Estimability and efficiency in nearly orthogonal 2[m1] x 3[m2] deletion designs

This article considers single replicate factorial experiments in incomplete blocks. A single replicate 2^m1 x 3^m2 deletion design in 3 incomplete blocks is obtained from a single replicate 3^m, where m = m_1 + m_2, preliminary design by deleting all runs (or treatment combinations) with the first m_1 factors at the level two. A systematic method for determining the unbiasedly estimable (u.e.) and not unbiasedly estimable (n.u.e.) factorial effects is provided. It is shown that for m_2 > 0 all factorial effects of the type F( α_1 · · · α_m_1 , α_(m_1 +1) · · · α_m), where α_i; = 0, l for i = 1, · · ·, m_1, α_i; = 0, 1, 2 for i = m_(1+1), · · ·, m, with (α_1 · · · α_m) != (0 · · · 0), and (α_m,+l · · · _m) != α(l · · · 1) for a= 1, 2, are u.e. and the remaining factorial effects are n.u.e. It is noted that (2^m_1 - 1) factorial effects of 2^m_1 factorial experiments and (3^m_2) factorial effects of 3^m, factorial experiments, which are embedded in 2^m_1 x 3^m, factorial experiments, are u. e. The 2 x 3m-l deletion designs were considered in the work of Voss (1986). Defining factorial effects of a 2^m_1 x 3^m, factorial experiment in a form different than in Voss (1986), we develop a simple representation of u.e. and n. u. e. factorial effects. In this representation, there are (2^(m_1 + 1) + 1) n. u. e. factorial effects of the type F( α_1 · · · α_m_1, α· · · α). This number is smaller than the corresponding number of n. u. e. factorial effects in the representation of Voss (1986). The relative efficiency expressions, and their bounds, in the estimation of factorial effects of 2^m_1 x 3^m_2 deletion designs are also given

The Ambiguity of Simplicity

A system's apparent simplicity depends on whether it is represented classically or quantally. This is not so surprising, as classical and quantum physics are descriptive frameworks built on different assumptions that capture, emphasize, and express different properties and mechanisms. What is surprising is that, as we demonstrate, simplicity is ambiguous: the relative simplicity between two systems can change sign when moving between classical and quantum descriptions. Thus, notions of absolute physical simplicity---minimal structure or memory---at best form a partial, not a total, order. This suggests that appeals to principles of physical simplicity, via Ockham's Razor or to the "elegance" of competing theories, may be fundamentally subjective, perhaps even beyond the purview of physics itself. It also raises challenging questions in model selection between classical and quantum descriptions. Fortunately, experiments are now beginning to probe measures of simplicity, creating the potential to directly test for ambiguity.Comment: 7 pages, 6 figures, http://csc.ucdavis.edu/~cmg/compmech/pubs/aos.ht

Frozen reaction fronts in steady flows: a burning-invariant-manifold perspective

The dynamics of fronts, such as chemical reaction fronts, propagating in two-dimensional fluid flows can be remarkably rich and varied. For time-invariant flows, the front dynamics may simplify, settling in to a steady state in which the reacted domain is static, and the front appears "frozen". Our central result is that these frozen fronts in the two-dimensional fluid are composed of segments of burning invariant manifolds---invariant manifolds of front-element dynamics in $xy\theta$-space, where $\theta$ is the front orientation. Burning invariant manifolds (BIMs) have been identified previously as important local barriers to front propagation in fluid flows. The relevance of BIMs for frozen fronts rests in their ability, under appropriate conditions, to form global barriers, separating reacted domains from nonreacted domains for all time. The second main result of this paper is an understanding of bifurcations that lead from a nonfrozen state to a frozen state, as well as bifurcations that change the topological structure of the frozen front. Though the primary results of this study apply to general fluid flows, our analysis focuses on a chain of vortices in a channel flow with an imposed wind. For this system, we present both experimental and numerical studies that support the theoretical analysis developed here.Comment: 21 pages, 30 figure

Prediction, Retrodiction, and The Amount of Information Stored in the Present

We introduce an ambidextrous view of stochastic dynamical systems, comparing their forward-time and reverse-time representations and then integrating them into a single time-symmetric representation. The perspective is useful theoretically, computationally, and conceptually. Mathematically, we prove that the excess entropy--a familiar measure of organization in complex systems--is the mutual information not only between the past and future, but also between the predictive and retrodictive causal states. Practically, we exploit the connection between prediction and retrodiction to directly calculate the excess entropy. Conceptually, these lead one to discover new system invariants for stochastic dynamical systems: crypticity (information accessibility) and causal irreversibility. Ultimately, we introduce a time-symmetric representation that unifies all these quantities, compressing the two directional representations into one. The resulting compression offers a new conception of the amount of information stored in the present.Comment: 17 pages, 7 figures, 1 table; http://users.cse.ucdavis.edu/~cmg/compmech/pubs/pratisp.ht

Mode-locking in advection-reaction-diffusion systems: an invariant manifold perspective

Fronts propagating in two-dimensional advection-reaction-diffusion (ARD) systems exhibit rich topological structure. When the underlying fluid flow is periodic in space and time, the reaction front can lock to the driving frequency. We explain this mode-locking phenomenon using so-called burning invariant manifolds (BIMs). In fact, the mode-locked profile is delineated by a BIM attached to a relative periodic orbit (RPO) of the front element dynamics. Changes in the type (and loss) of mode-locking can be understood in terms of local and global bifurcations of the RPOs and their BIMs. We illustrate these concepts numerically using a chain of alternating vortices in a channel geometry.Comment: 9 pages, 13 figure

Information Accessibility and Cryptic Processes: Linear Combinations of Causal States

We show in detail how to determine the time-reversed representation of a stationary hidden stochastic process from linear combinations of its forward-time $\epsilon$-machine causal states. This also gives a check for the $k$-cryptic expansion recently introduced to explore the temporal range over which internal state information is spread.Comment: 6 pages, 9 figures, 2 tables; http://users.cse.ucdavis.edu/~cmg/compmech/pubs/iacplcocs.ht