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.     

On processes which cannot be distinguished by finite observation

A function J defined on a family C of stationary processes is finitely observable if there is a sequence of functions s _n such that s _n (x ₁,…, x _n ) → J(X) in probability for every process X=(x _n ) ∈ C. Recently, Ornstein and Weiss proved the striking result that if C is the class of aperiodic ergodic finite valued processes, then the only finitely observable isomorphism invariant defined on C is entropy [8]. We sharpen this in several ways. Our main result is that if X → Y is a zero-entropy extension of finite entropy ergodic systems and C is the family of processes arising from generating partitions of X and Y, then every finitely observable function on C is constant. This implies Ornstein and Weiss’ result, and extends it to many other families of processes, e.g., it follows that there are no nontrivial finitely observable isomorphism invariants for processes arising from the class of Kronecker systems, the class of mild mixing zero entropy systems, or the class of strong mixing zero entropy systems. It also follows that for the class of processes arising from irrational rotations, every finitely observable isomorphism invariant must be constant for rotations belonging to a set of full Lebesgue measure. This research was supported by the Israel Science Foundation (grant No. 1333/04)     

An explicit compact universal space for real flows

The Kakutani–Bebutov Theorem (1968) states that any compact metric real flow whose fixed point set is homeomorphic to a subset of $\mathbb{R}$ embeds into the Bebutov flow, the $\mathbb{R}$-shift on $C(\mathbb{R},[0,1])$. An interesting fact is that this universal space is a function space. However, it is not compact, nor locally compact. We construct an explicit countable product of compact subspaces of the Bebutov flow which is a universal space for all compact metric real flows, with no restriction; namely, into which any compact metric real flow embeds. The result is compared to previously known universal spaces.     

Structural Insight into IAPP-Derived Amyloid Inhibitors and Their Mechanism of Action

Designed peptides derived from the islet amyloid polypeptide (IAPP) cross-amyloid interaction surface with Aβ (termed interaction surface mimics or ISMs) have been shown to be highly potent inhibitors of Aβ amyloid self-assembly. However, the molecular mechanism of their function is not well understood. Using solution-state and solid-state NMR spectroscopy in combination with ensemble-averaged dynamics simulations and other biophysical methods including TEM, fluorescence spectroscopy and microscopy, and DLS, we characterize ISM structural preferences and interactions. We find that the ISM peptide R3-GI is highly dynamic, can adopt a β-like structure, and oligomerizes into colloid-like assemblies in a process that is reminiscent of liquid–liquid phase separation (LLPS). Our results suggest that such assemblies yield multivalent surfaces for interactions with Aβ40. Sequestration of substrates into these colloid-like structures provides a mechanistic basis for ISM function and the design of novel potent anti-amyloid molecules.     

Simple Majority Consensus in Networks with Unreliable Communication

In this work, we analyze the performance of a simple majority-rule protocol solving a fundamental coordination problem in distributed systems—binary majority consensus—in the presence of probabilistic message loss. Using probabilistic analysis for a large-scale, fully-connected, network of 2n agents, we prove that the Simple Majority Protocol (SMP) reaches consensus in only three communication rounds, with probability approaching 1 as n grows to infinity. Moreover, if the difference between the numbers of agents that hold different opinions grows at a rate of n, then the SMP with only two communication rounds attains consensus on the majority opinion of the network, and if this difference grows faster than n, then the SMP reaches consensus on the majority opinion of the network in a single round, with probability converging to 1 as exponentially fast as n→∞. We also provide some converse results, showing that these requirements are not only sufficient, but also necessary.     

Isotopic thermal diffusion factors of the noble gases

A new set of isotopic thermal diffusion factors for Ar, Kr and Xe is reported. The data have been obtained from thermal diffusion column measurements in a column previously calibrated with Ne gas, and complete the α_T data for the binary mixtures of noble gases given previously. The results lead to the conclusion that the thermal diffusion properties of the noble gases and their mixtures are well described by the law of corresponding states first proposed by Kestin, Ro and Wakeham. It is also concluded that the calibration technique used makes the thermal diffusion column a valuable instrument for the determination of the thermal diffusion factors of isotopic and non-isotopic dilute gas mixtures.     

Auditory stream segregation in monkey auditory cortex: effects of frequency separation, presentation rate, and tone duration

Auditory stream segregation refers to the organization of sequential sounds into "perceptual streams" reflecting individual environmental sound sources. In the present study, sequences of alternating high and low tones, "...ABAB...," similar to those used in psychoacoustic experiments on stream segregation, were presented to awake monkeys while neural activity was recorded in primary auditory cortex (A1). Tone frequency separation (AF), tone presentation rate (PR), and tone duration (TD) were systematically varied to examine whether neural responses correlate with effects of these variables on perceptual stream segregation. "A" tones were fixed at the best frequency of the recording site, while "B" tones were displaced in frequency from "A" tones by an amount = delta F. As PR increased, "B" tone responses decreased in amplitude to a greater extent than "A" tone responses, yielding neural response patterns dominated by "A" tone responses occurring at half the alternation rate. Increasing TD facilitated the differential attenuation of "B" tone responses. These findings parallel psychoacoustic data and suggest a physiological model of stream segregation whereby increasing delta F, PR, or TD enhances spatial differentiation of "A" tone and "B" tone responses along the tonotopic map in A1.     

Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential

We study analytically and numerically the stability of the standing waves for a nonlinear Schrödinger equation with a point defect and a power type nonlinearity. A major difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves. This is overcome by a perturbation method and continuation arguments. Among others, in the case of a repulsive defect, we show that the standing-wave solution is stable in and unstable in under subcritical nonlinearity. Further we investigate the nature of instability: under critical or supercritical nonlinear interaction, we prove the instability by blowup in the repulsive case by showing a virial theorem and using a minimization method involving two constraints. In the subcritical radial case, unstable bound states cannot collapse, but rather narrow down until they reach the stable regime (a finite-width instability). In the nonradial repulsive case, all bound states are unstable, and the instability is manifested by a lateral drift away from the defect, sometimes in combination with a finite-width instability or a blowup instability.     

Pattern Matching with Swaps

Let a text string T of n symbols and a pattern string P of m symbols from alphabet Σ be given. A swapped version T′ of T is a length n string derived from T by a series of local swaps (i.e., t′_ℓ ← t_ℓ + 1 and t′_ℓ + 1 ← t_ℓ), where each element can participate in no more than one swap. The pattern matching with swaps problem is that of finding all locations i for which there exists a swapped version T′ of T with an exact matching of P in location i of T′. It has been an open problem whether swapped matching can be done in less than O(nm) time. In this paper we show the first algorithm that solves the pattern matching with swaps problem in time o(nm). We present an algorithm whose time complexity is O(nm^1/3 log m log σ) for a general alphabet Σ, where σ = min(m,Σ).