The effect of projections on fractal sets and measures in Banach spaces

We study the extent to which the Hausdorff dimension of a compact subset of an infinite-dimensional Banach space is affected by a typical mapping into a finite-dimensional space. It is possible that the dimension drops under all such mappings, but the amount by which it typically drops is controlled by the ‘thickness exponent’ of the set, which was defined by Hunt and Kaloshin (Nonlinearity 12 (1999), 1263–1275). More precisely, let $X$ be a compact subset of a Banach space $B$ with thickness exponent $\tau$ and Hausdorff dimension $d$. Let $M$ be any subspace of the (locally) Lipschitz functions from B to $\mathbb{R}^{m}$ that contains the space of bounded linear functions. We prove that for almost every (in the sense of prevalence) function f in M, the Hausdorff dimension of f(X) is at least min{m,d/(1 + tau)}. We also prove an analogous result for a certain part of the dimension spectra of Borel probability measures supported on X. The factor 1/(1 + tau) can be improved to 1/(1 + tau/2) if B is a Hilbert space. Since dimension cannot increase under a (locally) Lipschitz function, these theorems become dimension preservation results when tau = 0. We conjecture that many of the attractors associated with the evolution equations of mathematical physics have thickness exponent zero. We also discuss the sharpness of our results in the case tau > 0

Comment on "Long Time Evolution of Phase Oscillator Systems" [Chaos 19,023117 (2009), arXiv:0902.2773]

A previous paper (arXiv:0902.2773, henceforth referred to as I) considered a general class of problems involving the evolution of large systems of globally coupled phase oscillators. It was shown there that, in an appropriate sense, the solutions to these problems are time asymptotically attracted toward a reduced manifold of system states (denoted M). This result has considerable utility in the analysis of these systems, as has been amply demonstrated in recent papers. In this note, we show that the analysis of I can be modified in a simple way that establishes significant extensions of the range of validity of our previous result. In particular, we generalize I in the following ways: (1) attraction to M is now shown for a very general class of oscillator frequency distribution functions g(\omega), and (2) a previous restriction on the allowed class of initial conditions is now substantially relaxed

Bridging scholarly theory and forensic practice: toward a more pedagogical model of rhetorical criticism

Brian Ott was a professor in the Department of Speech Communication at Colorado State University.Includes bibliographical references (pages 71-74).In this essay, the author contends that competitors in the event of rhetorical criticism, or communication analysis (CA) as it is alternatively called, are locked into a model that poses serious questions about the educational value of the event. In an effort to narrow the ever widening gap between theory and practice and to heighten the pedagogical value of contest rhetorical criticism, the author proposes to chart briefly the chief features of the existing RC model, to identify the limitations posed by that model, and to suggest several viable alternatives

Synchronization in large directed networks of coupled phase oscillators

We extend recent theoretical approximations describing the transition to synchronization in large undirected networks of coupled phase oscillators to the case of directed networks. We also consider extensions to networks with mixed positive/negative coupling strengths. We compare our theory with numerical simulations and find good agreement