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Section (fiber bundle)

About: Section (fiber bundle) is a research topic. Over the lifetime, 3683 publications have been published within this topic receiving 66817 citations. The topic is also known as: section of a fiber bundle.


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TL;DR: In this article, the authors studied mathematical models for the unidirectional propagation of long waves in systems that manifest nonlinear and dispersive effects of a particular but common kind.
Abstract: Several topics are studied concerning mathematical models for the unidirectional propagation of long waves in systems that manifest nonlinear and dispersive effects of a particular but common kind. Most of the new material presented relates to the initial-value problem for the equation u$\_{t}$ + u$\_{x}$ + uu$\_{x}$ - u$\_{xxt}$ = 0, (a) whose solution u(x,t) is considered in a class of real nonperiodic functions defined for -$\infty $ < x < $\infty $, t $\geq $ 0. As an approximation derived for moderately long waves of small but finite amplitude in particular physical systems, this equation has the same formal justification as the Korteweg-de Vries equation u$\_{t}$ + u$\_{x}$ + uu$\_{x}$ + u$\_{xxx}$ = 0, (b) with which (a) is to be compared in various ways. It is contended that (a) is in important respects the preferable model, obviating certain problematical aspects of (b) and generally having more expedient mathematical properties. The paper divides into two parts where respectively the emphasis is on descriptive and on rigorous mathematics. In section 2 the origins and immediate properties of equations (a) and (b) are discussed in general terms, and the comparative shortcomings of (b) are reviewed. In the remainder of the paper (section section 3, 4) - which can be read independently of the preceding discussion - an exact theory of (a) is developed. In section 3 3 the existence of classical solutions is proved; and following our main result, theorem 1, several extensions and sidelights are presented. In section 4 solutions are shown to be unique, to depend continuously on their initial values, and also to depend continuously on forcing functions added to the right-hand side of (a). Thus the initial-value problem is confirmed to be classically well set in the Hadamard sense. In appendix 1 a generalization of (a) is considered, in which dispersive effects within a wide class are represented by an abstract pseudo-differential operator. The physical origins of such an equation are explained in the style of section 2, two examples are given deriving from definite physical problems, and an existence theory is outlined. In appendix 2 a technical fact used in section 3 is established.

1,856 citations

Journal ArticleDOI
TL;DR: In this article, the significance of the largest observation in a sample of size $n$ from a normal population was investigated and the authors proposed a new statistic, S^2_n/S^2, to test whether the two largest observations are too large.
Abstract: The problem of testing outlying observations, although an old one, is of considerable importance in applied statistics. Many and various types of significance tests have been proposed by statisticians interested in this field of application. In this connection, we bring out in the Histrical Comments notable advances toward a clear formulation of the problem and important points which should be considered in attempting a complete solution. In Section 4 we state some of the situations the experimental statistician will very likely encounter in practice, these considerations being based on experience. For testing the significance of the largest observation in a sample of size $n$ from a normal population, we propose the statistic $\frac{S^2_n}{S^2} = \frac{\sum^{n-1}_{i=1} (x_i - \bar x_n)^2}{\sum^n_{i=1} (x_i - \bar x)^2}$ where $x_1 \leq x_2 \leq \cdots \leq x_n, \bar x_n = \frac{1}{n - 1} \sum^{n-1}_{i=1} x_i$ and $\bar x = \frac{1}{n}\sum^{n}_{i=1} x_i.$ A similar statistic, $S^2_1/S^2$, can be used for testing whether the smallest observation is too low. It turns out that $\frac{S^2_n}{S^2} = 1 - \frac{1}{n - 1} \big(\frac{x_n - \bar x}{s}\big)^2 = 1 - \frac{1}{n - 1} T^2_n,$ where $s^2 = \frac{1}{n}\sigma(x_i - \bar x)^2,$ and $T_n$ is the studentized extreme deviation already suggested by E. Pearson and C. Chandra Sekar [1] for testing the significance of the largest observation. Based on previous work by W. R. Thompson [12], Pearson and Chandra Sekar were able to obtain certain percentage points of $T_n$ without deriving the exact distribution of $T_n$. The exact distribution of $S^2_n/S^2$ (or $T_n$) is apparently derived for the first time by the present author. For testing whether the two largest observations are too large we propose the statistic $\frac{S^2_{n-1,n}}{S^2} = \frac{\sum^{n-2}_{i=1} (x_i - \bar x_{n-1,n})^2}{\sum^n_{i=1} (x_i - \bar x)^2},\quad\bar x_{n-1,n} = \frac{1}{n - 2} \sum^{n-2}_{i=1} x_i$ and a similar statistic, $S^2_{1,2}/S^2$, can be used to test the significance of the two smallest observations. The probability distributions of the above sample statistics $S^2 = \sum^n_{i=1} (x_i - \bar x)^2 \text{where} \bar x = \frac{1}{n} \sum^n_{i=1} x_i$ $S^2_n = \sum^{n-1}_{i=1} (x_i - \bar x_n)^2 \text{where} \bar x_n = \frac{1}{n-1} \sum^{n-1}_{i=1} x_i$ $S^2_1 = \sum^n_{i=2} (x_i - \bar x_1)^2 \text{where} \bar x_1 = \frac{1}{n-1} \sum^n_{i=2} x_i$ are derived for a normal parent and tables of appropriate percentage points are given in this paper (Table I and Table V). Although the efficiencies of the above tests have not been completely investigated under various models for outlying observations, it is apparent that the proposed sample criteria have considerable intuitive appeal. In deriving the distributions of the sample statistics for testing the largest (or smallest) or the two largest (or two smallest) observations, it was first necessary to derive the distribution of the difference between the extreme observation and the sample mean in terms of the population $\sigma$. This probability$X_1 \leq x_2 \leq x_3 \cdots \leq x_n$ $s^2 = \frac{1}{n} \sum^n_{i=1} (x_i - \bar x)^2 \quad \bar x = \frac{1}{n} \sum^n_{i=1} x_i$ distribution was apparently derived first by A. T. McKay [11] who employed the method of characteristic functions. The author was not aware of the work of McKay when the simplified derivation for the distribution of $\frac{x_n - \bar x}{\sigma}$ outlined in Section 5 below was worked out by him in the spring of 1945, McKay's result being called to his attention by C. C. Craig. It has been noted also that K. R. Nair [20] worked out independently and published the same derivation of the distribution of the extreme minus the mean arrived at by the present author--see Biometrika, Vol. 35, May, 1948. We nevertheless include part of this derivation in Section 5 below as it was basic to the work in connection with the derivations given in Sections 8 and 9. Our table is considerably more extensive than Nair's table of the probability integral of the extreme deviation from the sample mean in normal samples, since Nair's table runs from $n = 2$ to $n = 9,$ whereas our Table II is for $n = 2$ to $n = 25$. The present work is concluded with some examples.

1,401 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that a random variable can be associated with another random variable if the test functions are either (a) binary or (b) bounded and continuous.
Abstract: It is customary to consider that two random variables $S$ and $T$ are associated if $\operatorname{Cov}\lbrack S, T\rbrack = EST - ES\cdot ET$ is nonnegative. If $\operatorname{Cov}\lbrack f(S), g(T)\rbrack \geqq 0$ for all pairs of nondecreasing functions $f, g$, then $S$ and $T$ may be considered more strongly associated. Finally, if $\operatorname{Cov}\lbrack f(S, T), g(S, T)\rbrack \geqq 0$ for all pairs of functions $f, g$ which are nondecreasing in each argument, then $S$ and $T$ may be considered still more strongly associated. The strongest of these three criteria has a natural multivariate generalization which serves as a useful definition of association: DEFINITION 1.1. We say random variables $T_1,\cdots, T_n$ are associated if \begin{equation*}\tag{1.1}\operatorname{Cov}\lbrack f(\mathbf{T}), g(\mathbf{T})\rbrack \geqq 0\end{equation*} for all nondecreasing functions $f$ and $g$ for which $Ef(\mathbf{T}), Eg(\mathbf{T}), Ef(\mathbf{T})g(\mathbf{T})$ exist. (Throughout, we use $\mathbf{T}$ for $(T_1,\cdots, T_n)$; also, without further explicit mention we consider only test functions $f, g$ for which $\operatorname{Cov}\lbrack f(\mathbf{T}), g(\mathbf{T})\rbrack$ exists.) In Section 2 we develop the fundamental properties of association: Association of random variables is preserved under (a) taking subsets, (b) forming unions of independent sets, (c) forming sets of nondecreasing functions, (d) taking limits in distribution. In Section 3 we develop some simpler criteria for association. We show that to establish association it suffices to take in (1.1) nondecreasing test functions $f$ and $g$ which are either (a) binary or (b) bounded and continuous. In Section 4 we develop the special properties of association that hold in the case of binary random variables, i.e., random variables that take only the values 0 or 1. These properties turn out to be quite useful in applications. We also discuss association in the bivariate case. We relate our concept of association in this case to several discussed by Lehmann (1966). Finally, in Section 5 applications in probability and statistics are presented yielding results by Robbins (1954), Marshall-Olkin (1966), and Kimball (1951). An application in reliability which motivated our original interest in association will be presented in a forthcoming paper.

1,246 citations

Journal ArticleDOI
TL;DR: In this article, the authors show that the treatment mean square and the treatment group interaction can be tested in the same approximate fashion by using the Box procedure, and that the conservative test would be $F(1, n - 1).
Abstract: The mixed model in a 2-way analysis of variance is characterized by a fixed classification, e.g., treatments, and a random classification, e.g., plots or individuals. If we consider $k$ different treatments each applied to everyone of $n$ individuals, and assume the usual analysis of variance assumptions of uncorrelated errors, equal variances and normality, an appropriate analysis for the set of $nk$ observations $x_{ij}, i = 1, 2, \cdots n, j = 1, 2, \cdots k$, is ???? where the $F$ ratio under the null hypothesis has the $F$ distribution with $(k - 1)$ and $(k - 1)(n - 1)$ degrees of freedom. As is well known, if we extend the situation so that the errors have equal correlations instead of being uncorrelated, the $F$ ratio has the same distribution. Under the null hypothesis, the numerator estimates the same quantity as the denominator, namely, $(1 - \rho)\sigma^2$, where $\rho$ is the constant correlation coefficient among the treatments. This case can also be considered as a sampling of $n$ vectors (individuals) from a $k$-variate normal population with variance-covariance matrix $$V = \sigma^2 \begin{pmatrix} 1 & \rho & \cdots & \rho \\ \rho & & & \vdots \\ \vdots & & & \rho \\ \rho & \cdots & \rho & 1\end{pmatrix}.$$ If we consider this type of formulation and suppose the $k$ treatment errors to have a multivariate normal distribution with unknown variance-covariance matrix (the same for each individual), then the usual test described above is valid for $k = 2$. For $k > 2$, and $n \geqq k$, Hotelling's $T^2$ is the appropriate test for the homogeneity of the treatment means. However, the working statistician is sometimes confronted with the case where $k > n$, or he does not have the adequate means for computing large order inverse matrices and would therefore like to use the original test ratio which in general does not have the requisite $F$ distribution. Box [1] and [2] has given an approximate distribution of the test ratio to be $F\lbrack(k - 1)\epsilon, (k - 1)(n - 1)\epsilon\rbrack$ where $\epsilon$ is a function of the population variances and covariances and may further be approximated by the sample variances and covariances. We show in Section 3 that $\epsilon \geqq (k - 1)^{-1}$, and therefore a conservative test would be $F(1, n - 1)$. Box referred only to one group of $n$ individuals. We shall extend his results to a frequently occurring case, namely, the analysis of $g$ groups where the $\alpha$th group has $n_\alpha$ individuals, $\alpha = 1, 2, \cdots g$, and $\Sigma^g_{\alpha = 1} n_\alpha = N$. We will show that the treatment mean square and the treatment $\times$ group interaction can be tested in the same approximate fashion by using the Box procedure.

1,102 citations

Journal ArticleDOI
A. Jayaraman1
TL;DR: The present status of high pressure research with the diamond anvil cell (DAC) is reviewed in this paper, mainly from an experimental aspect, with a view to illustrating the physics behind high-pressure phenomena, including metal-semiconductor transitions, electronic transitions, phonons and phase transitions.
Abstract: The present status of high-pressure research with the diamond anvil cell (DAC) is reviewed in this article, mainly from an experimental aspect. After a brief description of the different types of DAC's that are currently in vogue, the techniques used in conjunction with the DAC in modern high-pressure research are presented. These include techniques for low- and high-temperature studies, x-ray diffractometry, spectroscopy with the DAC, and other measurements. Results on selected materials, with a view to illustrating the physics behind high-pressure phenomena, are presented and discussed. These include metal-semiconductor transitions, electronic transitions, phonons and high-pressure lattice dynamics, and phase transitions. A whole section is devoted to the behavior of condensed gases, principally ${\mathrm{H}}_{2}$, ${\mathrm{D}}_{2}$, ${\mathrm{O}}_{2}$, ${\mathrm{N}}_{2}$, and rare-gas solids. The concluding section briefly deals with speculations on ultra-high-pressure research with the DAC in the future.

1,083 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20222
2021230
2020176
2019186
2018180
2017159