Asymptotically Optimal Bounds for (t,2) Broadcast Domination on Finite Grids

Let $G=(V,E)$ be a graph and $t,r$ be positive integers. The \emph{signal} that a tower vertex $T$ of signal strength $t$ supplies to a vertex $v$ is defined as $sig(T,v)=max(t-dist(T,v),0),$ where $dist(T,v)$ denotes the distance between the vertices $v$ and $T$. In 2015 Blessing, Insko, Johnson, and Mauretour defined a \emph{$(t,r)$ broadcast dominating set}, or simply a \emph{$(t,r)$ broadcast}, on $G$ as a set $\mathbb{T}\subseteq V$ such that the sum of all signals received at each vertex $v \in V$ from the set of towers $\mathbb{T}$ is at least $r$. The $(t,r)$ broadcast domination number of a finite graph $G$, denoted $\gamma_{t,r}(G)$, is the minimum cardinality over all $(t,r)$ broadcasts for $G$. Recent research has focused on bounding the $(t,r)$ broadcast domination number for the $m \times n$ grid graph $G_{m,n}$. In 2014, Grez and Farina bounded the $k$-distance domination number for grid graphs, equivalent to bounding $\gamma_{t,1}(G_{m,n})$. In 2015, Blessing et al. established bounds on $\gamma_{2,2}(G_{m,n})$, $\gamma_{3,2}(G_{m,n})$, and $\gamma_{3,3}(G_{m,n})$. In this paper, we take the next step and provide a tight upper bound on $\gamma_{t,2}(G_{m,n})$ for all $t>2$. We also prove the conjecture of Blessing et al. that their bound on $\gamma_{3,2}(G_{m,n})$ is tight for large values of $m$ and $n$.Comment: 8 pages, 4 figure

Structured penalties for functional linear models---partially empirical eigenvectors for regression

One of the challenges with functional data is incorporating spatial structure, or local correlation, into the analysis. This structure is inherent in the output from an increasing number of biomedical technologies, and a functional linear model is often used to estimate the relationship between the predictor functions and scalar responses. Common approaches to the ill-posed problem of estimating a coefficient function typically involve two stages: regularization and estimation. Regularization is usually done via dimension reduction, projecting onto a predefined span of basis functions or a reduced set of eigenvectors (principal components). In contrast, we present a unified approach that directly incorporates spatial structure into the estimation process by exploiting the joint eigenproperties of the predictors and a linear penalty operator. In this sense, the components in the regression are `partially empirical' and the framework is provided by the generalized singular value decomposition (GSVD). The GSVD clarifies the penalized estimation process and informs the choice of penalty by making explicit the joint influence of the penalty and predictors on the bias, variance, and performance of the estimated coefficient function. Laboratory spectroscopy data and simulations are used to illustrate the concepts.Comment: 29 pages, 3 figures, 5 tables; typo/notational errors edited and intro revised per journal review proces

Asymptotically Optimal Bounds for (t,2) Broadcast Domination on Finite Grids

Let G = (V,E) be a graph and t,r be positive integers. The signal that a tower vertex T of signal strength t supplies to a vertex v is defined as sig(T, v) = max(t − dist(T,v),0), where dist(T,v) denotes the distance between the vertices v and T. In 2015 Blessing, Insko, Johnson, and Mauretour defined a (t, r) broadcast dominating set, or simply a (t, r) broadcast, on G as a set T ⊆ V such that the sum of all signal received at each vertex v ∈ V from the set of towers T is at least r. The (t, r) broadcast domination number of a finite graph G, denoted γt,r(G), is the minimum cardinality over all (t,r) broadcasts for G. Recent research has focused on bounding the (t, r) broadcast domination number for the m×n grid graph Gm,n. In 2014, Grez and Farina bounded the k-distance domination number for grid graphs, equivalent to bounding γt,1(Gm,n). In 2015, Blessing et al. established bounds on γ2,2(Gm,n), γ3,2(Gm,n), and γ3,3(Gm,n). In this paper, we take the next step and provide a tight upper bound on γt,2(Gm,n) for all t \u3e 2. We also prove the conjecture of Blessing et al. that their bound on γ3,2(Gm,n) is tight for large values of m and n

Statistical methods for tissue array images - algorithmic scoring and co-training

Recent advances in tissue microarray technology have allowed immunohistochemistry to become a powerful medium-to-high throughput analysis tool, particularly for the validation of diagnostic and prognostic biomarkers. However, as study size grows, the manual evaluation of these assays becomes a prohibitive limitation; it vastly reduces throughput and greatly increases variability and expense. We propose an algorithm - Tissue Array Co-Occurrence Matrix Analysis (TACOMA) - for quantifying cellular phenotypes based on textural regularity summarized by local inter-pixel relationships. The algorithm can be easily trained for any staining pattern, is absent of sensitive tuning parameters and has the ability to report salient pixels in an image that contribute to its score. Pathologists' input via informative training patches is an important aspect of the algorithm that allows the training for any specific marker or cell type. With co-training, the error rate of TACOMA can be reduced substantially for a very small training sample (e.g., with size 30). We give theoretical insights into the success of co-training via thinning of the feature set in a high-dimensional setting when there is "sufficient" redundancy among the features. TACOMA is flexible, transparent and provides a scoring process that can be evaluated with clarity and confidence. In a study based on an estrogen receptor (ER) marker, we show that TACOMA is comparable to, or outperforms, pathologists' performance in terms of accuracy and repeatability.Comment: Published in at http://dx.doi.org/10.1214/12-AOAS543 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Exponential Dichotomy and Mild Solutions of Nonautonomous Equations in Banach Spaces

We prove that the exponential dichotomy of a strongly continuous evolution family on a Banach space is equivalent to the existence and uniqueness of continuous bounded mild solutions of the corresponding inhomogeneous equation. This result addresses nonautonomous abstract Cauchy problems with unbounded coefficients. The technique used involves evolution semigroups. Some applications are given to evolution families on scales of Banach spaces arising in center manifolds theory. © 1998 Plenum Publishing Corporation