Mathematics

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    Orders of accumulation of entropy and random subshifts of finite type
    (2011) McGoff, Kevin Alexander; Boyle, McBlaine M; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    For a continuous map T of a compact metrizable space X with finite topological entropy, the order of accumulation of entropy of T is a countable ordinal that arises in the context of entropy structure and symbolic extensions. We show that every countable ordinal is realized as the order of accumulation of some dynamical system. Our proof relies on the functional analysis of metrizable Choquet simplices and a realization theorem of Downarowicz and Serafin. Further, if M is a metrizable Choquet simplex, we bound the ordinals that appear as the order of accumulation of entropy of a dynamical system whose simplex of invariant measures is affinely homeomorphic to M. These bounds are given in terms of the Cantor-Bendixson rank of F, the closure of the extreme points of M, and the relative Cantor-Bendixson rank of F with respect to the extreme points of M. We address the optimality of these bounds. Given any compact manifold M and any countable ordinal alpha, we construct a continuous, surjective self-map of M having order of accumulation of entropy alpha. If the dimension of M is at least 2, then the map can be chosen to be a homeomorphism. The realization theorem of Downarowicz and Serafin produces dynamical systems on the Cantor set; by contrast, our constructions work on any manifold and provide a more direct dynamical method of obtaining systems with prescribed entropy properties. Next we consider random subshifts of finite type. Let X be an irreducible shift of finite type (SFT) of positive entropy with its set of words of length n denoted B_n(X). Define a random subset E of B_n(X) by independently choosing each word from B_n(X) with some probability alpha. Let X_E be the (random) SFT built from the set E. For each alpha in [0,1] and n tending to infinity, we compute the limit of the likelihood that X_E; is empty, as well as the limiting distribution of entropy for X_E. For alpha near 1 and n tending to infinity, we show that the likelihood that X_E contains a unique irreducible component of positive entropy converges exponentially to 1.
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    Spectral viscosity approximations to multidimensional scalar conservation laws
    (American Mathematical Society, 1993-10) Chen, Gui-Qiang; Du, Qiang; Tadmor, Eitan
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    Estimation theory of a location parameter in small samples
    (2008-04-22) Yu, Tinghui; Kagan, Abram M; Mathematical Statistics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    The topic of this thesis is estimation of a location parameter in small samples. Chapter 1 is an overview of the general theory of statistical estimates of parameters, with a special attention on the Fisher information, Pitman estimator and their polynomial versions. The new results are in Chapters 2 and 3 where the following inequality is proved for the variance of the Pitman estimator t_n from a sample of size n from a population F(x−\theta): nVar(t_n) >= (n+1)Var(t_{n+1}) for any n >= 1, only under the condition of finite second moments(even the absolute continuity of F is not assumed). The result is much stronger than the known Var(t_n) >= Var(t_{n+1}). Among other new results are (i) superadditivity of 1/Var(t_n) with respect to the sample size: 1/Var(t_{m+n}) >= 1/Var(t_m) + 1/Var(t_n), proved as a corollary of a more general result; (ii) superadditivity of Var(t_n) for a fixed n with respect to additive perturbations; (iii) monotonicity of Var(t_n) with respect to the scale parameter of an additive perturbation when the latter belongs to the class of self-decomposable random variables. The technically most difficult result is an inequality for Var(t_n), which is a stronger version of the classical Stam inequality for the Fisher information. As a corollary, an interesting property of the conditional expectation of the sample mean given the residuals is discovered. Some analytical problems arising in connection with the Pitman estimators are studied. Among them, a new version of the Cauchy type functional equation is solved. All results are extended to the case of polynomial Pitman estimators and to the case of multivariate parameters. In Chapter 4 we collect some open problems related to the theory of location parameters.
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    Zd Symbolic Dynamics: Coding with an Entropy Inequality
    (2006-04-28) Desai, Angela Veronica; Boyle, Michael; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    In this paper we discuss subsystem and coding results in Zd symbolic dynamics for d greater than 1. We prove that any Zd shift of finite type with positive topological entropy has a family of subsystems of finite type whose entropies are dense in the interval from zero to the entropy of the original shift. We show a similar result for Zd sofic shifts, and also show every Zd sofic shift can be covered by a Zd shift of finite type arbitrarily close in entropy. We also show that if a Z² shift of finite type with entropy greater than log N satisfies a certain mixing condition, then it must factor onto the full N-shift.