Even Kakutani Equivalence And The Loose Block Independence Property For Positive Entropy Zᵈ Actions

In this paper we define the loose block independence property for positive entropy Zᵈ actions and extend some of the classical results to higher dimensions. In particular, we prove that two loose block independent actions are even Kakutani equivalent if and only if they have the same entropy. We also prove that for d \u3e 1 the ergodic, isometric extensions of the positive entropy loose block independent Zᵈ actions are also loose block independent

Auslander Systems

The authors generalize the dynamical system constructed by J. Auslander in 1959, resulting in perhaps the simplest family of examples of minimal but not strictly ergodic systems. A characterization of unique ergodicity and mean-L-stability is given. The new systems are also shown to have zero topological entropy and fail to be weakly rigid. Some results on the set of idempotents in the enveloping semigroup are also achieved

The Decomposition Theorem For Two-Dimensional Shifts Of Finite Type

A one-dimensional shift of finite type can be described as the collection of bi-infinite walks along an edge graph. The Decomposition Theorem states that every conjugacy between two shifts of finite type can be broken down into a finite sequence of splittings and amalgamations of their edge graphs. When dealing with two-dimensional shifts of finite type, the appropriate edge graph description is not as clear; we turn to Nasu\u27s notion of a textile system for such a description and show that all two-dimensional shifts of finite type can be so described. We then define textile splittings and amalgamations and prove that every conjugacy between two-dimensional shifts of finite type can be broken down into a finite sequence of textile splittings, textile amalgamations, and a third operation called an inversion

Putting The Pieces Together: Understanding Robinson’s Nonperiodic Tilings

A discussion of Robinson\u27s nonperiodic tilings and nonperiodic tilings with nonsquare tiles (Penrose and pinwheel)

Renewal Systems, Sharp-Eyed Snakes, And Shifts Of Finite Type

Johnson and Madden look at collections of bi-infinite strings of symbols that occur in several different areas of mathematics and ask whether these collections are the same in some sense. A dynamical systems property called entropy can be used to show that the shifts of finite type are not all conjugate to uniquely decipherable renewal systems

Projectional Entropy In Higher Dimensional Shifts Of Finite Type

Any higher dimensional shift space (X, ℤᵈ) contains many lower dimensional shift spaces obtained by projection onto r-dimensional sublattices L of ℤᵈ where r \u3c d. We show here that any projectional entropy is bounded below by the ℤᵈ entropy and, in the case of certain shifts of finite type satisfying a mixing condition, equality is achieved if and only if the shift of finite type is the infinite product of a lower dimensional projection

The Art of Teaching Mathematics

On June 10–12, 2007, Harvey Mudd College hosted A Conference on the Art of Teaching Mathematics. The conference brought together approximately thirty mathematicians from the Claremont Colleges, Denison, DePauw, Furman, Middlebury, Penn State, Swarthmore, and Vassar to explore the topic of teaching as an art. Assuming there is an element of artistic creativity in teaching mathematics, in what ways does it surface and what should we be doing to develop this creativity

Characterizing the Effects of Chronic 2G Centrifugation on the Rat Skeletal System

During weightlessness, the skeletal system of astronauts is negatively affected by decreased calcium absorption and bone mass loss. Therefore, it is necessary to counteract these changes for long-term skeletal health during space flights. Our long-term plan is to assess artificial gravity (AG) as a possible solution to mitigate these changes. In this study, we aim to determine the skeletal acclimation to chronic centrifugation. We hypothesize that a 2G hypergravity environment causes an anabolic response in growing male rats. Specifically, we predict chronic 2G to increase tissue mineral density, bone volume fraction of the cancellous tissue and to increase overall bone strength. Systemically, we predict that bone formation markers (i.e., osteocalcin) are elevated and resorption markers (i.e., tartrate resistant acid phosphatase) are decreased or unchanged from controls. The experiment has three groups, each with an n8: chronic 2g, cage control (housed on the centrifuge, but not spun), and a vivarium control (normal rat caging). Pre-pubescent, male Long-Evans rats were used to assess our hypothesis. This group was subject to 90 days of 2G via centrifugation performed at the Chronic Acceleration Research Unit (CARU) at University of California Davis. After 90 days, animals were euthanized and tissues collected. Blood was drawn via cardiac puncture and the right leg collected for structural (via microcomputed tomography) and strength quantification. Understanding how counteract these skeletal changes will have major impacts for both the space-faring astronauts and the people living on Earth