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Covariance matrix

In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of a given random vector.

Intuitively, the covariance matrix generalizes the notion of variance to multiple dimensions. As an example, the variation in a collection of random points in two-dimensional space cannot be characterized fully by a single number, nor would the variances in the $$x$$ and $$y$$ directions contain all of the necessary information; a $$2 \times 2$$ matrix would be necessary to fully characterize the two-dimensional variation.

Any covariance matrix is symmetric and positive semi-definite and its main diagonal contains variances (i.e., the covariance of each element with itself).

The covariance matrix of a random vector $$\mathbf{X}$$ is typically denoted by $$\operatorname{K}_{\mathbf{X}\mathbf{X}}$$, $$\Sigma$$ or $$S$$.

Definition
Throughout this article, boldfaced unsubscripted $$\mathbf{X}$$ and $$\mathbf{Y}$$ are used to refer to random vectors, and Roman subscripted $$X_i$$ and $$Y_i$$ are used to refer to scalar random variables.

If the entries in the column vector $$\mathbf{X} = (X_1, X_2, \dots, X_n)^\mathsf{T}$$ are random variables, each with finite variance and expected value, then the covariance matrix $$\operatorname{K}_{\mathbf{X}\mathbf{X}}$$ is the matrix whose $$(i,j)$$ entry is the covariance $$\operatorname{K}_{X_i X_j} = \operatorname{cov}[X_i, X_j] = \operatorname{E}[(X_i - \operatorname{E}[X_i])(X_j - \operatorname{E}[X_j])]$$ where the operator $$\operatorname{E}$$ denotes the expected value (mean) of its argument.

Conflicting nomenclatures and notations
Nomenclatures differ. Some statisticians, following the probabilist William Feller in his two-volume book An Introduction to Probability Theory and Its Applications, call the matrix $$\operatorname{K}_{\mathbf{X}\mathbf{X}}$$ the variance of the random vector $$\mathbf{X}$$, because it is the natural generalization to higher dimensions of the 1-dimensional variance. Others call it the covariance matrix, because it is the matrix of covariances between the scalar components of the vector $$\mathbf{X}$$. $$ \operatorname{var}(\mathbf{X}) = \operatorname{cov}(\mathbf{X},\mathbf{X}) = \operatorname{E} \left[ (\mathbf{X} - \operatorname{E} [\mathbf{X}]) (\mathbf{X} - \operatorname{E} [\mathbf{X}])^\mathsf{T} \right]. $$

Both forms are quite standard, and there is no ambiguity between them. The matrix $$\operatorname{K}_{\mathbf{X}\mathbf{X}}$$ is also often called the variance-covariance matrix, since the diagonal terms are in fact variances.

By comparison, the notation for the cross-covariance matrix between two vectors is $$ \operatorname{cov}(\mathbf{X},\mathbf{Y}) =\operatorname{K}_{\mathbf{X}\mathbf{Y}}= \operatorname{E} \left[ (\mathbf{X} - \operatorname{E}[\mathbf{X}]) (\mathbf{Y} - \operatorname{E}[\mathbf{Y}])^\mathsf{T} \right]. $$

Relation to the autocorrelation matrix
The auto-covariance matrix $$\operatorname{K}_{\mathbf{X}\mathbf{X}}$$ is related to the autocorrelation matrix $$\operatorname{R}_{\mathbf{X}\mathbf{X}}$$ by $$\operatorname{K}_{\mathbf{X}\mathbf{X}} = \operatorname{E}[(\mathbf{X} - \operatorname{E}[\mathbf{X}])(\mathbf{X} - \operatorname{E}[\mathbf{X}])^\mathsf{T}] = \operatorname{R}_{\mathbf{X}\mathbf{X}} - \operatorname{E}[\mathbf{X}] \operatorname{E}[\mathbf{X}]^\mathsf{T}$$ where the autocorrelation matrix is defined as $$\operatorname{R}_{\mathbf{X}\mathbf{X}} = \operatorname{E}[\mathbf{X} \mathbf{X}^\mathsf{T}]$$.

Relation to the correlation matrix
An entity closely related to the covariance matrix is the matrix of Pearson product-moment correlation coefficients between each of the random variables in the random vector $$\mathbf{X}$$, which can be written as $$\operatorname{corr}(\mathbf{X}) = \big(\operatorname{diag}(\operatorname{K}_{\mathbf{X}\mathbf{X}})\big)^{-\frac{1}{2}} \, \operatorname{K}_{\mathbf{X}\mathbf{X}} \, \big(\operatorname{diag}(\operatorname{K}_{\mathbf{X}\mathbf{X}})\big)^{-\frac{1}{2}},$$ where $$\operatorname{diag}(\operatorname{K}_{\mathbf{X}\mathbf{X}})$$ is the matrix of the diagonal elements of $$\operatorname{K}_{\mathbf{X}\mathbf{X}}$$ (i.e., a diagonal matrix of the variances of $$X_i$$  for $$i = 1, \dots, n$$).

Equivalently, the correlation matrix can be seen as the covariance matrix of the standardized random variables $$X_i/\sigma(X_i)$$ for $$i = 1, \dots, n$$. $$ \operatorname{corr}(\mathbf{X}) = \begin{bmatrix} 1 & \frac{\operatorname{E}[(X_1 - \mu_1)(X_2 - \mu_2)]}{\sigma(X_1)\sigma(X_2)} & \cdots & \frac{\operatorname{E}[(X_1 - \mu_1)(X_n - \mu_n)]}{\sigma(X_1)\sigma(X_n)} \\ \\ \frac{\operatorname{E}[(X_2 - \mu_2)(X_1 - \mu_1)]}{\sigma(X_2)\sigma(X_1)} & 1 & \cdots & \frac{\operatorname{E}[(X_2 - \mu_2)(X_n - \mu_n)]}{\sigma(X_2)\sigma(X_n)} \\ \\ \vdots & \vdots & \ddots & \vdots \\ \\ \frac{\operatorname{E}[(X_n - \mu_n)(X_1 - \mu_1)]}{\sigma(X_n)\sigma(X_1)} & \frac{\operatorname{E}[(X_n - \mu_n)(X_2 - \mu_2)]}{\sigma(X_n)\sigma(X_2)} & \cdots & 1 \end{bmatrix}. $$

Each element on the principal diagonal of a correlation matrix is the correlation of a random variable with itself, which always equals 1. Each off-diagonal element is between −1 and +1 inclusive.

Inverse of the covariance matrix
The inverse of this matrix, $$\operatorname{K}_{\mathbf{X}\mathbf{X}}^{-1}$$, if it exists, is the inverse covariance matrix (or inverse concentration matrix), also known as the precision matrix (or concentration matrix).

Just as the covariance matrix can be written as the rescaling of a correlation matrix by the marginal variances: $$\operatorname{cov}(\mathbf{X}) = \begin{bmatrix} \sigma_{x_1} & & & 0\\ & \sigma_{x_2}\\ & & \ddots\\ 0 & & & \sigma_{x_n} \end{bmatrix} \begin{bmatrix} 1 & \rho_{x_1, x_2} & \cdots & \rho_{x_1, x_n}\\ \rho_{x_2, x_1} & 1 & \cdots & \rho_{x_2, x_n}\\ \vdots & \vdots & \ddots & \vdots\\ \rho_{x_n, x_1} & \rho_{x_n, x_2} & \cdots & 1\\ \end{bmatrix} \begin{bmatrix} \sigma_{x_1} & & & 0\\ & \sigma_{x_2}\\ & & \ddots\\ 0 & & & \sigma_{x_n} \end{bmatrix}$$

So, using the idea of partial correlation, and partial variance, the inverse covariance matrix can be expressed analogously: $$\operatorname{cov}(\mathbf{X})^{-1} = \begin{bmatrix} \frac{1}{\sigma_{x_1|x_2...}} & & & 0\\ & \frac{1}{\sigma_{x_2|x_1,x_3...}}\\ & & \ddots\\ 0 & & & \frac{1}{\sigma_{x_n|x_1...x_{n-1}}} \end{bmatrix} \begin{bmatrix} 1 & -\rho_{x_1, x_2\mid x_3...} & \cdots & -\rho_{x_1, x_n\mid x_2...x_{n-1}}\\ -\rho_{x_2, x_1\mid x_3...} & 1 & \cdots & -\rho_{x_2, x_n\mid x_1,x_3...x_{n-1}}\\ \vdots & \vdots & \ddots & \vdots\\ -\rho_{x_n, x_1\mid x_2...x_{n-1}} & -\rho_{x_n, x_2\mid x_1,x_3...x_{n-1}} & \cdots & 1\\ \end{bmatrix} \begin{bmatrix} \frac{1}{\sigma_{x_1|x_2...}} & & & 0\\ & \frac{1}{\sigma_{x_2|x_1,x_3...}}\\ & & \ddots\\ 0 & & & \frac{1}{\sigma_{x_n|x_1...x_{n-1}}} \end{bmatrix}$$ This duality motivates a number of other dualities between marginalizing and conditioning for Gaussian random variables.

Basic properties
For $$\operatorname{K}_{\mathbf{X}\mathbf{X}}=\operatorname{var}(\mathbf{X}) = \operatorname{E} \left[ \left( \mathbf{X} - \operatorname{E}[\mathbf{X}] \right) \left( \mathbf{X} - \operatorname{E}[\mathbf{X}] \right)^\mathsf{T} \right]$$ and $$ \boldsymbol{\mu}_\mathbf{X} = \operatorname{E}[\textbf{X}]$$, where $$\mathbf{X} = (X_1,\ldots,X_n)^\mathsf{T}$$ is a $$n$$-dimensional random variable, the following basic properties apply:
 * 1) $$ \operatorname{K}_{\mathbf{X}\mathbf{X}} = \operatorname{E}(\mathbf{X X^\mathsf{T}}) - \boldsymbol{\mu}_\mathbf{X}\boldsymbol{\mu}_\mathbf{X}^\mathsf{T} $$
 * 2) $$ \operatorname{K}_{\mathbf{X}\mathbf{X}} \,$$ is positive-semidefinite, i.e. $$\mathbf{a}^T \operatorname{K}_{\mathbf{X}\mathbf{X}} \mathbf{a} \ge 0 \quad \text{for all } \mathbf{a} \in \mathbb{R}^n$$
 * 3) $$ \operatorname{K}_{\mathbf{X}\mathbf{X}} \,$$ is symmetric, i.e. $$\operatorname{K}_{\mathbf{X}\mathbf{X}}^\mathsf{T} = \operatorname{K}_{\mathbf{X}\mathbf{X}}$$
 * 4) For any constant (i.e. non-random) $$m \times n$$ matrix $$\mathbf{A}$$ and constant $$m \times 1$$ vector $$\mathbf{a}$$, one has $$ \operatorname{var}(\mathbf{A X} + \mathbf{a}) = \mathbf{A}\, \operatorname{var}(\mathbf{X})\, \mathbf{A}^\mathsf{T} $$
 * 5) If $$\mathbf{Y}$$ is another random vector with the same dimension as $$\mathbf{X}$$, then $$\operatorname{var}(\mathbf{X} + \mathbf{Y}) = \operatorname{var}(\mathbf{X}) + \operatorname{cov}(\mathbf{X},\mathbf{Y}) + \operatorname{cov}(\mathbf{Y}, \mathbf{X}) + \operatorname{var}(\mathbf{Y}) $$ where $$\operatorname{cov}(\mathbf{X}, \mathbf{Y})$$ is the cross-covariance matrix of $$\mathbf{X}$$ and $$\mathbf{Y}$$.

Block matrices
The joint mean $$\boldsymbol\mu$$ and joint covariance matrix $$\boldsymbol\Sigma$$ of $$\mathbf{X}$$ and $$\mathbf{Y}$$ can be written in block form $$ \boldsymbol\mu = \begin{bmatrix} \boldsymbol{\mu}_X \\ \boldsymbol{\mu}_Y \end{bmatrix}, \qquad \boldsymbol\Sigma = \begin{bmatrix} \operatorname{K}_\mathbf{XX} & \operatorname{K}_\mathbf{XY} \\ \operatorname{K}_\mathbf{YX} & \operatorname{K}_\mathbf{YY} \end{bmatrix} $$ where $$ \operatorname{K}_\mathbf{XX} = \operatorname{var}(\mathbf{X}) $$, $$ \operatorname{K}_\mathbf{YY} = \operatorname{var}(\mathbf{Y}) $$ and $$ \operatorname{K}_\mathbf{XY} = \operatorname{K}^\mathsf{T}_\mathbf{YX} = \operatorname{cov}(\mathbf{X}, \mathbf{Y}) $$.

$$ \operatorname{K}_\mathbf{XX} $$ and $$ \operatorname{K}_\mathbf{YY} $$ can be identified as the variance matrices of the marginal distributions for $$ \mathbf{X} $$ and $$ \mathbf{Y} $$ respectively.

If $$\mathbf{X}$$ and $$\mathbf{Y}$$ are jointly normally distributed, $$ \mathbf{X}, \mathbf{Y} \sim\ \mathcal{N}(\boldsymbol\mu, \operatorname{\boldsymbol\Sigma}), $$ then the conditional distribution for $$\mathbf{Y}$$ given $$\mathbf{X}$$ is given by $$ \mathbf{Y} \mid \mathbf{X} \sim\ \mathcal{N}(\boldsymbol{\mu}_\mathbf{Y|X}, \operatorname{K}_\mathbf{Y|X}), $$ defined by conditional mean $$ \boldsymbol{\mu}_{\mathbf{Y}|\mathbf{X}} = \boldsymbol{\mu}_\mathbf{Y} + \operatorname{K}_\mathbf{YX} \operatorname{K}_\mathbf{XX}^{-1} \left( \mathbf{X} - \boldsymbol{\mu}_\mathbf{X} \right) $$ and conditional variance $$ \operatorname{K}_\mathbf{Y|X} = \operatorname{K}_\mathbf{YY} - \operatorname{K}_\mathbf{YX} \operatorname{K}_\mathbf{XX}^{-1} \operatorname{K}_\mathbf{XY}. $$

The matrix $$ \operatorname{K}_\mathbf{YX} \operatorname{K}_\mathbf{XX}^{-1} $$ is known as the matrix of regression coefficients, while in linear algebra $$ \operatorname{K}_\mathbf{Y|X} $$ is the Schur complement of $$ \operatorname{K}_\mathbf{XX} $$ in $$ \boldsymbol\Sigma $$.

The matrix of regression coefficients may often be given in transpose form, $$ \operatorname{K}_\mathbf{XX}^{-1} \operatorname{K}_\mathbf{XY} $$, suitable for post-multiplying a row vector of explanatory variables $$ \mathbf{X}^\mathsf{T} $$ rather than pre-multiplying a column vector $$ \mathbf{X} $$. In this form they correspond to the coefficients obtained by inverting the matrix of the normal equations of ordinary least squares (OLS).

Partial covariance matrix
A covariance matrix with all non-zero elements tells us that all the individual random variables are interrelated. This means that the variables are not only directly correlated, but also correlated via other variables indirectly. Often such indirect, common-mode correlations are trivial and uninteresting. They can be suppressed by calculating the partial covariance matrix, that is the part of covariance matrix that shows only the interesting part of correlations.

If two vectors of random variables $$\mathbf{X}$$ and $$\mathbf{Y}$$ are correlated via another vector $$\mathbf{I}$$, the latter correlations are suppressed in a matrix $$ \operatorname{K}_\mathbf{XY \mid I} = \operatorname{pcov}(\mathbf{X},\mathbf{Y} \mid \mathbf{I}) = \operatorname{cov}(\mathbf{X},\mathbf{Y}) - \operatorname{cov}(\mathbf{X},\mathbf{I}) \operatorname{cov}(\mathbf{I},\mathbf{I})^{-1} \operatorname{cov}(\mathbf{I},\mathbf{Y}). $$ The partial covariance matrix $$\operatorname{K}_\mathbf{XY \mid I}$$ is effectively the simple covariance matrix $$\operatorname{K}_\mathbf{XY}$$ as if the uninteresting random variables $$\mathbf{I}$$ were held constant.

Covariance matrix as a parameter of a distribution
If a column vector $$ \mathbf{X} $$ of $$ n $$ possibly correlated random variables is jointly normally distributed, or more generally elliptically distributed, then its probability density function $$ \operatorname{f}(\mathbf{X}) $$ can be expressed in terms of the covariance matrix $$ \boldsymbol{\Sigma} $$ as follows $$ \operatorname{f}(\mathbf{X}) = (2 \pi)^{-n/2} |\boldsymbol{\Sigma}|^{-1/2} \exp \left ( - \tfrac{1}{2} \mathbf{(X - \mu)^\mathsf{T} \Sigma^{-1} (X - \mu)} \right ), $$ where $$ \boldsymbol{\mu} = \operatorname{E}[\mathbf{X}] $$ and $$ |\boldsymbol{\Sigma}| $$ is the determinant of $$ \boldsymbol{\Sigma} $$.

Covariance matrix as a linear operator
Applied to one vector, the covariance matrix maps a linear combination c of the random variables X onto a vector of covariances with those variables: $$\mathbf c^\mathsf{T} \Sigma = \operatorname{cov}(\mathbf c^\mathsf{T} \mathbf X, \mathbf X)$$. Treated as a bilinear form, it yields the covariance between the two linear combinations: $$\mathbf d^\mathsf{T} \boldsymbol\Sigma \mathbf c = \operatorname{cov}(\mathbf d^\mathsf{T} \mathbf X, \mathbf c^\mathsf{T} \mathbf X)$$. The variance of a linear combination is then $$\mathbf c^\mathsf{T} \boldsymbol\Sigma \mathbf c$$, its covariance with itself.

Similarly, the (pseudo-)inverse covariance matrix provides an inner product $$\langle c - \mu| \Sigma^+ |c - \mu\rangle$$, which induces the Mahalanobis distance, a measure of the "unlikelihood" of c.

Which matrices are covariance matrices?
From the identity just above, let $$\mathbf{b}$$ be a $$(p \times 1)$$ real-valued vector, then $$\operatorname{var}(\mathbf{b}^\mathsf{T}\mathbf{X}) = \mathbf{b}^\mathsf{T} \operatorname{var}(\mathbf{X}) \mathbf{b},\,$$ which must always be nonnegative, since it is the variance of a real-valued random variable, so a covariance matrix is always a positive-semidefinite matrix.

The above argument can be expanded as follows:$$ \begin{align} & w^\mathsf{T} \operatorname{E} \left[(\mathbf{X} - \operatorname{E}[\mathbf{X}]) (\mathbf{X} - \operatorname{E}[\mathbf{X}])^\mathsf{T}\right] w = \operatorname{E} \left[w^\mathsf{T}(\mathbf{X} - \operatorname{E}[\mathbf{X}]) (\mathbf{X} - \operatorname{E}[\mathbf{X}])^\mathsf{T}w\right] \\ &= \operatorname{E} \big[\big( w^\mathsf{T}(\mathbf{X} - \operatorname{E}[\mathbf{X}]) \big)^2 \big] \geq 0, \end{align} $$where the last inequality follows from the observation that $$w^\mathsf{T}(\mathbf{X} - \operatorname{E}[\mathbf{X}])$$ is a scalar.

Conversely, every symmetric positive semi-definite matrix is a covariance matrix. To see this, suppose $$M$$ is a $$p \times p$$ symmetric positive-semidefinite matrix. From the finite-dimensional case of the spectral theorem, it follows that $$M$$ has a nonnegative symmetric square root, which can be denoted by M1/2. Let $$\mathbf{X}$$ be any $$p \times 1$$ column vector-valued random variable whose covariance matrix is the $$p \times p$$ identity matrix. Then $$\operatorname{var}(\mathbf{M}^{1/2} \mathbf{X}) = \mathbf{M}^{1/2} \, \operatorname{var}(\mathbf{X}) \, \mathbf{M}^{1/2} = \mathbf{M}.$$

Complex random vectors
The variance of a complex scalar-valued random variable with expected value $$\mu$$ is conventionally defined using complex conjugation: $$ \operatorname{var}(Z) = \operatorname{E}\left[ (Z - \mu_Z)\overline{(Z - \mu_Z)} \right], $$ where the complex conjugate of a complex number $$z$$ is denoted $$\overline{z}$$; thus the variance of a complex random variable is a real number.

If $$\mathbf{Z} = (Z_1,\ldots,Z_n) ^\mathsf{T}$$ is a column vector of complex-valued random variables, then the conjugate transpose $$\mathbf{Z}^\mathsf{H}$$ is formed by both transposing and conjugating. In the following expression, the product of a vector with its conjugate transpose results in a square matrix called the covariance matrix, as its expectation: $$ \operatorname{K}_{\mathbf{Z}\mathbf{Z}} = \operatorname{cov}[\mathbf{Z},\mathbf{Z}] = \operatorname{E} \left[ (\mathbf{Z} - \boldsymbol{\mu}_\mathbf{Z})(\mathbf{Z} - \boldsymbol{\mu}_\mathbf{Z})^\mathsf{H} \right], $$ The matrix so obtained will be Hermitian positive-semidefinite, with real numbers in the main diagonal and complex numbers off-diagonal.

 * Properties
 * The covariance matrix is a Hermitian matrix, i.e. $$\operatorname{K}_{\mathbf{Z}\mathbf{Z}}^\mathsf{H} = \operatorname{K}_{\mathbf{Z}\mathbf{Z}}$$.
 * The diagonal elements of the covariance matrix are real.

Pseudo-covariance matrix
For complex random vectors, another kind of second central moment, the pseudo-covariance matrix (also called relation matrix) is defined as follows: $$ \operatorname{J}_{\mathbf{Z}\mathbf{Z}} = \operatorname{cov}[\mathbf{Z},\overline{\mathbf{Z}}] = \operatorname{E} \left[ (\mathbf{Z} - \boldsymbol{\mu}_\mathbf{Z})(\mathbf{Z} - \boldsymbol{\mu}_\mathbf{Z})^\mathsf{T} \right] $$

In contrast to the covariance matrix defined above, Hermitian transposition gets replaced by transposition in the definition. Its diagonal elements may be complex valued; it is a complex symmetric matrix.

Estimation
If $$\mathbf{M}_{\mathbf{X}}$$ and $$\mathbf{M}_{\mathbf{Y}}$$ are centered data matrices of dimension $$p \times n$$ and $$q \times n$$ respectively, i.e. with n columns of observations of p and q rows of variables, from which the row means have been subtracted, then, if the row means were estimated from the data, sample covariance matrices $$\mathbf{Q}_{\mathbf{XX}}$$ and $$\mathbf{Q}_{\mathbf{XY}}$$ can be defined to be $$ \mathbf{Q}_{\mathbf{XX}} = \frac{1}{n-1} \mathbf{M}_{\mathbf{X}} \mathbf{M}_{\mathbf{X}}^\mathsf{T}, \qquad \mathbf{Q}_{\mathbf{XY}} = \frac{1}{n-1} \mathbf{M}_{\mathbf{X}} \mathbf{M}_{\mathbf{Y}}^\mathsf{T} $$ or, if the row means were known a priori, $$ \mathbf{Q}_{\mathbf{XX}} = \frac{1}{n} \mathbf{M}_{\mathbf{X}} \mathbf{M}_{\mathbf{X}}^\mathsf{T}, \qquad \mathbf{Q}_{\mathbf{XY}} = \frac{1}{n} \mathbf{M}_{\mathbf{X}} \mathbf{M}_{\mathbf{Y}}^\mathsf{T}. $$

These empirical sample covariance matrices are the most straightforward and most often used estimators for the covariance matrices, but other estimators also exist, including regularised or shrinkage estimators, which may have better properties.

Applications
The covariance matrix is a useful tool in many different areas. From it a transformation matrix can be derived, called a whitening transformation, that allows one to completely decorrelate the data or, from a different point of view, to find an optimal basis for representing the data in a compact way (see Rayleigh quotient for a formal proof and additional properties of covariance matrices). This is called principal component analysis (PCA) and the Karhunen–Loève transform (KL-transform).

The covariance matrix plays a key role in financial economics, especially in portfolio theory and its mutual fund separation theorem and in the capital asset pricing model. The matrix of covariances among various assets' returns is used to determine, under certain assumptions, the relative amounts of different assets that investors should (in a normative analysis) or are predicted to (in a positive analysis) choose to hold in a context of diversification.

Use in optimization
The evolution strategy, a particular family of Randomized Search Heuristics, fundamentally relies on a covariance matrix in its mechanism. The characteristic mutation operator draws the update step from a multivariate normal distribution using an evolving covariance matrix. There is a formal proof that the evolution strategy's covariance matrix adapts to the inverse of the Hessian matrix of the search landscape, up to a scalar factor and small random fluctuations (proven for a single-parent strategy and a static model, as the population size increases, relying on the quadratic approximation). Intuitively, this result is supported by the rationale that the optimal covariance distribution can offer mutation steps whose equidensity probability contours match the level sets of the landscape, and so they maximize the progress rate.

Covariance mapping
In covariance mapping the values of the $$ \operatorname{cov}(\mathbf{X}, \mathbf{Y}) $$ or $$ \operatorname{pcov}(\mathbf{X}, \mathbf{Y} \mid \mathbf{I}) $$ matrix are plotted as a 2-dimensional map. When vectors $$ \mathbf{X} $$ and $$ \mathbf{Y} $$ are discrete random functions, the map shows statistical relations between different regions of the random functions. Statistically independent regions of the functions show up on the map as zero-level flatland, while positive or negative correlations show up, respectively, as hills or valleys.

In practice the column vectors $$ \mathbf{X}, \mathbf{Y} $$, and $$ \mathbf{I} $$ are acquired experimentally as rows of $$ n $$ samples, e.g. $$ \left[\mathbf{X}_1, \mathbf{X}_2, \dots, \mathbf{X}_n\right] = \begin{bmatrix} X_1(t_1) & X_2(t_1) & \cdots & X_n(t_1) \\ \\ X_1(t_2) & X_2(t_2) & \cdots & X_n(t_2) \\ \\ \vdots  &   \vdots & \ddots & \vdots   \\ \\ X_1(t_m) & X_2(t_m) & \cdots & X_n(t_m) \end{bmatrix} , $$ where $$ X_j(t_i) $$ is the i-th discrete value in sample j of the random function $$ X(t) $$. The expected values needed in the covariance formula are estimated using the sample mean, e.g. $$ \langle \mathbf{X} \rangle = \frac{1}{n} \sum_{j=1}^{n} \mathbf{X}_j $$ and the covariance matrix is estimated by the sample covariance matrix $$ \operatorname{cov}(\mathbf{X},\mathbf{Y}) \approx \langle \mathbf{XY^\mathsf{T}} \rangle - \langle \mathbf{X} \rangle \langle \mathbf{Y}^\mathsf{T} \rangle , $$ where the angular brackets denote sample averaging as before except that the Bessel's correction should be made to avoid bias. Using this estimation the partial covariance matrix can be calculated as $$ \operatorname{pcov}(\mathbf{X},\mathbf{Y} \mid \mathbf{I}) = \operatorname{cov}(\mathbf{X},\mathbf{Y}) - \operatorname{cov}(\mathbf{X},\mathbf{I}) \left (   \operatorname{cov}(\mathbf{I},\mathbf{I})  \backslash \operatorname{cov}(\mathbf{I},\mathbf{Y}) \right ), $$ where the backslash denotes the left matrix division operator, which bypasses the requirement to invert a matrix and is available in some computational packages such as Matlab.

Fig. 1 illustrates how a partial covariance map is constructed on an example of an experiment performed at the FLASH free-electron laser in Hamburg. The random function $$ X(t) $$ is the time-of-flight spectrum of ions from a Coulomb explosion of nitrogen molecules multiply ionised by a laser pulse. Since only a few hundreds of molecules are ionised at each laser pulse, the single-shot spectra are highly fluctuating. However, collecting typically $$ m=10^4 $$ such spectra, $$ \mathbf{X}_j(t) $$, and averaging them over $$ j $$ produces a smooth spectrum $$ \langle \mathbf{X}(t) \rangle $$, which is shown in red at the bottom of Fig. 1. The average spectrum $$ \langle \mathbf{X} \rangle $$ reveals several nitrogen ions in a form of peaks broadened by their kinetic energy, but to find the correlations between the ionisation stages and the ion momenta requires calculating a covariance map.

In the example of Fig. 1 spectra $$ \mathbf{X}_j(t) $$ and $$ \mathbf{Y}_j(t) $$ are the same, except that the range of the time-of-flight $$ t $$ differs. Panel a shows $$ \langle \mathbf{XY^\mathsf{T}} \rangle $$, panel b shows $$ \langle \mathbf{X} \rangle \langle \mathbf{Y}^\mathsf{T} \rangle $$ and panel c shows their difference, which is $$ \operatorname{cov}(\mathbf{X},\mathbf{Y}) $$ (note a change in the colour scale). Unfortunately, this map is overwhelmed by uninteresting, common-mode correlations induced by laser intensity fluctuating from shot to shot. To suppress such correlations the laser intensity $$ I_j $$ is recorded at every shot, put into $$ \mathbf{I} $$ and $$ \operatorname{pcov}(\mathbf{X},\mathbf{Y} \mid \mathbf{I}) $$ is calculated as panels d and e show. The suppression of the uninteresting correlations is, however, imperfect because there are other sources of common-mode fluctuations than the laser intensity and in principle all these sources should be monitored in vector $$ \mathbf{I} $$. Yet in practice it is often sufficient to overcompensate the partial covariance correction as panel f shows, where interesting correlations of ion momenta are now clearly visible as straight lines centred on ionisation stages of atomic nitrogen.

Two-dimensional infrared spectroscopy
Two-dimensional infrared spectroscopy employs correlation analysis to obtain 2D spectra of the condensed phase. There are two versions of this analysis: synchronous and asynchronous. Mathematically, the former is expressed in terms of the sample covariance matrix and the technique is equivalent to covariance mapping.

title: Covariance matrix


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Mine plow

A mine plow (plough in British English) is a device designed to clear a lane through a minefield, allowing other vehicles to follow. A mine plow is typically mounted to a tank or military engineering vehicle. Buried land mines are plowed up and pushed outside the tank's track path or tipped over. Since modern anti-tank mines rely on a focused explosion to destroy armored vehicles, they are useless when turned upside-down; as the tank runs over the mine, it will expend its blast down instead of upwards, causing insignificant damage, if any.

History
Towards the end of the First World War, the French mounted a plow on their Renault FT tank.

The British started work on plow designs in 1937, and a successful design was introduced for the Matilda Mk I tank though it was not used.

The first recorded combat use is by a "Bullshorn" plow on a Churchill tank of the British 79th Armoured Division, on Sword Beach during the Allied invasion of Normandy (this was one of "Hobart's Funnies" specialized vehicles). The "Bullshorn" was one of a number of plow designs which were tested and used by the British Army.

The mine plow is still in use by many combat engineering units. The Royal Engineers have deployed the Trojan to Afghanistan where it was usually fitted with a plow on the front, which enables it to clear mines, either detonating them on contact, or pushing them out of the way to clear a safe channel for following vehicles.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Mine plow


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Grace in Christianity

In Western Christian theology, grace is created by God who gives it as help to one because God desires one to have it, not necessarily because of anything one has done to earn it. It is understood by Western Christians to be a spontaneous gift from God to people – "generous, free and totally unexpected and undeserved" – that takes the form of divine favor, love, clemency, and a share in the divine life of God. In the Eastern Orthodox Church, grace is the uncreated Energies of God. Among Eastern Christians generally, grace is considered to be the partaking of the Divine Nature described in 2 Peter 1:4 and grace is the working of God himself, not a created substance of any kind that can be treated like a commodity.

As an attribute of God it manifests most in the salvation of sinners and Western Christianity holds that the initiative in the relationship of grace between God and an individual is always on the side of God.

The question of the means of grace has been called "the watershed that divides Catholicism from Protestantism, Calvinism from Arminianism, modern theological liberalism from theological conservatism." The Catholic Church holds that it is because of the action of Christ and the Holy Spirit in transforming into the divine life what is subjected to God's power that "the sacraments confer the grace they signify": "the power of Christ and his Spirit acts in and through [each sacrament], independently of the personal holiness of the minister. Nevertheless, the fruits of the sacraments also depend on the disposition of the one who receives them." The Sacred Mysteries (sacraments) are seen as a means of partaking of divine grace because God works through his Church.

Catholics, Eastern Orthodox and Protestants agree that faith is a gift from God, as in Ephesians 2:8: "For by grace you have been saved through faith, and that not of yourselves; it is the gift of God." Lutherans hold that the means of grace are "the gospel in Word and sacraments." That the sacraments are means of grace is also the teaching of John Wesley, who described the Eucharist as "the grand channel whereby the grace of his Spirit was conveyed to the souls of all the children of God".

Calvinists emphasize "the utter helplessness of people apart from grace." But God reaches out with "first grace" or "prevenient grace". The Calvinist doctrine known as irresistible grace states that, since all persons are by nature spiritually dead, no one desires to accept this grace until God spiritually enlivens them by means of regeneration. God regenerates only individuals whom he has predestined to salvation. Arminians understand the grace of God as cooperating with one's free will in order to bring an individual to salvation. According to Evangelical theologian Charles C. Ryrie, modern liberal theology "gives an exaggerated place to the abilities of people to decide their own fate and to effect their own salvation entirely apart from God's grace."

Old and New Testaments of the Christian Bible
Grace is the English translation of the Greek χάρις meaning "that which brings delight, joy, happiness, or good fortune."

Old Testament
The Septuagint translates as χάρις the Hebrew word חֵ֖ן as found in Genesis 6:8 to describe why God saved Noah from the flood. The Old Testament use of the word includes the concept that those showing favor do gracious deeds, or acts of grace, such as being kind to the poor and showing generosity. Descriptions of God's graciousness abound in the Torah/Pentateuch, for example in Deuteronomy 7:8 and Numbers 6:24–27. In the Psalms, examples of God's grace include teaching the Law (Psalm 119:29) and answering prayers (Psalm 27:7). Another example of God's grace appears in Psalm 85, a prayer for restoration, forgiveness, and the grace and mercy of God to bring about new life following the Exile.

Roman Catholicism
In the definition of the Catechism of the Catholic Church, "grace is favour, the free and undeserved help that God gives us to respond to his call to become children of God, adoptive sons, partakers of the divine nature and of eternal life". Grace is a participation in the life of God, which is poured unearned into human beings, whom it heals of sin and sanctifies.

The means by which God grants grace are many. They include the entirety of revealed truth, the sacraments and the hierarchical ministry. Among the principal means of grace are the sacraments (especially the Eucharist), prayers and good works. The sacramentals also are means of grace. The sacraments themselves, not the persons who administer or those who receive them, are "the means of grace", although lack of the required dispositions on the part of the recipient will block the effectiveness of the sacrament.

The Catholic Church holds that "by grace alone, in faith in Christ's saving work and not because of any merit on our part, we are accepted by God and receive the Holy Spirit, who renews our hearts while equipping and calling us to good works." Both the Council of Orange (529) and the Council of Trent affirmed that we are "justified gratuitously, because none of the things that precede justification, whether faith or works, merit the grace of justification".

The Council of Trent declared that the free will of man, moved and excited by God, can by its consent co-operate with God, who excites and invites its action; and that it can thereby dispose and prepare itself to obtain the grace of justification. The will can resist grace if it chooses. It is not like a lifeless thing, which remains purely passive. Weakened and diminished by Adam's fall, free will is yet not destroyed in the race (Sess. VI, cap. i and v).

The joint declaration between Catholics and Lutherans on the doctrine of justification affirms:

"We confess together that all persons depend completely on the saving grace of God for their salvation. Justification takes place solely by God's grace. When Catholics say that persons 'cooperate' in preparing for and accepting justification by consenting to God's justifying action, they see such personal consent as itself an effect of grace, not as an action arising from innate human abilities."

Sanctifying and actual grace
According to a commonly accepted categorization, made by St. Thomas Aquinas in his Summa Theologiae, grace can be given either to make the person receiving it pleasing to God (gratia gratum faciens) – so that the person is sanctified and justified – or else to help the receiver lead someone else to God (gratia gratis data). The former type of grace, gratia gratum faciens, in turn, can be described as sanctifying (or habitual) grace – when it refers to the divine life which, according to the Church, infuses a person's soul once they are justified; or else as actual grace – when it refers to those punctual (not habitual) helps that are directed to the production of sanctifying grace where it does not already exist, or its maintenance and increase it where it is already present. According to the Catechism of the Catholic Church:

"Sanctifying grace is an habitual gift, a stable and supernatural disposition that perfects the soul itself to enable it to live with God, to act by his love. Habitual grace, the permanent disposition to live and act in keeping with God's call, is distinguished from actual graces which refer to God's interventions, whether at the beginning of conversion or in the course of the work of sanctification."

The infusion of sanctifying grace, says the Church, transforms a sinner into a holy child of God, and in this way a person participates in the Divine Sonship of Jesus Christ and receives the indwelling of the Holy Spirit. Sanctifying grace remains permanently in the soul as long as one does not reject one's adopted sonship by committing a mortal sin, which severs one's friendship with God. Less serious sins, venial sin, although they "allow charity to subsist, they offend and wound it." However, God is infinitely merciful, and sanctifying grace can always be restored to the penitent heart, normatively in the Sacrament of Reconciliation (or Sacrament of Penance).

Augustine versus Pelagius
In the early 5th century, Pelagius, an ascetic who is said to have come from Britain, was concerned about the moral laxity of society that he witnessed in Rome. He blamed this laxity on the theology of divine grace preached by Augustine of Hippo, among others. He strongly affirmed that humans had free will and were able to choose good as well as evil. Augustine, drawing on the exaggerated statements of the followers of Pelagius rather than on Pelagius' own writings, began a debate that was to have long-reaching effects on subsequent developments of the doctrine in Western Christianity. Pelagianism was repudiated by the Council of Carthage in 418, largely at Augustine's insistence. However, what Pelagius taught was likely what has come to be called semi-pelagianism.

In semi-Pelagian thought, both God and the human person always participate in the salvation process. Humans make free will choices, which are aided by God through creation, natural grace, "supernatural" grace, and God's restrictions on demonic influences. God continually brings the human person to real choices, which God also aids, in the process of spiritual growth and salvation. Semi-Pelagianism is similar to synergism, which is the traditional patristic doctrine. John Cassian, in continuity with patristic doctrine, taught that though grace is required for persons to save themselves at the beginning, there is no such thing as total depravity, but there remains a moral or noetic ability within humans that is unaffected by original sin, and that persons must work together (synergism) with divine grace to be saved. This position is held by the Eastern Orthodox Church and by many Reformed Protestants, and in the Catholic Church has been especially associated with the Society of Jesus.

Catholic versus Protestant
In 1547, the Council of Trent, which sought to address and condemn Protestant objections, aimed to purge the Roman Catholic Church of controversial movements and establish an orthodox Roman Catholic teaching on grace and justification, as distinguished from the Protestant teachings on those concepts. It taught that justification and sanctification are elements of the same process. The grace of justification is bestowed through the merit of Christ's passion, without any merits on the part of the person justified, who is enabled to cooperate only through the grace of God. The grace of justification may be lost through mortal sin, but can also be restored by the sacrament of Penance. The sacraments are, together with revealed truth, the principal means of the grace, a treasury of grace, that Christ has merited by his life and death and has given to the Church. This does not mean that other groups of Christians have no treasury of grace at their disposal, for, as the Second Vatican Council declared, "many elements of sanctification and of truth are found outside of (the Catholic Church's) visible structure".

Jansenists versus Jesuits
At about the same time that Calvinists and Arminians were debating the meaning of grace in Protestantism, in Catholicism a similar debate was taking place between the Jansenists and the Jesuits. Cornelius Jansen's 1640 work Augustinus sought to refocus Catholic theology on the themes of original sin, human depravity, the necessity of divine grace, and predestination, as he found them in the works of Augustine. The Jansenists, like the Puritans, believed themselves to be members of a gathered church called out of worldly society, and banded together in institutions like the Port-Royal convents seeking to lead lives of greater spiritual intensity. Blaise Pascal attacked what he called moral laxity in the casuistry of the Jesuits. Jansenist theology remained a minority party within Catholicism, and during the second half of the 17th and 18th centuries it was condemned as a heresy for its similarities to Calvinism, though its style remained influential in ascetic circles.

Grace and merit
Citing the Council of Trent, the Catechism of the Catholic Church states:

"With regard to God, there is no strict right to any merit on the part of man. Between God and us there is an immeasurable inequality, for we have received everything from him, our Creator. The merit of man before God in the Christian life arises from the fact that God has freely chosen to associate man with the work of his grace. The fatherly action of God is first on his own initiative, and then follows man's free acting through his collaboration, so that the merit of good works is to be attributed in the first place to the grace of God, then to the faithful. Man's merit, moreover, itself is due to God, for his good actions proceed in Christ, from the predispositions and assistance given by the Holy Spirit. [...] The charity of Christ is the source in us of all our merits before God. Grace, by uniting us to Christ in active love, ensures the supernatural quality of our acts and consequently their merit before God and before men. The saints have always had a lively awareness that their merits were pure grace."

Eastern Christianity
In the Eastern Orthodox Church, grace is identified with the uncreated Energies of God. Among Eastern Christians generally, grace is considered to be the partaking of the Divine Nature described in 2 Peter 1:4. The Holy Mysteries (Latin, "sacraments") are seen as a means of partaking of divine grace because God works through his Church, not just because specific legalistic rules are followed; and grace is the working of God himself, not a created substance of any kind that can be treated like a commodity. <ref

Orthodox theologians reject Augustine's formulation of original sin and actively oppose the content and implications of John Calvin's conceptions of total depravity and irresistible grace, characteristic of Reformed Protestantism, as well as the Thomistic and scholastic theology which would become official Roman Catholic pedagogy until the Second Vatican Council. Eastern Christians typically view scholasticism and similarly discursive, systematic theologies as rationalistic corruptions of the theology of the Cappadocian and early Desert Fathers that led the Western Church astray into heresy. Orthodoxy teaches that it is possible and necessary for the human will to cooperate with divine grace for the individual to be saved, or healed from the disease of sin. This cooperation is called synergism (see also semipelagianism and monergism), so that humans may become deified in conformity to the divine likeness – a process called theosis – by merging with the uncreated Energies of God (revealed to the senses as the Tabor Light of transfiguration), notably through a method of prayer called hesychasm.

Protestant Reformation
The Protestant Reformation reacted against the concepts of grace and merit as they were understood in late medieval Catholic theology.

Luther and Lutheran theology
Martin Luther's posting of his ninety-five theses to the church door in Wittenberg in 1517 was a direct consequence of the perfunctory sacramentalism and treasury doctrines of the medieval church. The act was precipitated by the arrival of Johann Tetzel, authorized by the Vatican to sell indulgences.

The effectiveness of these indulgences was predicated on the doctrine of the treasury of grace proclaimed by Pope Clement VI. The theory was that merit earned by acts of piety could augment the believer's store of sanctifying grace. Gifts to the Church were acts of piety. The Church, moreover, had a treasury full of grace above and beyond what was needed to get its faithful into heaven. The Church was willing to part with some of its surplus in exchange for earthly gold. Martin Luther's anger against this practice, which seemed to him to involve the purchase of salvation, began a swing of the pendulum back towards the Pauline vision of grace, as opposed to James's.

Luther taught that men were helpless and without a plea before God's justice, and their acts of piety were utterly inadequate before his infinite holiness. Were God just, and not merciful, everyone would go to hell, because everyone, even the best of mankind, deserves to go to hell. Mankind's inability to achieve salvation by its own effort suggests that even the best intentions are somehow tainted by mankind's sinful nature. This doctrine is sometimes called total depravity, a term derived from Calvinism and its relatives.

It is by faith alone (sola fide) and by grace alone (sola gratia) that men are saved. Good works are something the believers should undertake out of gratitude towards their Savior; but they are not sufficient for salvation and cannot earn anyone salvation; there is no room for the notion of "merit" in Luther's doctrine of redemption. (There may, however, be degrees of reward for the redeemed in heaven.) Only the unearned, unmerited grace of God can save anyone. No one can have a claim of entitlement to God's grace, and it is only by his generosity that salvation is even possible.

As opposed to the treasury of grace from which believers can make withdrawals, in Lutheranism salvation becomes a declaration of spiritual bankruptcy, in which penitents acknowledge the inadequacy of their own resources and trust only in God to save them. Accepting Augustine's concern for legal justification as the base metaphor for salvation, the believers are not so much made righteous in Lutheranism as they are considered covered by Christ's righteousness. Acknowledging that they have no power to make themselves righteous, the penalty for their sins is discharged because Jesus has already paid for it with his blood. His righteousness is credited to those who believe in and thus belong to him.

Calvin and Reformed theology
Calvin and Luther believed free will does not co-operate with God's grace which, according to them, cannot be rejected (see monergism). The Lutheran Augsburg Confession says of baptism, "Lutherans teach that it is necessary to salvation and that by baptism the grace of God is offered and that children are to be baptized, who by baptism, being offered to God, are received into God's favor." The French reformer John Calvin expanded and further developed these Augustinian themes in his systematic Institutes of the Christian Religion in 1536.

The logical structure of Calvinism is often expressed as the acronym, TULIP. These five categories do not comprise Calvinism in its entirety; they simply encapsulate its central, definitive doctrines.

 * Total depravity (also known as total inability, which is inexorably tied to a strong doctrine of original sin as having completely enslaved the human will)
 * Unconditional election
 * Limited Atonement (also known as definite atonement or particular redemption)
 * Irresistible Grace
 * Perseverance of the Saints (colloquially known as "once saved, always saved" or, as interpreted a distinct way among Reformed or Strict Baptists as well as non-Calvinist General Baptists, eternal security)

The notion that God has foreordained who will be saved is generally called predestination. The concept of predestination peculiar to Calvinism, "double-predestination", (in conjunction with limited atonement) is the most controversial expression of the doctrine. According to Reformed theology, the "good news" of the gospel of Christ is that God has freely granted the gift of salvation to those the Holy Spirit causes to believe; what he freely grants to some (the "elect" individuals), he withholds from others (the "reprobate" individuals).

Calvin sought to provide assurance to the faithful that God would actually save them. His teaching implied what came to be known as the doctrine of the perseverance of the saints, the notion that God would actually save those who were his Elect. The actual status and ultimate state of any man's soul were unknown except to God. When assurance of election was rigorously pressed as an experience to be sought, especially by the Puritans, this led to a legalism as rigid as the one Protestantism sought to reject, as men were eager to demonstrate that they were among the chosen by the conspicuous works-righteousness of their lives.

The relatively radical positions of Reformed theology provoked a strong reaction from both Roman Catholics and Lutherans.

Classical and Wesleyan Arminian theology
In the beginning of the 17th century, the Dutch theologian Jacobus Arminius formulated Arminianism and departed from Calvin's theology in particular on election and predestination. Arminianism affirms the compatibility between human free will and divine foreknowledge, and its incompatibility with theological determinism. Predestination in Arminianism is based on divine foreknowledge, unlike in Calvinism. Thus, the offer of salvation through grace does not act irresistibly in a purely cause-effect, deterministic method but rather in an influence-and-response fashion that can be both freely accepted and freely denied. In Arminianism, God takes initiative in the salvation process and his grace comes to all people. This is done through prevenient grace which acts on all people to convince them of the Gospel, draw them strongly towards salvation, and enable the possibility of sincere faith. As Roger Olson put it: "[Arminius]' evangelical synergism reserves all the power, ability and efficacy in salvation to grace, but allows humans the God-granted ability to resist or not resist it. The only "contribution" humans make is nonresistance to grace."

Later, John Wesley also rejected the Calvinist doctrine of predestination, and had the same Arminian understanding as expressed in Wesleyan theology. It remains the standard teaching of Methodist churches. Wesley also appealed to prevenient grace, stating that God makes the initial move in salvation, but human beings are free to respond or reject God's graceful initiative. The doctrine of prevenient grace remains one of Methodism's most important doctrines.

John Wesley distinguished three kinds of divine grace in the process of salvation: 1. "Prevenient grace" which is an enabling grace preceding regeneration ("prevenient" means preceding). 2. "Justifying grace" which can bring regeneration but which is resistible. 3. "Sustaining grace" which helps a person to remain into regeneration, and to reach sanctification and final salvation. In particular Wesley taught that Christian believers are to participate in the means of grace and to continue to grow in the Christian life, assisted by God's sustaining grace.

The Protestant Reformation and ecclesiology
Protestantism in all three major schools of theology – Lutheran, Calvinist, and Arminian – emphasize God's initiative in the work of salvation, which is achieved by grace alone through faith alone, in either stream of thinking – although these terms are understood differently, according to the differences in systems.

Classical Calvinism teaches that the sacraments are "signs and seals of the covenant of grace" and "effectual means of salvation", and Lutheranism teaches that new life, faith, and union with Christ are granted by the Holy Spirit working through the sacraments. However, for a large portion of the Protestant world, the sacraments largely lost the importance that Luther (and to a slightly lesser degree, Calvin) attributed to them. This happened under the influence of ideas of the Anabaptists which were ideas also seen in the Donatists in North Africa in 311 AD, and these ideas then spread to Calvinists through the Congregationalist and Baptist movements, and to Lutherans through Pietism (although much of Lutheranism recoiled against the Pietist movement after the mid-19th century).

Where the sacraments are de-emphasized, they become "ordinances", acts of worship which are required by Scripture, but whose effect is limited to the voluntary effect they have on the worshipper's soul. This belief finds expression in the Baptist and Anabaptist practice of believer's baptism, given not to infants as a mark of membership in a Christian community, but to adult believers after they have achieved the age of reason and have professed their faith. These ordinances are never considered works-righteousness. The ritual as interpreted in light of such ideas does not at all bring about salvation, nor does its performance bring about the forgiveness of sins; the forgiveness which the believer has received by faith is merely pictured, not effectively applied, by baptism; salvation and participation in Christ is memorialized ("this do in remembrance of me" in the Lord's Supper and baptism picturing a Christian's rebirth as death to sin and alive in Christ), not imparted, by the Eucharist. The Church to the Baptists becomes an assembly of true believers in Christ Jesus who gather together for worship and fellowship and remembering what Christ did for them.

Churches of Christ
The Churches of Christ believe that the grace of God that saves is the plan of salvation, rather than salvation itself. This plan includes two parts, 1) the perfect life, death, burial, and resurrection of Jesus the Christ, 2) the gospel/New Testament/the faith.

Concerning Ephesians 2:8 which states: "For by grace are ye saved through faith; and that not of yourselves: it is the gift of God", it is noted that the word "it" is a pronoun and refers back to a noun. As the word "saved" is a verb, "it" does not refer to "saved" but to grace, giving the definition of grace as "the gift of God". Furthermore, as the Book of James distinguishes between a dead faith (a faith without works) and a living faith (a faith accompanied by works of obedience), it is believed that God's gift operates through an individual's living faith resulting in that individual being saved.

 * 1) Grace is contrasted with the Law of Moses (Romans 6:14; Hebrews 10:4; John 1:17) and the church of Christ believes that Paul's contrast between work and faith is as described under the Efforts to resolve the tension section, a contrast between works of the Old Covenant and obedient faith under the New Covenant.
 * 2) Grace saves (Ephesians 2:5); justifies (Romans 3:24; Titus 3:7).
 * 3) Grace can not be added to (Galatians 5:4).
 * 4) Grace teaches (Titus 2:11); can be preached (Ephesians 3:8).
 * 5) Grace calls mankind (2 Timothy 1:9; Galatians 1:15).
 * 6) Grace is brought by revelation (1 Peter 1:13).
 * 7) Grace and truth came by Jesus Christ (John 1:17)
 * 8) Grace is sufficient for mankind (2 Corinthians 12:9)

The Galatians were removed from the calling of the gospel (Galatians 1:6,7; 2 Thessalonians 2:14) unto another gospel (another message) which verse 7 says is not a gospel at all but a perversion.

The Church of Christ believes that grace provides the following plan, which, if followed, results in salvation:

 * One must hear the gospel/word (Romans 10:17).
 * Believe the gospel (Mark 16:15–16).
 * Repent of their past sins (Acts 2:38).
 * Confess their faith in Christ before men (Matthew 10:32; Romans 10:9–10).
 * Be immersed in water into Christ for the remission of those sins (1 Peter 3:21; Romans 6:3–18; John 3:3,5; 1 John 5:6,8; Acts 2:38; Mark 16:16; etc.)
 * Live faithfully even to the point of death (Revelation 2:10; Romans 11:17–22; James 5:19–20).

Orthodox

 * Bishop Kallistos (Ware), The Inner Kingdom: The Collected Works (St. Vladimir's Seminary, 2000) ISBN 0-88141-209-0
 * The Way of a Pilgrim and A Pilgrim Continues on His Way, Olga Savin, trans. (Shambhala, 2001) ISBN 1-57062-807-6

Roman Catholic

 * Catholic Teaching on Sin & Grace (Center for Learning, 1997), ISBN 1-56077-521-1
 * George Hayward Joyce, The Catholic Doctrine of Grace (Newman, 1950),
 * Stephen J. Duffy, The Graced Horizon: Nature and Grace in Modern Catholic Thought (HPAC, 1992), ISBN 0-8146-5705-2
 * Stephen J. Duffy, The Graced Horizon: Nature and Grace in Modern Catholic Thought (HPAC, 1992), ISBN 0-8146-5705-2
 * Stephen J. Duffy, The Graced Horizon: Nature and Grace in Modern Catholic Thought (HPAC, 1992), ISBN 0-8146-5705-2

Protestant

 * Dietrich Bonhoeffer, The Cost of Discipleship, Fuller and Booth, trans. (Touchstone, 1995).
 * John Calvin, "Institutes of the Christian Religion, Book 2 Chapter 4"
 * Randy Maddox, Responsible Grace (Kingswood, 1994) ISBN 0-687-00334-2
 * Alister McGrath, Iustitia Dei: A History of the Christian Doctrine of Justification (Cambridge, 1998) ISBN 0-521-62481-9
 * Glen Pettigrove, "Forgiveness and Grace", in Forgiveness and Love (Oxford University Press, 2012) 124–150.
 * R. C. Sproul, Grace Unknown: The Heart of Reformed Theology (Baker Book House, 1999) ISBN 0-8010-1121-3
 * Ulasien, Paul, The Power of a Grace Perspective  (Infinity, 2011) ISBN 0-7414-6729-1,
 * Philip Yancey, What's So Amazing About Grace? (Zondervan, 1997) ISBN 0-310-24565-6
 * Paul F. M. Zahl, Grace in Practice: A Theology of Everyday Life (Eerdmans, 2007) ISBN 978-0-8028-2897-2
 * Paul F. M. Zahl, Grace in Practice: A Theology of Everyday Life (Eerdmans, 2007) ISBN 978-0-8028-2897-2

title: Grace in Christianity


text:
Mine roller

A mine roller or mine trawl is a demining device mounted on a tank or armoured personnel carrier, designed to detonate anti-tank mines. It allows engineers to clear a lane through a minefield which is protected by enemy fire.

The device is usually composed of a fork or two push arm assemblies fitted to the front of a tank hull, with two banks of rollers that can be lowered in front of the tank's tracks. Each roller bank has several heavy wheels studded with short projecting steel girders, which apply a higher ground pressure than the tank's tracks. This ensures the explosion of pressure-fused anti-tank mines, which would otherwise explode under the track itself.

History
At the end of the First World War, the British Army Engineers Major Giffard Le Quesne Martel and Major Charles Inglis experimented with tank bridges and mine rollers based on the Mark V tank. Three special tank battalions were mustered for trials at Christchurch in Hampshire, England, in 1918. Because of the Armistice, these were never tested in battle, but some development work continued with the Experimental Bridging Company until 1925.

After great difficulties caused by minefields in the Winter War against Finland, the Soviet Red Army assigned P.M. Mugalev at the Dormashina Factory in Nikolayev to design a mine-clearing vehicle. Prototypes were tested based on the T-28 medium tank in 1940. Development was interrupted by the start of World War II, but resumed in 1942. T-60 and KV tank chassis underwent trials, but only the T-34 was deemed to have a sufficiently robust transmission and clutch.

Experimental detachments of PT-34 mine roller tanks were formed in May 1942, and saw action at Voronezh in August. The first Independent Engineer Tank Regiment with eighteen mine rollers was fielded in October 1943. At least five regiments were formed during the war.

The PT-34's huge roller fork was semi-permanently mounted on a T-34 or T-34-85 tank. The rollers were usually removed for travel, and only installed for mine clearing operations. Adaptations for later tanks consisted of two lighter arms. The Mugalev system was adopted by U.S. and Israeli forces in the 1980s.

During WWII, the Germans created the experimental Räumer S 4-wheeled armoured mine clearance vehicle, built by Krupp. Weighing 130 tons, the prototype was captured by the Americans at the end of the war.

The British developed the Anti Mine Roller Attachment (AMRA) for their Matilda II, Valentine, and Crusader tanks in the desert. The rollers only covered the width of each track rather than clearing a tank-width path for subsequent vehicles and troops. To these were added a Churchill tank with the evocatively named "Canadian Indestructible Roller Device" (CIRD). The British used mine rollers to detect the presence of minefields and then used mine flails for the clearance.

During World War II, the mine roller most used by US forces was the T1E3 Mine Exploder unit, attached to the M4 Sherman medium tank. It was nicknamed Aunt Jemima because of its pancake-like appearance. It had two sets of five disc rollers, 10 ft in diameter each.

Post WW2
Since 1945, a number of countries have developed mine rollers including:
 * Russia - Stankomash Ltd. - TMT.
 * UK - Armtrac Limited - Armtrac Roller - capable of operating at 50 kph.
 * UK - Pearson Engineering - Light Weight Mine Roller (LWMR).
 * USA - Axion Corporation - fits to Abrams tank.

title: Mine roller


text:
Name binding

In programming languages, name binding is the association of entities (data and/or code) with identifiers. An identifier bound to an object is said to reference that object. Machine languages have no built-in notion of identifiers, but name-object bindings as a service and notation for the programmer is implemented by programming languages. Binding is intimately connected with scoping, as scope determines which names bind to which objects – at which locations in the program code (lexically) and in which one of the possible execution paths (temporally).

Use of an identifier id in a context that establishes a binding for id is called a binding (or defining) occurrence. In all other occurrences (e.g., in expressions, assignments, and subprogram calls), an identifier stands for what it is bound to; such occurrences are called applied occurrences.

Binding time

 * Static binding (or early binding) is name binding performed before the program is run.
 * Dynamic binding (or late binding or virtual binding) is name binding performed as the program is running.

An example of a static binding is a direct C function call: the function referenced by the identifier cannot change at runtime.

An example of dynamic binding is dynamic dispatch, as in a C++ virtual method call. Since the specific type of a polymorphic object is not known before runtime (in general), the executed function is dynamically bound. Take, for example, the following Java code:

is an interface, so  must refer to a subtype of it. may reference a, an  , or some other subtype of. The method referenced by  is not known until runtime. In C, which does not have dynamic binding, a similar goal may be achieved by a call to a function pointed to by a variable or expression of a function pointer type, whose value is unknown until it is evaluated at run-time.

Rebinding and mutation
Rebinding should not be confused with mutation or assignment.

 * Rebinding is a change to the referencing identifier.
 * Assignment is a change to (the referenced) variable.
 * Mutation is a change to an object in memory, possibly referenced by a variable or bound to an identifier.

Consider the following Java code:

The identifier  is bound to a variable in the first line; in the second, a reference an object (a linked list of strings) is assigned to the variable. The linked list referenced by the variable is then mutated, adding a string to the list. Next, the variable is assigned the constant. In the last line, the identifier is rebound for the scope of the block. Operations within the block access a new variable and not the variable previously bound to.

Late static
Late static binding is a variant of binding somewhere between static and dynamic binding. Consider the following PHP example:

In this example, the PHP interpreter binds the keyword  inside   to class , and so the call to   produces the string "hello". If the semantics of  had been based on late static binding, then the result would have been "bye".

Beginning with PHP version 5.3, late static binding is supported. Specifically, if  in the above were changed to   as shown in the following block, where the keyword   would only be bound at runtime, then the result of the call to   would be "bye":

title: Name binding


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Special team

Special team may refer to:
 * Special teams in American football, units that are on the field during kicking plays
 * Special teams in ice hockey, players on the ice during a power play or penalty kill.
 * Police Special Operation Teams, a law enforcement agency in Turkey

title: Special team


text:
Talk:Redwood (disambiguation)

definition
I added a general definition here, since so many of the pages linked here do not refer to a specific species, just to Redwood in general, as in "under the redwoods" or "a redwood casket". I left most of the detail to the individual articles; just wanted a basic overview for the casual link-clicker (like me). Catherine

Redwood Regions
A place where you can find redwoods in america is near the rockies. to go to a link full of redwood information studies statistics and pictures click []http://www.myspace.com/lilbeezee0527 —Preceding unsigned comment added by 99.172.32.193 (talk) 00:46, 29 October 2008 (UTC)

Proposed page move
I propose redirecting this title to Sequoioideae, the subfamily of trees which covers the three species that are primarily known as "Redwoods", and moving the information on this page to Redwood (disambiguation). I am fairly confident that these trees, collectively, comprise the primary meaning of the word. Cheers! bd2412 T 15:17, 4 January 2011 (UTC)
 * Done. Cheers! bd2412  T 14:37, 5 January 2011 (UTC)

title: Talk:Redwood (disambiguation)


text:
Talk:Grace in Christianity

"Grace and Merit" needs a Catholic Theologian to revise and edit
It would seem that there is too much biased Point of View in this section. For example:

According to Eusebius, the Roman emperor Constantine I was not baptised until shortly before his death in the year 337. To some (Who are we referencing here?) this might suggest that his commitment to Christianity was lukewarm; in an attempt to rebut this suggestion, a contrary suggestion was made (by whom?). Christians at the time of Constantine, or at least at the time this explanation was devised, (What times are we talking about? Reference please.) believed that the performance of the ritual itself (where does this italic come from? - Don't Catholics believe that it is by the 'Sacrament' itself, not the 'ritual') conferred forgiveness of sins. This, however, was a one-shot deal; post-baptism sins cannot be forgiven in a second Baptism not "ritual", and could only be resolved by (please add) another Sacrament called penance. By postponing baptism until the last illness, it made it unlikely that the believer committed a serious sin between baptism and death. Another explanation (from what source?) is that many men at that time followed a very strict interpretation of the passages in 1 John that said Christians do not sin; since they thought themselves unlikely to stop sinning upon their conversion, they put off their conversion and baptisms until shortly before death. Thus, postponing their baptisms was understood as an act of humility. (Where does this come from? No Catholic I know speaks of this.)

Again

From a contemporary perspective, it is impossible to tell what Constantine intended. But the theology assumed in this explanation suggests that the concept of grace as understood by Constantine may have been altered into something Protestants find hard to fit into the New Testament's treatment of the concept. (This is a biased 'Point of View' paragraph.)

Later

Rather than God's property to be offered at His sole discretion, (Point of View bias) in Medieval Western Christianity at least, grace became a sort of spiritual currency, and the Church was its banker. Believers acquired grace by participating in the Church's sacraments.

And further:

The Church can offer the excess merit in its treasury to be applied to the deficits in merit suffered by its penitent sinners. Pope Clement VI proclaimed this to be a doctrine of the Roman Catholic Church in 1343.(reference please)

To my eyes this whole section (and indeed the entire "Grace in Roman Catholicism" sub-section could do with a thorough revision by a competent Catholic Theologian. At present it lacks a certain objectivity with too much bias and a point of view which seems to lack sympathy or understanding of the Catholic viewpoit it is attempting to describe to the researcher (PatrickGuadalupe (talk) 13:21, 25 May 2012 (UTC)).

Are ANY Catholic theologians reading this? Enlighten me if I am mistaken. Can a regular editor put in some "reference please" tags? PatrickGuadalupe (talk) 04:22, 26 October 2012 (UTC)
 * No professional theologian is needed. All that is needed is to replace the original-research disquisition with well-sourced statements of Catholic teaching on the subject.  Esoglou (talk) 10:04, 16 January 2013 (UTC)

Thanks for the input, Esoglou Esoglou. I think the changes you made on 16 January, 2013, at 10;05 go a long way in improving the article (PatrickGuadalupe (talk) 08:15, 31 January 2013 (UTC)).

"Divine Childship" of Jesus
Is this to avoid "Divine Sonship"? Is it necessary? --Richardson mcphillips (talk) 19:09, 9 February 2013 (UTC)
 * It's not necessary, and it's not the phrase that's commonly used. It gets 200 results in Google Books vs. 60,000 for "Divine Sonship". Also, the paragraph also needs to be rewritten to avoid the "we". StAnselm (talk) 19:36, 9 February 2013 (UTC)

Suggested merge of Actual grace
In the section Sanctifying grace, this article states: "Grace has been divided by some theologians into two forms, Sanctifying Grace and Actual Grace."

It feels unbalanced that actual grace is split out into its own article that's only three short paragraphs long, while sanctifying grace is a section in this article. — Hex    (❝  ?!  ❞)   18:42, 9 March 2013 (UTC)
 * Support. The other article, the one to be merged, is not only unsourced but also quite inaccurate.  Esoglou (talk) 18:56, 9 March 2013 (UTC)

Since the article on actual grace seems to have been deleted, I updated the title of the section originally called "sanctifying grace" to include actual grace. AthanasiusOfAlex (talk) 18:35, 5 February 2014 (UTC)

Catholicism: sanctifying and actual grace
The article currently states, “Grace has been divided by some theologians into two forms, Sanctifying Grace and Actual Grace.” This conceptual division is actually referred to in the Catechism (see No. 2000), so it is more than just “some theologians.” I went ahead and rephrased it so as to reflect the presence in the Catechism. (In any case, I think it would be necessary to mention which theologians make this division: as it is now, the sentence would merit the "who?" tag.)

I also added a number of citations, and references to the Summa Theologiae and the Council of Trent, removing some phrases that seem to have come from an old article. (For example, “its excellence, dignity, and importance become immediately apparent; for holiness and the sonship of God depend solely upon the possession of sanctifying grace, wherefore it is frequently called simply grace without any qualifying word to accompany it as, for instance, in the phrases ‘to live in grace’ or ‘to fall from grace.’”)

AthanasiusOfAlex (talk) 18:30, 5 February 2014 (UTC)

St. Paul and Grace
Saul, on the road to Damascus, experienced Grace and was forever changed. Today anyone who receives (experiences) Grace is likewise forever changed. Grace as a theological concept is interesting but there would be no one to care were it not for the experience.

Grace creates the freedom (and the peace and the joy) of which Jesus spoke. Grace provides the power that propels the follower of Christ through a life dedicated to loving God and loving one's neighbor. Without bathing daily in Grace, we simply become too burdened by guilt and fear to ever realize peace, joy and freedom.

No amount of theological knowledge, specialized terminology and learned discourse will ever motivate a sinner to fall on his knees and allow Grace to wash over him. And when he does, no explanation or rationalization is needed.Mpm1212 (talk) 21:57, 24 May 2016 (UTC)

Grace is the essence of God that reveals His virtues for His creation. Oluwaseyi229 (talk) 04:42, 19 April 2020 (UTC)
 * For insertion into the article, you need to cite WP:RS. WP:OR is not enough. Bealtainemí (talk) 08:05, 19 April 2020 (UTC)

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Grace in protestant theology
Is this sentence really accurate? "Protestants almost universally believe that grace is given by God based on the faith of the believer." I'm not a Protestant and so I might be missing something, but I thought that Protestants (like Catholics) believe that faith is impossible without grace--that is, that grace enables faith. This sentence suggests the opposite, that faith is a prerequisite to grace. Is there a nuance that I am missing? PStrait (talk) 17:56, 21 April 2021 (UTC)
 * It sounds extremely dubious, plus it's unsourced, so I have removed it. StAnselm (talk) 19:32, 21 April 2021 (UTC)

Description of the views of 'modern liberal theology' in the lede
The lede says 'According to Evangelical theologian Charles C. Ryrie, modern liberal theology "gives an exaggerated place to the abilities of people to decide their own fate and to effect their own salvation entirely apart from God's grace."' It seems problematic that the views of a theological current, in this case 'modern liberal theology', should be represented by someone who is clearly critical of it and describes it in a negative way (accusing it of 'exaggeration'). 62.73.69.121 (talk) 18:43, 9 November 2023 (UTC)

title: Talk:Grace in Christianity


text:
Talk:Docetism

old talk
Should there be a link to the "1st papal crusade"? Is there an article about it yet? Wesley

It is not surprising that the Muslims share the Docetic heresy: Gnosticism persisted in the Persian Empire long after it was eliminated in the Greco-Roman one. Gnostics were Mohammed's source where Jesus was concerned.

Not all Gnostics favored docetism. Whoever wrote this article is rather ignorant of the diversity of Gnostic beliefs.

Wesley, can you name one of Muhammad's Gnostic sources? Meanwhile, let me say something here.

The Qur'anic verse quoted here at the end of the article should be elaborated at more. Let me bring another translation of the verse (4:157). The translation of the verse in The Noble Qur'an (trans. by Khan and Hilali) says as follow:

(And because of their [the Jews] saying (in boast), "We killed Messiah 'Iesa (Jesus), son of Maryam (Mary), the Messenger of Allâh," - but they killed him not, nor crucified him, but the resemblance of 'Iesa (Jesus) was put over another man (and they killed that man), and those who differ therein are full of doubts. They have no (certain) knowledge, they follow nothing but conjecture. For surely; they killed him not.) The brackets are mine since the verse is a continuation of the previous one. Verse (4:157) says in Arabic:

وما قتلوه وما صلبوه ولكن شبه لهم وإن الذين اختلفوا فيه لفي شك منه ما لهم به من علم إلا اتباع الظن وما قتلوه يقيناً

The word شبه shubbiha doesn't mean 'imagine' or 'seem' but 'resemble.' The Qur'an here is saying here that someone was crucified instead of Jesus who was raised to Heaven (as verse 4:158) says, and his resemblance descended on someone else. Alathiri 01:02, 27 November 2006 (UTC)

A little correction here!
Docetismism? How does that compare to Chritianityity or Americanismism or humanitarianarian? THIS IS OUR COMPETENT WIKIPEDIA? —Preceding unsigned comment added by 98.144.120.119 (talk) 14:49, 12 February 2010 (UTC)

Claims of the dominance of docetism and late authorship of the New Testament
I corrected the text's assertion that docetism was the dominant Christology from AD 70, for about 100 years, and that the New Testament was written later. It's far too contentious of an assertion. I think it suffices to say that it was quite common, and that certain New Testament passages appear to have been written to contradict it. Asserting which was more prevalent seems beyond anyone's current expertise, especially that of the 1907 source which was cited authoritatively. Meanwhile, since 1907, much research has addressed the reasons why some scholars had supposed that most of the New Testament must have had a late second-century origin. Again, arguments can be made both ways, but it seems beyond the scope and expertise of this article to affirm decisively which is superior. —Preceding unsigned comment added by 98.204.183.211 (talk) 12:13, 14 May 2009 (UTC)

Buddhist analogy taken off
In the 3rd paragraph was this one phrase discussion the idea of temporary phenomenon and its relation to illusion:

"It could be compared to how a Buddhist speaks about illusion: illusion is everything that is temporary, not everything that is not real."

This quote is not an adequate and accurate representation of the Buddhist view of reality, for Buddhist do not deny the existance of a phenomenon based on its transientness. It blurrs the point of the article and may be misleading for readers. Thus this sentence is taken down.

Theosophical deletion
I deleted the coment which I assume had triggered the warning about 'original research'. It was simply a statement that a theosophical interpretation of Christ was right and everything else in the article was wrong. If I could have rescued a neutral statement of what theosophical belief in this area was I would have done, but it wasn't expressed well enough; I suggest that someone with that knowledge creates it if it is useful. I also deleted the warning, and only afterwards wondered if I was meant to - apologies if that is a breach of etiquette.

Gospel of John
It is unclear to me why the Gospel of John has been placed under the "Texts including Docetism" section. The passage cited has little bearing on central Docetist notion of the illusion of the crucifixion, unlike, for example, the passage cited from the Qu'ran below. From my researches, it seems that most scholars consider the Gospel of John to be opposed to Docetist ideas. I have removed this part of the article before, but someone has replaced it; if someone could clear up why it is there, I would much appreciate it. In the mean time I have removed it again. Parthenias (talk) 22:27, 1 June 2008 (UTC)

I'll find a source, but I guess you are coming from the angle that the Gospel of John never had any insertions, etc. I mean the passages docetism speaks for itself(docetism simply refers to Jesus not having a physical flesh & blood body during any time of his ministry, it's not strictly crucifixion related). ʄ! •¿talk?  08:34, 2 June 2008 (UTC)

I don't know that 'insertions' are really at issue here. My problem is that the text does not explicitly embrace Docetism: saying "Do not touch me" really makes no statement about the reality or illusion of one's body. If you have scholarly evidence, that should go alongside the quote; perhaps even a section "Possible Docetism in the New Testament" would be good, especially as I am confident that this is a minority opinion. In the meantime, I am going to remove the reference until there is some further explanation, preferably in the article itself. Parthenias (talk) 05:36, 6 June 2008 (UTC)

Docetism and Islam
The article should perhaps mention the historically alleged links between Docetism and Islam, since both will typically reject the reality of Jesus's crucifixion. Some Muslims scholars also cite related docetist literature in order to reject the crucifixion and the resurrection. ADM (talk) 20:55, 10 April 2009 (UTC)

 * This section should be removed. The Islamic view that Jesus was not crucified is in no way connected to docetism.  The koran holds that Jesus was the messiah, and he was taken up to heaven, but he was not God or the son of God.  —Preceding unsigned comment added by 86.135.70.167 (talk) 18:30, 16 May 2010 (UTC)

 * The Islamic connection has nothing to do with Jesus' status, but rather with his manner of death -- i.e. that he did not actually die on a cross, but only his eidolon appeared to do so. Such a belief can be fairly called "Docetic"... AnonMoos (talk) 14:48, 28 November 2011 (UTC)

 * I've reverted the section blanking, as the main work cited does characterize Islamic views as Docetic. However I'm open to further discussion of the text there as I do not see the particular passage being referred to there; I suspect there's some WP:OR going on. Mangoe (talk) 15:55, 10 May 2012 (UTC)

Islam is docetism manichean, because Bahira, mentor from Muhammad was manichean. — Preceding unsigned comment added by 200.8.22.136 (talk) 22:22, 3 November 2013 (UTC)

Physical Body?
Item amended to "his human form was an illusion".

"Broadly it is taken as the belief that Jesus only seemed to be human, and that his physical body was a phantasm." This is a loaded definition and could be seen to be self-contradictory.

To a Docetist, Christ is a divine being and therefore he simply doesn't have a 'physical (material) body'.

Nicea
Item Removed  — Preceding unsigned comment added by 62.25.109.197 (talk) 10:50, 17 December 2013 (UTC)

"Docetism was unequivocally rejected at the First Council of Nicaea in 325[9]" Is there any proof of this because the text of the council doesn't even mention the Docetism? It appears to be and unfounded assumption and I suggest its removed. Docetism was possibly condemned by local synods and councils, perhaps these should be referenced.

I'm guessing what was meant is that the Council of Nicea rejected Docetism by implication of the Nicene Creed which states: "Who for us men, and for our salvation, came down and was incarnate and was made man; He suffered, and the third day he rose again, ascended into heaven..." but a better source should be found for when it was condemned. The earliest council I know of that condemned it was Chalcedon. — Preceding unsigned comment added by 2601:480:4101:B790:8C8D:90FF:C2DC:65AC (talk) 14:03, 5 October 2015 (UTC)

Monophysites
"While these characteristics (abstaining from the Eucharist) fit a Monophysite framework" Actually they don't, monophysites partake of the Eucharists as do their modern representatives the Miaphysites. — Preceding unsigned comment added by 62.25.109.197 (talk) 12:16, 26 September 2012 (UTC)

Apparition or myth?
The article is criss -crossing two different conceptions of docetism. We should be careful with word use, since ideas and terms get crossed very easily. The article is a witness to this. There is fatal difference between Jesus seeming to have a body, and Jesus not having a body at all. The former fits more along the lines of an external appearance and hallucination of a sort, that is, jesus appeared to them in the form of a body, which was not physical. The latter is best understood as an internal unembodied presence, which wholly lacks any form of body. Note that docetism can be interpreted as the doctrine in which Jesus is believed to have had a spiritual body which appeared external to the disciples. Docetism can also be surmised instead as the negation of that belief, which is that jesus never had any form or type of body whatsoever, be it spiritual or physical. I am unsure of any scholarly material that etches out this distinction.

There should perhaps be an article that details this distinction to help sand out interpretative issues. Or maybe previous entries can be edited in order that the article is more objective in classification. — Preceding unsigned comment added by 174.92.30.129 (talk) 20:34, 29 November 2014 (UTC)

 * If according to both theories Jesus did not undergo any bodily suffering, then they're quasi-equivalent from the point of view of mainstream Christian doctrine... AnonMoos (talk) 23:58, 2 December 2014 (UTC)

The theory in which Christ is an apparition is compatible with a suffering savior. In this case, prior to the crucifixion Jesus can be interpreted as a physically existing human who suffered on the cross, had died, and was resurrected. Prior to his ascension he appeared to the disciples in a new spiritual form, giving messages and teachings, and was then brought into heaven by god, where he awaits the second coming. Therefore Jesus can be interpreted as being an apparition after the Resurrection but prior to the ascension, with one still being able to maintain the belief without contradiction that he was human and suffered on the cross.

To say that the two theories are quasi-equivalent is an oversimplification and only works to obfuscate intellectual thinking, as well as cloud Jesus theories and lead them into some false dichotomy of orthodoxy and heterodoxy.Saintobalys (talk) 21:56, 11 January 2015 (UTC)

Ignatius of Antioch
in this edit you said you were removing OR and UNDUE. Maybe, but you added new material including two citations from a thesis by an (otherwise unpublished?) author, Daniel Streett. The new material itself, analysing the target of Ignatius of Antioch's writing, seems to be given undue weight in this article on docetism. Can you justify keeping it here? – Fayenatic  L ondon 08:30, 8 September 2015 (UTC)

In Popular Culture?
Surely Docetism's "cameo" in Treehouse of Horror XXV deserves a mention? Gambaguru (talk) 21:29, 18 May 2017 (UTC)

 * Must every reference that Simpsons makes be added to Wikipedia? 76.32.229.126 (talk) 04:27, 27 September 2022 (UTC)

title: Talk:Docetism


text:
Compile time

In computer science, compile time (or compile-time) describes the time window during which a language's statements are converted into binary instructions for the processor to execute. The term is used as an adjective to describe concepts related to the context of program compilation, as opposed to concepts related to the context of program execution (runtime). For example, compile-time requirements are programming language requirements that must be met by source code before compilation and compile-time properties are properties of the program that can be reasoned about during compilation. The actual length of time it takes to compile a program is usually referred to as compilation time.

Compile time/Early binding vs Run time
The determination of execution model have been set during the compile time stage. Run time- the method of execution and allocation - have been set during the run time and are based on the run time dynamicity.

Overview
Most compilers have at least the following compiler phases (which therefore occur at compile-time): syntax analysis, semantic analysis, and code generation. During optimization phases, constant expressions in the source code can also be evaluated at compile-time using compile-time execution, which reduces the constant expressions to a single value. This is not necessary for correctness, but improves program performance during runtime.

Programming language definitions usually specify compile time requirements that source code must meet to be successfully compiled. For example, languages may stipulate that the amount of storage required by types and variables can be deduced.

Properties of a program that can be reasoned about at compile time include range-checks (e.g.,  proving that an array index will not exceed the array bounds), deadlock freedom in concurrent languages, or timings  (e.g., proving that a sequence of code takes no more than an allocated amount of time).

Compile-time occurs before link time (when the output of one or more compiled files are joined) and runtime (when a program is executed). Although in the case of dynamic compilation, the final transformations into machine language happen at runtime.

There is a trade-off between compile-time and link-time in that many compile time operations can be deferred to link-time without incurring run-time cost.

title: Compile time


text:
Talk:Microsoft Windows

Semi-protected edit request on 6 June 2024
75.154.76.226 (talk) 20:25, 6 June 2024 (UTC) No information is verbalized here
 * Red question icon with gradient background.svg Not done: it's not clear what changes you want to be made. Please mention the specific changes in a "change X to Y" format and provide a reliable source if appropriate.  JTP (talk • contribs) 21:38, 6 June 2024 (UTC)

"Windows®" listed at Redirects for discussion
The redirect [//en.wikipedia.org/w/index.php?title=Windows%C2%AE&redirect=no Windows®] has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at  until a consensus is reached.  cogsan (nag me)  (stalk me) 16:20, 8 July 2024 (UTC)

Current related
Since Windows is going down due to the July 2024 global cyber outages, a should be added to the top of the page

-- 65.92.247.96 (talk) 09:39, 19 July 2024 (UTC)
 * ✅ megamanfan3 (talk) 10:20, 19 July 2024 (UTC)

title: Talk:Microsoft Windows


text:
AOC (disambiguation)

AOC commonly refers to Alexandria Ocasio-Cortez (born 1989), a Democratic member of the United States House of Representatives.

AOC or AoC may also refer to:

Aeronautics

 * Aeronautical operational control, communications between an aircraft and ground operators
 * Air Operations Center, a command center operated by the United States Air Force
 * Air officer commanding, a title in some air forces
 * Air operator's certificate
 * Aircraft Operating Company, a former British aerial photography company
 * Aircraft Operations Center, a part of the US NOAA
 * Aviation Officer Candidate, a rank at the U.S. Navy Officer Candidate School
 * Leipzig–Altenburg Airport's IATA code

Law

 * Appellation d'origine contrôlée, a protected designation of origin for French agricultural products
 * Appellation d'origine contrôlée (Switzerland), a protected designation of origin for Swiss wines

Organizations

 * AOC International (formerly Admiral Overseas Corporation), an electronics company in Taiwan
 * Adelaide Ornithologists Club, a club in Australia
 * Akan Orthography Committee, a committee for the Akan language
 * Aliança Operário-Camponesa or Worker–Peasant Alliance, a front of the Communist Party of Portugal
 * Alliance of Civilizations, a UN anti-extremism organization
 * Architect of the Capitol, a US agency
 * Army Ordnance Corps (United Kingdom), the predecessor of the Royal Army Ordnance Corps of the British Army
 * Arts Orange County, an arts council in California
 * Association of Colleges, an association in the UK
 * Association of Old Crows, an electronic warfare defense organization in the US
 * American Olympic Committee, a former and colloquial name for the United States Olympic & Paralympic Committee
 * Australian Olympic Committee
 * Athabasca Oil Corporation

Science and technology

 * Active optical cable, a cable used in 10 Gigabit Ethernet
 * Advice of Charge, a part of the mobile GSM services standard
 * Antibody-oligonucleotide conjugate
 * Axiom of choice, a concept in set theory.

Other uses

 * Act of Contrition, genre of Christian prayer
 * Advent of Code, annual set of computer programming challenges
 * Anglican Orthodox Church, a Continuing Anglican denomination in the United States
 * Atatürk Orman Çiftliği, a public recreational farming area in Turkey
 * Attestation of Compliance, a component of the self-assessment questionnaire in the Payment Card Industry Data Security Standard
 * Great Lakes Areas of Concern, designated portions of the Great Lakes Basin with environmental degradation issues
 * Hyrule Warriors: Age of Calamity, a 2020 video game released for the Nintendo Switch

title: AOC (disambiguation)


text:
User:MGee

HI ;)

That's it for now.

title: User:MGee


text:
Wikipedia talk:Pages by alphabetical order

I don't think the page is feasible. Do we really want people to put a new entry to this pages? -- Taku 17:59 Mar 4, 2003 (UTC)

It may sometimes be useful and/or interesting, to get a partial alphabetical list.

It surprised me, however, that http://www.wikipedia.org/w/wiki.phtml?title=Special:Allpages&limit=1&offset=100000 gives Loa loa filaria for article number 100,001. This would suggest that not all pages are alphabetically numbered. It is rather recent, from Feb 22, 2003, so may be new articles are first put at the end, and later put in alphabetic order.

I also wonder if such requests are time-consuming for the server - Patrick 18:38 Mar 4, 2003 (UTC)

 * It does seem to take a while sometimes...Maybe there is a way to keep the pages cached (put the results in a wikipage!)? Again, I wasn't expecting this page to get someone immediately to the page they wish given that the absence of such a feature led me to believe it would have been done already if it could be automatic. Brettz9 23:58 Mar 4, 2003 (UTC)

As Special:Allpages has an new layout, can I redirect this page there? Docu 00:02 Apr 16, 2003 (UTC)

 * Sounds good to me...As an aside...I think the Allpages would benefit from anchors among other pages (imho anchors would be a great addition...if articles get too large, they can simply be broken up; it's too useful to be able to browse through without needing to click and open up a separate view). - Brettz9 06:06 Apr 18, 2003 (UTC)