Piecewise-linear maps with heterogeneous chaos

Chaotic dynamics can be quite heterogeneous in the sense that in some regions the dynamics are unstable in more directions than in other regions. When trajectories wander between these regions, the dynamics is complicated. We say a chaotic invariant set is heterogeneous when arbitrarily close to each point of the set there are different periodic points with different numbers of unstable dimensions. We call such dynamics heterogeneous chaos (or hetero-chaos), While we believe it is common for physical systems to be hetero-chaotic, few explicit examples have been proved to be hetero-chaotic. Here we present two more explicit dynamical systems that are particularly simple and tractable with computer. It will give more intuition as to how complex even simple systems can be. Our maps have one dense set of periodic points whose orbits are 1D unstable and another dense set of periodic points whose orbits are 2D unstable. Moreover, they are ergodic relative to the Lebesgue measure.Comment: 16 pages, 9 figure

A Study of Secondary School Principals\u27 Perceptions Regarding Current Teacher Evaluation Procedures in Three Midwestern States

In recent years, new methods of evaluating teachers have been introduced and recommended by educational experts. This study was conducted to determine the methods and procedures currently used in three midwestern states to evaluate secondary education teachers and to investigate secondary school principals\u27 perceptions regarding those methods and procedures. The study, which took place during the spring of 1996, included a survey of a random sample of 300 secondary school principals in Illinois, Iowa and Indiana through the National Association of Secondary School Principals. One hundred and fifty-six principals responded to the survey which found that although principals agreed that alternative evaluation methods such as portfolios and video tapes were excellent ways to evaluate teachers, only 17% actually used them. The respondents also indicated that student test scores should not be used to evaluate teachers, but that multiple methods of evaluation for tenured teachers should be used

Atomic and Electronic Structures of Unreconstructed Polar MgO(111) Thin Film on Ag(111)

Atomic and electronic structures of a polar surface of MgO formed on Ag(111) was investigated by using reflection high energy electron diffraction (RHEED), Auger electron spectroscopy, electron energy loss spectroscopy (EELS), and ultraviolet photoemission spectroscopy (UPS). A rather flat unreconstructed polar MgO(111) 1$\times$1 surface could be grown by alternate adsorption of Mg and O$_{2}$ on Ag(111). The stability of the MgO(111) surface was discussed in terms of interaction between Ag and Mg atoms at the interface, and charge state of the surface atoms. EELS of this surface did not show a band gap region, and finite density of states appeared at the Fermi level in UPS. These results suggest that a polar MgO(111) surface was not an insulating surface but a semiconducting or metallic surface.Comment: 6 figures, to be published in Phys. Rev.

Eating behaviors of American adults during the COVID-19 pandemic

The objective was to assess the impact of the COVID-19 pandemic on eating behaviors of a cross section of the US adult population. Participants (N=844) completed a survey via Amazon Mechanical Turk which evaluated frequency of consumption of five food categories - mixed dishes, salty, sweet, creamy foods and beverages during perceived stress. Physical, emotional and psychological self-care and demographics were also collected. Males eating behaviors especially mixed dishes, sweet foods, and creamy foods decreased when stressed. However, in females there was an increase in the consumption of sweet and creamy foods. When compared to their counterpart, there was a reduction in food choices among minorities and Hispanics. Participants with higher BMI had increased eating behaviors of mixed dishes, salty, and sweet foods. The physical self-care scale revealed there was limited consumption of sweet foods but the psychological self-care demonstrated that there was an increase in creamy food intak

Hausdorff dimension of heteroclinic intersections for some partially hyperbolic sets

We introduce a $C^2$-open set of diffeomorphisms of $\mathbb R^3$ which have two transitive hyperbolic sets, one is of index 1 (the dimension of the unstable subbundle) and the other is of index 2. We prove that: the unstable set of the first hyperbolic set and the stable set of the second are of Hausdorff dimension nearly 2; the intersection of these unstable and stable sets contains a set of Hausdorff dimension nearly 1.Comment: 28 pages, 8 figure