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A common approach to evaluating the significance of a collection of $p$-values combines them with a pooling function, in particular when the original data are not available. These pooled $p$-values convert a sample of $p$-values into a single number which behaves like a univariate $p$-value. To clarify discussion of these functions, a telescoping series of alternative hypotheses are introduced that communicate the strength and prevalence of non-null evidence in the $p$-values before general pooling formulae are discussed. A pattern noticed in the UMP pooled $p$-value for a particular alternative motivates the definition and discussion of central and marginal rejection levels at $\alpha$. It is proven that central rejection is always greater than or equal to marginal rejection, motivating a quotient to measure the balance between the two for pooled $p$-values. A combining function based on the $\chi^2_{\kappa}$ quantile transformation is proposed to control this quotient and shown to be robust to mis-specified parameters relative to the UMP. Different powers for different parameter settings motivate a map of plausible alternatives based on where this pooled $p$-value is minimized.

Let $G$ be a graph and $S\subseteq V(G)$ with $|S|\geq 2$. Then the trees $T_1, T_2, \cdots, T_\ell$ in $G$ are \emph{internally disjoint Steiner trees} connecting $S$ (or $S$-Steiner trees) if $E(T_i) \cap E(T_j )=\emptyset$ and $V(T_i)\cap V(T_j)=S$ for every pair of distinct integers $i,j$, $1 \leq i, j \leq \ell$. Similarly, if we only have the condition $E(T_i) \cap E(T_j )=\emptyset$ but without the condition $V(T_i)\cap V(T_j)=S$, then they are \emph{edge-disjoint Steiner trees}. The \emph{generalized $k$-connectivity}, denoted by $\kappa_k(G)$, of a graph $G$, is defined as $\kappa_k(G)=\min\{\kappa_G(S)|S \subseteq V(G) \ \textrm{and} \ |S|=k \}$, where $\kappa_G(S)$ is the maximum number of internally disjoint $S$-Steiner trees. The \emph{generalized local edge-connectivity} $\lambda_{G}(S)$ is the maximum number of edge-disjoint Steiner trees connecting $S$ in $G$. The {\it generalized $k$-edge-connectivity} $\lambda_k(G)$ of $G$ is defined as $\lambda_k(G)=\min\{\lambda_{G}(S)\,|\,S\subseteq V(G) \ and \ |S|=k\}$. These measures are generalizations of the concepts of connectivity and edge-connectivity, and they and can be used as measures of vulnerability of networks. It is, in general, difficult to compute these generalized connectivities. However, there are precise results for some special classes of graphs. In this paper, we obtain the exact value of $\lambda_{k}(S(n,\ell))$ for $3\leq k\leq \ell^n$, and the exact value of $\kappa_{k}(S(n,\ell))$ for $3\leq k\leq \ell$, where $S(n, \ell)$ is the Sierpi\'{n}ski graphs with order $\ell^n$. As a direct consequence, these graphs provide additional interesting examples when $\lambda_{k}(S(n,\ell))=\kappa_{k}(S(n,\ell))$. We also study the some network properties of Sierpi\'{n}ski graphs.

Brown and Walker (1997) showed that GMRES determines a least squares solution of $ A x = b $ where $ A \in {\bf R}^{n \times n} $ without breakdown for arbitrary $ b, x_0 \in {\bf R}^n $ if and only if $A$ is range-symmetric, i.e. $ {\cal R} (A^{\rm T}) = {\cal R} (A) $, where $ A $ may be singular and $ b $ may not be in the range space ${\cal R} A)$ of $A$. In this paper, we propose applying GMRES to $ A C A^{\rm T} z = b $, where $ C \in {\bf R}^{n \times n} $ is symmetric positive definite. This determines a least squares solution $ x = CA^{\rm T} z $ of $ A x = b $ without breakdown for arbitrary (singular) matrix $A \in {\bf R}^{n \times n}$ and $ b \in {\bf R}^n $. To make the method numerically stable, we propose using the pseudoinverse with an appropriate threshold parameter to suppress the influence of tiny singular values when solving the severely ill-conditioned Hessenberg systems which arise in the Arnoldi process of GMRES when solving inconsistent range-symmetric systems. Numerical experiments show that the method taking $C$ to be the identity matrix gives the least squares solution even when $A$ is not range-symmetric, including the case when $ {\rm index}(A) >1$.

We describe Bayes factors functions based on z, t, $\chi^2$, and F statistics and the prior distributions used to define alternative hypotheses. The non-local alternative prior distributions are centered on standardized effects, which index the Bayes factor function. The prior densities include a dispersion parameter that models the variation of effect sizes across replicated experiments. We examine the convergence rates of Bayes factor functions under true null and true alternative hypotheses. Several examples illustrate the application of the Bayes factor functions to replicated experimental designs and compare the conclusions from these analyses to other default Bayes factor methods.

Let ${\mathcal P}$ be a family of probability measures on a measurable space $(S,{\mathcal A}).$ Given a Banach space $E,$ a functional $f:E\mapsto {\mathbb R}$ and a mapping $\theta: {\mathcal P}\mapsto E,$ our goal is to estimate $f(\theta(P))$ based on i.i.d. observations $X_1,\dots, X_n\sim P, P\in {\mathcal P}.$ In particular, if ${\mathcal P}=\{P_{\theta}: \theta\in \Theta\}$ is an identifiable statistical model with parameter set $\Theta\subset E,$ one can consider the mapping $\theta(P)=\theta$ for $P\in {\mathcal P}, P=P_{\theta},$ resulting in a problem of estimation of $f(\theta)$ based on i.i.d. observations $X_1,\dots, X_n\sim P_{\theta}, \theta\in \Theta.$ Given a smooth functional $f$ and estimators $\hat \theta_n(X_1,\dots, X_n), n\geq 1$ of $\theta(P),$ we use these estimators, the sample split and the Taylor expansion of $f(\theta(P))$ of a proper order to construct estimators $T_f(X_1,\dots, X_n)$ of $f(\theta(P)).$ For these estimators and for a functional $f$ of smoothness $s\geq 1,$ we prove upper bounds on the $L_p$-errors of estimator $T_f(X_1,\dots, X_n)$ under certain moment assumptions on the base estimators $\hat \theta_n.$ We study the performance of estimators $T_f(X_1,\dots, X_n)$ in several concrete problems, showing their minimax optimality and asymptotic efficiency. In particular, this includes functional estimation in high-dimensional models with many low dimensional components, functional estimation in high-dimensional exponential families and estimation of functionals of covariance operators in infinite-dimensional subgaussian models.

Matrix-product constructions giving rise to locally recoverable codes are considered, both the classical $r$ and $(r,\delta)$ localities. We study the recovery advantages offered by the constituent codes and also by the defining matrices of the matrix product codes. Finally, we extend these methods to a particular variation of matrix-product codes and quasi-cyclic codes. Singleton-optimal locally recoverable codes and almost Singleton-optimal codes, with length larger than the finite field size, are obtained, some of them with superlinear length.

The Fast Fourier Transform (FFT) over a finite field $\mathbb{F}_q$ computes evaluations of a given polynomial of degree less than $n$ at a specifically chosen set of $n$ distinct evaluation points in $\mathbb{F}_q$. If $q$ or $q-1$ is a smooth number, then the divide-and-conquer approach leads to the fastest known FFT algorithms. Depending on the type of group that the set of evaluation points forms, these algorithms are classified as multiplicative (Math of Comp. 1965) and additive (FOCS 2014) FFT algorithms. In this work, we provide a unified framework for FFT algorithms that include both multiplicative and additive FFT algorithms as special cases, and beyond: our framework also works when $q+1$ is smooth, while all known results require $q$ or $q-1$ to be smooth. For the new case where $q+1$ is smooth (this new case was not considered before in literature as far as we know), we show that if $n$ is a divisor of $q+1$ that is $B$-smooth for a real $B>0$, then our FFT needs $O(Bn\log n)$ arithmetic operations in $\mathbb{F}_q$. Our unified framework is a natural consequence of introducing the algebraic function fields into the study of FFT.

A periodic temporal graph $\mathcal{G}=(G_0, G_1, \dots, G_{p-1})^*$ is an infinite periodic sequence of graphs $G_i=(V,E_i)$ where $G=(V,\cup_i E_i)$ is called the footprint. Recently, the arena where the Cops and Robber game is played has been extended from a graph to a periodic graph; in this case, the copnumber is also the minimum number of cops sufficient for capturing the robber. We study the connections and distinctions between the copnumber $c(\mathcal{G})$ of a periodic graph $\mathcal{G}$ and the copnumber $c(G)$ of its footprint $G$ and establish several facts. For instance, we show that the smallest periodic graph with $c(\mathcal{G}) = 3$ has at most $8$ nodes; in contrast, the smallest graph $G$ with $c(G) = 3$ has $10$ nodes. We push this investigation by generating multiple examples showing how the copnumbers of a periodic graph $\mathcal{G}$, the subgraphs $G_i$ and its footprint $G$ can be loosely tied. Based on these results, we derive upper bounds on the copnumber of a periodic graph from properties of its footprint such as its treewidth.

We give a strongly explicit construction of $\varepsilon$-approximate $k$-designs for the orthogonal group $\mathrm{O}(N)$ and the unitary group $\mathrm{U}(N)$, for $N=2^n$. Our designs are of cardinality $\mathrm{poly}(N^k/\varepsilon)$ (equivalently, they have seed length $O(nk + \log(1/\varepsilon)))$; up to the polynomial, this matches the number of design elements used by the construction consisting of completely random matrices.

A code $C \subseteq \{0, 1, 2\}^n$ of length $n$ is called trifferent if for any three distinct elements of $C$ there exists a coordinate in which they all differ. By $T(n)$ we denote the maximum cardinality of trifferent codes with length. $T(5)=10$ and $T(6)=13$ were recently determined. Here we determine $T(7)=16$, $T(8)=20$, and $T(9)=27$. For the latter case $n=9$ there also exist linear codes attaining the maximum possible cardinality $27$.