Mean dimension and an embedding problem: An example

For any positive integer D, we construct a minimal dynamical system with mean dimension equal to D/2 that cannot be embedded into (([0, 1] ^D )^?, shift).     

Lowering topological entropy

The main result we prove in this paper is that for any finite dimensional dynamical system (with topological entropyh), and for any factor with strictly lower entropyh′, there exists an intermediate factor of entropyh″ for everyh″∈[h′, h]. Two examples, one of them minimal, show that this is not the case for infinite dimensional systems.     

A Genetic Algorithm for Solving a Capacitated p-Median Problem

Facility-location problems have several applications, such as telecommunications, industrial transportation and distribution. One of the most well-known facility-location problems is the p-median problem. This work addresses an application of the capacitated p-median problem to a real-world problem. We propose a genetic algorithm (GA) to solve the capacitated p-median problem. The proposed GA uses not only conventional genetic operators, but also a new heuristic “hypermutation” operator suggested in this work. The proposed GA is compared with a tabu search algorithm.     

Synthesis and characterization of oxorhenium(v) diamido pyridine complexes that catalyze oxygen atom transfer reactions

The detailed syntheses of complexes 1-4, Re(O)(X)(DAP) (X = Me, 1; Cl, 2; I, 3; OTf (triflate), 4) incorporating the diamido pyridine (DAP) ancillary ligand (2,6-bis((mesitylamino)methyl)pyridine) are described and shown to be effective catalysts for oxygen atom transfer (OAT) reactions of PyO to PPh(3). The catalytic activities are as follows: 4≈3 > 2 > 1. The observed electronic trend is consistent with the turnover limiting reduction of the proposed Re(VII) dioxo intermediate, Re(O)(2)(X)(DAP), during the catalytic cycle. The catalytic activity of complexes 1-3 was compared to previously published diamido amine (DAAm) oxorhenium complexes of the type Re(O)(X)(DAAm) (X = Me, 5; Cl, 6; I, 7 and DAAm = N,N-bis(2-arylaminoethyl)methylamine) which exhibit hydrolytic degradation during the catalytic reaction. Complexes 1-3 displayed higher turnover frequencies compared to 5-7. This higher catalytic activity was attributed to the more rigid DAP ligand backbone, which makes the complexes less susceptible to decomposition. However, another decomposition pathway was proposed for this catalytic system due to the observation of Re(O)(3)((MesNCH(2))(MesNCH)NC(5)H(3)) 8 in which one arm of the DAP ligand is oxidized.     

Observation of inductive effects that cause a change in the rate-determining step for the conversion of rhenium azides to imido complexes

The cationic oxorhenium(V) complex [Re(O)(hoz)(2)(CH(3)CN)][B(C(6)F(5))(4)] [1; Hhoz = 2-(2'-hydroxyphenyl)-2-oxazoline] reacts with aryl azides (N(3)Ar) to give cationic cis-rhenium(VII) oxoimido complexes of the general formula [Re(O)(NAr)(hoz)(2)][B(C(6)F(5))(4)] [2a-2f; Ar = 4-methoxyphenyl, 4-methylphenyl, phenyl, 3-methoxyphenyl, 4-chlorophenyl, and 4-(trifluoromethyl)phenyl]. The kinetics of formation of 2 in CH(3)CN are first-order in both azide (N(3)Ar) and oxorhenium(V) complex 1, with second-order rate constants ranging from 3.5 × 10(-2) to 1.7 × 10(-1) M(-1) s(-1). A strong inductive effect is observed for electron-withdrawing substituents, leading to a negative Hammett reaction constant ρ = -1.3. However, electron-donating substituents on phenyl azide deviate significantly from this trend. Enthalpic barriers (ΔH(?)) determined by the Eyring-Polanyi equation are in the range 14-19 kcal mol(-1) for all aryl azides studied. However, electron-donating 4-methoxyphenyl azide exhibits a large negative entropy of activation, ΔS(?) = -21 cal mol(-1) K(-1), which is in sharp contrast to the near zero ΔS(?) observed for phenyl azide and 4-(trifluoromethyl)phenyl azide. The Hammett linear free-energy relationship and the activation parameters support a change in the mechanism between electron-withdrawing and electron-donating aryl azides. Density functional theory predicts that the aryl azides coordinate via N(α) and extrude N(2) directly. For the electron-withdrawing substituents, N(2) extrusion is rate-determining, while for the electron-donating substituents, the rate-determining step becomes the initial attack of the azide. The barriers for these two steps are inverted in their order with respect to the Hammett σ values; thus, the Hammett plot appears with a break in its slope.     

Mechanism for reduction catalysis by metal oxo: hydrosilation of organic carbonyl groups catalyzed by a rhenium(V) oxo complex

The rhenium oxo complex [Re(O)(hoz)2][TFPB], 1 (where hoz = 2-(2'-hydroxyphenyl)-2-oxazoline(-) and TFPB = tetrakis(pentafluorophenyl)borate) catalyzes the hydrosilation of aldehydes and ketones under ambient temperature and atmosphere. The major organic product is the protected alcohol as silyl ether. Isolated yields range from 86 to 57%. The reaction requires low catalyst loading (0.1 mol %) and proceeds smoothly in CH2Cl2 as well as neat without solvent. In the latter condition, the catalyst precipitates at the end of reaction, allowing easy separation and catalyst recycling. Re(O)(hoz)(H), 3, was prepared, and its involvement in an ionic hydrosilation mechanism was evaluated. Complex 3 was found to be less hydridic than Et3SiH, refuting its participation in catalysis. A viable mechanism that is consistent with experimental findings, rate measurements, and kinetic isotope effects (Et3SiH/Et3SiD = 1.3 and benzaldehyde-H/benzaldehyde-D = 1.0) is proposed. Organosilane is activated via eta2-coordination to rhenium, and the organic carbonyl adds across the coordinated Si-H bond [2 + 2] to afford the organic reduction product.     

Invariant sets and measures of nonexpansive group automorphisms

We write a formula for the LMO invariant of a rational homology sphere presented as a rational surgery on a link inS ³. Our main tool is a careful use of the Århus integral and the (now proven) “Wheels” and “Wheeling” conjectures of B-N, Garoufalidis, Rozansky and Thurston. As steps, side benefits and asides we give explicit formulas for the values of the Kontsevich integral on the Hopf link and on Hopf chains, and for the LMO invariant of lens spaces and Seifert fibered spaces. We find that the LMO invariant does not separate lens spaces, is far from separating general Seifert fibered spaces, but does separate Seifert fibered spaces which are integral homology spheres.     

Mean topological dimension

In this paper we present some results and applications of a new invariant for dynamical systems that can be viewed as a dynamical analogue of topological dimension. This invariant has been introduced by M. Gromov, and enables one to assign a meaningful quantity to dynamical systems of infinite topological dimension and entropy. We also develop an alternative approach that is metric dependent and is intimately related to topological entropy.     

Mean dimension of $${\mathbb{Z}^k}$$-actions

Mean dimension is a topological invariant for dynamical systems that is meaningful for systems with infinite dimension and infinite entropy. Given a \({\mathbb{Z}^k}\)-action on a compact metric space X, we study the following three problems closely related to mean dimension.

1. (1)
   When is X isomorphic to the inverse limit of finite entropy systems?
    
2. (2)
   Suppose the topological entropy \({h_{\rm top}(X)}\) is infinite. How much topological entropy can be detected if one considers X only up to a given level of accuracy? How fast does this amount of entropy grow as the level of resolution becomes finer and finer?
    
3. (3)
   When can we embed X into the \({\mathbb{Z}^k}\)-shift on the infinite dimensional cube \({([0,1]^D)^{\mathbb{Z}^k}}\)?
    

These were investigated for \({\mathbb{Z}}\)-actions in Lindenstrauss (Inst Hautes Études Sci Publ Math 89:227–262, 1999), but the generalization to \({\mathbb{Z}^k}\) remained an open problem. When X has the marker property, in particular when X has a completely aperiodic minimal factor, we completely solve (1) and a natural interpretation of (2), and give a reasonably satisfactory answer to (3).A key ingredient is a new method to continuously partition every orbit into good pieces.