that a compact, connected set which has uniform oscillations at all points and at all scales has dimension strictly larger than 1. We also show that limit sets of certain Kleinian groups have this property. More generally, we show that ifG is a non-elementary, analytically finite Kleinian group, and its limit set Λ(G) is connected, then Λ(G) is either a circle or has dimension strictly bigger than 1. The first author is partially supported by NSF Grant DMS 95-00577 and an Alfred P. Sloan research fellowship. The second author is partially supported by NSF grant DMS-94-23746.