Full Shift Research Articles

Spectral characteristics of Schrödinger operators generated by product systems

Motivated by the question of what spectral properties of dynamically defined Schrödinger operators may be preserved under periodic perturbations, we study ergodic Schrödinger operators defined over product dynamical systems in which one factor is periodic and the other factor is either a subshift over a finite alphabet or an irrational rotation of the circle. The scenario given by a periodic background potential corresponds to a separable structure in which the sampling function is the sum of two pieces, each of which depends only on a single factor of the product system. However, in each case that we study, our methods apply more generally to sampling functions that allow non-trivial dependencies between the product factors. In the case in which one factor is a Boshernitzan subshift, we prove that either the resulting operators are periodic or the resulting spectra must be Cantor sets. The main ingredient is a suitable stability result for the Boshernitzan condition under taking products. We also discuss the stability of purely singular continuous spectrum, which, given the zero-measure spectrum result, amounts to stability results for eigenvalue exclusion. In particular, we examine situations in which the existing criteria for the exclusion of eigenvalues are stable under periodic perturbations. As a highlight of this, we show that any simple Toeplitz subshift over a binary alphabet exhibits uniform absence of eigenvalues on the hull for any periodic perturbation whose period is commensurate with the coding sequence. This is new, even in the case in which the periodic background vanishes entirely. In the case of a full shift, we give an effective criterion to compute exactly the spectrum of a random Anderson model perturbed by a potential of period two, and we further show that the naive generalization of this criterion does not hold for period three. Next, we consider quasi-periodic potentials with potentials generated by trigonometric polynomials with periodic background. We show that the quasiperiodic cocycle induced by passing to blocks of period length is subcritical when the coupling constant is small and supercritical when the coupling constant is large. Thus, the spectral type is absolutely continuous for small coupling and pure point (for a.e. frequency and phase) when the coupling is large.

114 Development of High Volume Personal Aerosol Samplers

Abstract The tendency for decreasing a series of occupational exposure limit values is seen the last years. Especially metal species are linked to a larger health effect in latest toxicological assessments. These changes imply further reductions of the limits of quantification for both the sampling and analysis methods. The personal sampling methods (as a set of sampler, filter and pump) used today comprise air flows up to 10 l/min. In order to achieve higher amounts of sampled aerosols an airflow of 20 l/min is aimed for a new set of samplers for the inhalable and respirable dust fraction. This also enables shorter sampling times than a full shift, e.g. task based exposure assessment. A small set of prototypes was designed to go into testing. A filter size of 47 mm was chosen to realise a higher dust load and an acceptable pressure drop for a personal air sampling pump. Such a pump was recently developed and is now available. Samplers for the inhalable fraction with different numbers and orientation of orifices are tested in a calm air dust chamber and a wind tunnel according to the test standard EN 13205 in the particle size range up to 100 µm. A cyclone design according to the GK family of cyclones (Mesalabs) was chosen to meet the respirable convention. First test results will be shown, either in comparison with validated samplers or in terms of the sampling efficiency in dependence of the aerodynamic diameter.

Zero Temperature Limits for Quotients of Potentials in Countable Markov Shifts

Zero Temperature Limits for Quotients of Potentials in Countable Markov Shifts

Binary factors of shifts of finite type

AbstractWe construct two new classes of topological dynamical systems; one is a factor of a one-sided shift of finite type while the second is a factor of the two-sided shift. The data are a finite graph which presents the shift of finite type, a second finite directed graph and a pair of embeddings of it into the first, satisfying certain conditions. The factor is then obtained from a simple idea based on binary expansion of real numbers. In both cases, we construct natural metrics on the factors and, in the second case, this makes the system a Smale space, in the sense of Ruelle. We compute various algebraic invariants for these systems, including the homology for Smale space developed by the author and the K-theory of various $C^{*}$ -algebras associated to them, in terms of the pair of original graphs.

Japanese Pharmacists’ Perceptions of Self-Development Skills and Continuing Professional Development

The role of healthcare professionals, including pharmacists, is changing. Lifelong learning and continuing professional development (CPD) are more critical than ever for both current and future pharmacists in the face of global health challenges and new technologies, services and therapies that are continually and rapidly introduced into their daily practice. Currently, Japanese pharmacists' licences are not renewable, although most developed countries have a renewal system. Therefore, understanding Japanese pharmacists' perceptions of CPD is the first step in reviewing undergraduate and postgraduate education. The target population was Japanese pharmacists, i.e., community pharmacy pharmacists and hospital pharmacists. The participants were administered a questionnaire with 18 items related to continuing professional development. Our study found that regarding item "Q16 Do you think you need further education in your undergraduate education to continue your professional development?", (a) the ability to identify one's own problems and issues, (b) the ability to make plans to solve problems and issues, (c) the ability to carry out plans to solve problems and issues and (d) the ability to repeat steps of self-development, approximately 60% of pharmacists answered that these aspects were "necessary" or "quite necessary". As part of universities' responsibility for the lifelong education of pharmacists, it is necessary to systematically conduct teaching seminars or undergraduate education or postgraduate education on self-development while training pharmacists to meet the needs of citizens.

Continuous orbit equivalence rigidity for left-right wreath product actions

Drimbe and Vaes proved an orbit equivalence superrigidity theorem for left-right wreath product actions in the measurable setting. We establish the counterpart result in the topological setting for continuous orbit equivalence. This gives us minimal, topologically free actions that are continuous orbit equivalence superrigid. One main ingredient for the proof is to show continuous cocycle superrigidity for certain generalized full shifts, extending our previous result with Chung.

Spectrum, algebraicity and normalization in alternate bases

The first aim of this article is to give information about the algebraic properties of alternate bases β=(β0,…,βp−1) determining sofic systems. We show that a necessary condition is that the product δ=∏i=0p−1βi is an algebraic integer and all of the bases β0,…,βp−1 belong to the algebraic field Q(δ). On the other hand, we also give a sufficient condition: if δ is a Pisot number and β0,…,βp−1∈Q(δ), then the system associated with the alternate base β=(β0,…,βp−1) is sofic. The second aim of this paper is to provide an analogy of Frougny's result concerning normalization of real bases representations. We show that given an alternate base β=(β0,…,βp−1) such that δ is a Pisot number and β0,…,βp−1∈Q(δ), the normalization function is computable by a finite Büchi automaton, and furthermore, we effectively construct such an automaton. An important tool in our study is the spectrum of numeration systems associated with alternate bases. The spectrum of a real number δ>1 and an alphabet A⊂Z was introduced by Erdős et al. For our purposes, we use a generalized concept with δ∈C and A⊂C and study its topological properties.

Subgroups of continuous full groups and relative continuous orbit equivalences of one-sided topological Markov shifts

Let A be an irreducible non-permutation matrix with entries in {0,1}. We study a family ΓA,f of subgroups of the continuous full group ΓA of the one-sided topological Markov shift (XA,σA). The subgroup ΓA,f is indexed by an integer valued continuous function f on XA such that ΓA,0=ΓA the non-amenable continuous full group and ΓA,1=ΓAAF the amenable AF full group. We will first prove that a spatial realization theorem for the subgroups ΓA,f under certain conditions on the functions f. We will second introduce relative versions of continuous orbit equivalence to study classification of the subgroups ΓA,f related to their associated groupoids and C⁎-algebras.

Continuity of isomorphisms applied to rigidity problems of entropy spectra

For a fixed topological Markov shift, we consider measure-preserving dynamical systems of Gibbs measures for 2-locally constant functions on the shift. We also consider isomorphisms between two such systems. We study the set of all 2-locally constant functions f on the shift such that all those isomorphisms defined on the system associated with f are induced from automorphisms of the shift. We prove that this set contains a full-measure open set of the space of all 2-locally constant functions on the shift. We apply this result to rigidity problems of entropy spectra and show that the strong non-rigidity occurs if and only if so does the weak non-rigidity.

Conditioned limit theorems for hyperbolic dynamical systems

AbstractLet $({\mathbb X}, T)$ be a subshift of finite type equipped with the Gibbs measure $\nu $ and let f be a real-valued Hölder continuous function on ${\mathbb X}$ such that $\nu (f) = 0$ . Consider the Birkhoff sums $S_n f = \sum _{k=0}^{n-1} f \circ T^{k}$ , $n\geqslant 1$ . For any $t \in {\mathbb R}$ , denote by $\tau _t^f$ the first time when the sum $t+ S_n f$ leaves the positive half-line for some $n\geqslant 1$ . By analogy with the case of random walks with independent and identically distributed increments, we study the asymptotic as $ n\to \infty $ of the probabilities $ \nu (x\in {\mathbb X}: \tau _t^f(x)>n) $ and $ {\nu (x\in {\mathbb X}: \tau _t^f(x)=n) }$ . We also establish integral and local-type limit theorems for the sum $t+ S_n f(x)$ conditioned on the set $\{ x \in {\mathbb X}: \tau _t^f(x)>n \}.$

Dimension of invariant measures for affine iterated function systems

Let {Si}i∈Λ be a finite contracting affine iterated function system (IFS) on Rd. Let (Σ,σ) denote the two-sided full shift over the alphabet Λ, and let π:Σ→Rd be the coding map associated with the IFS. We prove that the projection of an ergodic σ-invariant measure on Σ under π is always exact dimensional, and its Hausdorff dimension satisfies a Ledrappier–Young-type formula. Furthermore, the result extends to average contracting affine IFSs. This completes several previous results and answers a folklore open question in the community of fractals. Some applications are given to the dimension of self-affine sets and measures.

Effect of mobile phone-based health insurance contribution payment system on retention of coverage in the National Health Insurance Scheme in Ghana: an evaluation study

BackgroundGhana introduced a mobile phone-based contribution payment system in its national health insurance scheme (NHIS) in December 2018 to improve the process of enrolment. We evaluated the effect of this digital health intervention on retention of coverage in the Scheme, one year after its implementation.MethodsWe used NHIS enrolment data for the period, 1 December 2018–31 December 2019. Descriptive statistics and propensity-score matching method were performed to examine a sample of 57,993 members’ data.ResultsProportion of members who renewed their membership in the NHIS via the mobile phone-based contribution payment system increased from 0% to 8.5% whilst those who did so through the office-based system only grew from 4.7% to 6.4% over the study period. The chance of renewing membership was higher by 17.4 percentage points for users of the mobile phone-based contribution payment system, compared to those who used the office-based contribution payment system. The effect was greater for the informal sector workers, males and the unmarried.ConclusionsThe mobile phone-based health insurance renewal system is improving coverage in the NHIS particularly for members who hitherto were less likely to renew their membership. Policy makers need to devise an innovative way for new members and all member categories to enrol using this payment system to accelerate progress towards attainment of universal health coverage. Further study needs to be conducted using mixed-method design with inclusion of more variables.

Machine Learning Full NMR Chemical Shift Tensors of Silicon Oxides with Equivariant Graph Neural Networks.

The nuclear magnetic resonance (NMR) chemical shift tensor is a highly sensitive probe of the electronic structure of an atom and furthermore its local structure. Recently, machine learning has been applied to NMR in the prediction of isotropic chemical shifts from a structure. Current machine learning models, however, often ignore the full chemical shift tensor for the easier-to-predict isotropic chemical shift, effectively ignoring a multitude of structural information available in the NMR chemical shift tensor. Here we use an equivariant graph neural network (GNN) to predict full 29Si chemical shift tensors in silicate materials. The equivariant GNN model predicts full tensors to a mean absolute error of 1.05 ppm and is able to accurately determine the magnitude, anisotropy, and tensor orientation in a diverse set of silicon oxide local structures. When compared with other models, the equivariant GNN model outperforms the state-of-the-art machine learning models by 53%. The equivariant GNN model also outperforms historic analytical models by 57% for isotropic chemical shift and 91% for anisotropy. The software is available as a simple-to-use open-source repository, allowing similar models to be created and trained with ease.

Multifractal analysis for Markov interval maps with countably many branches

We study multifractal decompositions based on Birkhoff averages for sequences of functions belonging to certain classes of symbolically continuous functions. We do this for an expanding interval map with countably many branches, which we assume can be coded by a topologically mixing countable Markov shift. This generalises previous work on expanding maps with finitely many branches, and expanding maps with countably many branches where the coding is assumed to be the full shift. When the infimum of the derivative on each branch approaches infinity in the limit, we can directly generalise the results of the full countable shift case. However when this does not hold, we show that there can be different behaviour, in particular in cases where the coding has finite topological entropy.

High pointwise emergence via saturated sets

In this article, we prove that the set of points whose pointwise emergence have same degree in with repect to polynomial has strong dynamical complexity in sense of entropy, residuality and density, and distributional chaos for a β-shift, a C 1 diffeomorphism having a nontrival homoclinic class, or a diffeomorphism preserving a hyperbolic ergodic measure. In this process, we construct nonempty compact connected set with fixed box-counting dimension in the space of invariant measures. Then combining with the relation between pointwise emergence of saturated set and box-counting dimension, we get the goal from two frameworks about dynamical complexity of saturated set. One framework was introduced by C. Pfister and W. Sullivan and can be applicable to transitive Anosov diffeomorphisms, mixing expanding maps, β-shifts, transitive subshifts of finite type, mixing sofic subshifts, and mixing S-gap shifts. The other framework is applicable to general homoclinic classes and diffeomorphisms preserving hyperbolic ergodic measures. We also get some combined observations with irregular set.

Generalized Fibonacci shifts in the Lorenz attractor

In this article we deal with symmetric Lorenz attractors having a homoclinic loop that exhibits a well ordered orbit. We show the symmetry implies a very regular behaviour on the dynamic in the topological and metric sense. Let ([−1,1],f) be the one-dimensional reduction Lorenz map satisfying a well ordered orbit and ([−1,0],f̃) be the quotient map, given by the equivalence relation x∼−x, the dynamic of f˜ is described explicitly as a subshift of finite type which generalizes the Fibonacci shifts and this fact is used to compute topological entropy of f.Moreover we show that in general ([−1,0],f̃) is related to a factor of the k-bonacci shift. In particular we found that the 1-dimensional Lorenz map replicates an interesting duplicating behaviour of the k-bonacci shift found in Sirvent (1996, 2011).

The entropy of multiplicative subshifts on trees

The aim of this paper is to study the topological entropy of a q-multiplicative subshift on tree (q-MST), that is, every boundary of the tree follows the rule of some one-dimensional multiplicative integer systems defined by Kenyon et al. [20]. Supposing the number of vertices in the tree T grows exponentially, the explicit formula for the topological entropy of the q-MST is presented in Theorem 1.1. Next, we consider the coupled q-MST, namely, every boundary of T satisfies the q-multiple constraint and is characterized by subshift of finite type. The comparison of topological entropy for the coupled and non-coupled q-MSTs is provided in Theorem 1.4. Furthermore, the explicit formula of some coupled q-MSTs are also obtained herein. Finally, the entropy formula for the general q-MSTs, known as XΩ(S), is described in Theorem 1.6.

Construction and applications of proximal maps for typical cocycles

AbstractFor typical cocycles over subshifts of finite type, we show that for any given orbit segment, we can construct a periodic orbit such that it shadows the given orbit segment and that the product of the cocycle along its orbit is a proximal linear map. Using this result, we show that suitable assumptions on the periodic orbits have consequences over the entire subshift.

Topological dynamics of Markov multi-maps of the interval

We study Markov multi-maps of the interval from the point of view of topological dynamics. Specifically, we define the forward trajectory system of a Markov multi-map, and we investigate whether it has properties such as topological transitivity, topological mixing, density of periodic points, and specification. To each Markov multi-map we associate a shift of finite type (SFT), and our main results relate the properties of the SFT with those of the forward trajectory system. Under a general coding condition, we establish necessary and sufficient conditions for topological transitivity and topological mixing of the forward trajectory system. These results complement existing work showing a relationship between the topological entropy of a Markov multi-map and its associated SFT. We also characterize when the inverse limit systems associated to the Markov multi-maps have the properties mentioned above.

Zeta Function for Commuting Matrix Shift of Finite Type

In this paper we show two-dimensional shifts of finite type whose transition matrices commutes and such that the entries of their product also consist of 0’s and 1’s. In particular, the form of associated zeta function of the two-dimensional shift was obtained. Unlike in the one-dimensional shifts of finite type; we gather that zeta functions for them have not closed form. This result derived by obtaining a formula for the number of periodic points of the subshift of finite type. This formula is then incorporated in the corresponding zeta function.