For integer $k\ge 1$, the entrywise power has entries $X_{ij}^k$ and equals the repeated Hadamard product. The zero power is the all-ones matrix, even at zero entries, because it represents the constant function one. Every nonnegative integer Hadamard power of a positive semidefinite matrix is positive semidefinite by the Schur product theorem. It is not an ordinary matrix power.

In the Gaussian white noise model, estimate a Haar scaling function coefficient by $\int {\varphi }_{J,k}dY$. The resulting Haar projection estimator is an unbiased estimator with independent coefficient errors of variance $1/n$. On each cell it is the observed path increment divided by the cell length. This gives a finite-dimensional estimator without imposing smoothness on the drift.

For a condensate with uniform bulk complex field one, its convergent localized momentum isThe subtraction removes the uniform-background phase-gradient contribution. If $\psi -1=O(r^{-1})$ and $\nabla \psi =O(r^{-2})$, its integrand is $O(r^{-3})$, integrable in two dimensions. Under decaying variations its functional derivative is $\delta p_j/\delta {\psi }^{\ast }=-i{\partial }_j\psi$; the subtraction is a boundary correction and does not change the bulk equation.

With $E=\frac{1}{2}\int [|\nabla \psi {|}^2+\frac{1}{2}(1-|\psi {|}^2{)}^2]$ and the renormalized momentum of a condensate, a Gross–Pitaevskii solitary wave is a critical point of $E-v{p}_x$. Along a differentiable solution family, fixed bulk normalization and vanishing variation boundary terms give $dE/d\lambda =vd{p}_x/d\lambda$. Therefore $dE/d{p}_x=v$ wherever $d{p}_x/d\lambda \ne 0$. At a cusp or turning point the parametrized identity is the appropriate statement; one must not assume a globally single-valued dispersion branch.

For a graph state $|G⟩={\prod }_{ij\in E}C{Z}_{ij}|+{⟩}^{\otimes |V|}$, measuring vertex $v$ in the computational basis with result $r$ gives the normalized stateEach Controlled-Z gate from $v$ to a neighbour acts as $Z_u^r$ after projecting $v$ onto $|r⟩$; all other edges remain unchanged. The projection contributes $1/\sqrt{2}$, so either result has probability $1/2$. For a four-cycle, deleting one vertex leaves a path and adds byproducts on its two endpoints.

Let $1$ be a leaf with neighbour $2$ and let the graph on the other vertices be generated by the Controlled-Z gate product $E_{\mathrm{rest}}$. If the leaf has Pauli Z gate frame $Z_1^r$, measuring it at angle $\alpha$ with result $s$ applies one-bit teleportation and leavesAny known Pauli Z gate frames on other vertices remain as additional factors before those vertices' input states. The identity $J(\alpha )Z^r=X^{r}J(\alpha )$ combines the leaf byproduct with the measurement result.

If global dimension is zero, every module is a projective module. Each short exact sequence splits because its quotient is projective. Conversely, splitting every short exact sequence makes each module a direct summand of a free module and therefore a projective module. Splitting also makes every module an injective module: for $A\hookrightarrow B$ choose a retraction $B\to A$ and compose it with any prescribed map out of $A$.

For reference Brownian motion $R=x+W$, stopped at $T=t\wedge {\tau }_{(a,b)}$ with $0<a<x<b$, this expression is the Radon-Nikodym derivative of the stopped Bessel process law of dimension $\delta$. The bounded stopped drift verifies the Novikov condition. The comparison laws must have the same initial value and the same stopping rule.

For $a>0$, the positive amplitude ordinary differential equation $R^{\prime }=aR-R^3$ has solution $R(t)=\sqrt{a}[1+(a/R_0^2-1)e^{-2at}{]}^{-1/2}$. This follows from the logistic differential equation for $R^2$. Every $R_0>0$ approaches $\sqrt{a}$ as a monotone function, and the positive distance to this limit decreases; an oscillating complex wave carrying this amplitude has no ordinary monotone ordering.

When an observation mean depends on the neighboring spins, changing a spin affects observations centered at its neighbors. Its own observation need not involve that spin. Selecting precisely the affected factors gives a correct local full conditional distribution without recomputing the entire posterior.

Two equally numerous groups of independent counts with Poisson distributions have areas $a$ and $2a$. Under common density their group means are in ratio $1:2$. With group sums $S_1,S_2$ and $T=S_1+S_2$, the maximized alternative-to-null likelihood ratio isUse zero-count limits and $\Lambda =1$ when $T=0$. The regular Wilks theorem calibration has one degree of freedom. Exactly conditioning on $T$ gives $S_1\sim \mathrm{Binomial}(T,1/3)$ under the null; this supports small-count testing without the large-sample approximation.

For a commutative complex unital Banach algebra, its Gelfand transform is a continuous unital algebra homomorphism into the continuous functions on its compact character space. The values of the transform of $a$ are exactly ${\sigma }_A(a)$, so $\Vert \hat{a}{\Vert }_{\infty }=r(a)$. Its kernel is the intersection of the kernels of the algebra characters. For a commutative C-star algebra, spectral radius norm equality for normal elements and the Stone-Weierstrass theorem upgrade this to an isometric C-star homomorphism onto $C(\Delta (A))$.

Suppose $B$ is a nonzero separable Banach space, the Cameron-Martin space of a Gaussian random variable in a Banach space is a dense subset of $B$, and every centered ball of positive radius has positive probability. Then $t\mapsto \mathbb{P}(\Vert X{\Vert }_B\le t)$ is strictly increasing on $(0,\infty )$. For $0<a<b$, choose $h$ in that Cameron-Martin space of a Gaussian random variable in a Banach space with $\Vert h{\Vert }_B=(a+b)/2$ and $0<\delta <(b-a)/2$. The closed ball of radius $\delta$ centered at $h$ lies in the annulus $a<\Vert x{\Vert }_B<b$ and has positive probability by the symmetric Gaussian translation lower bound. The nonzero assumption is necessary: on $B=\{0\}$ the distribution function equals one everywhere.

For bounded smooth positive $h$ bounded away from zero, differentiation and Gaussian integration by parts give the displayed identity. The Ornstein-Uhlenbeck gradient commutation identity and weighted Cauchy-Schwarz inequality bound its right side by $e^{-2t}\int |h^{\prime }{|}^2/hd\gamma$. Integrating and setting $h=f^2$ proves the Gaussian logarithmic Sobolev inequality. Positive clipped approximation extends it to functions of finite Gaussian Dirichlet energy.

Independent Gaussian random intercepts $u_j$ and random slopes $v_j$ give marginal within-group covariance ${\tau }_0^2+{\tau }_1^2{t}_{ij}{t}_{kj}+{\sigma }^2{1}_{\{i=k\}}$. In lme4, separate terms (1 | group) and (0 + x | group) impose zero intercept-slope covariance; (1 + x | group) estimates it. This independence restriction depends on the predictor origin: replacing $t$ by $t-c$ transforms the intercept effect to $u_j+c{v}_j$, generally correlated with $v_j$.

For $\kappa >0$, this error correlation decreases exponentially with elapsed time and permits irregular observation times. It is the stationary correlation of an Ornstein-Uhlenbeck process. Within a Gaussian linear mixed model, apply it to the errors conditional on random effects; the marginal correlation also includes those effects. nlme::corCAR1 parametrizes the same correlation by $\rho =e^{-\kappa }\in (0,1)$ and permits separate grouped time series.

In a Gaussian linear mixed model with $V=ZD{Z}^T+R$, the conditional multivariate normal distribution of $b$ given $Y$ has mean $D{Z}^T{V}^{-1}(Y-X\beta )$ and covariance matrix $D-D{Z}^T{V}^{-1}ZD$. When nonsingular, its mean is also its mode. Estimated parameters yield empirical conditional modes and shrink group deviations toward zero. At a zero variance component the associated effect is degenerate at zero; the covariance formula still applies, whereas formulas involving $D^{-1}$ require limits.

With known covariance parameters, best linear unbiased prediction minimizes prediction-error variance over linear predictors that are unbiased over both observation errors and random effects, for every fixed-effect value. In a Gaussian linear mixed model, the random-effect predictor is $D{Z}^T{V}^{-1}(Y-X{\hat{\beta }}_{\mathrm{GLS}})$, where generalized least squares estimates the fixed effects. Plugging in covariance estimates gives an empirical predictor; its uncertainty must also account for estimating those covariance parameters. It differs from a best linear unbiased estimator of an unknown fixed coefficient.

A variance component is a nonnegative parameter multiplying a specified covariance contribution. In a Gaussian linear mixed model with independent standardized group effects, ${\tau }_k^2{Z}_k{Z}_k^T$ describes the covariance induced by one set of random effects. Testing whether a component is zero is a variance-component likelihood-ratio test at a boundary; ordinary regular chi-squared likelihood-ratio calibration need not apply.

For a quadratic determinant in two effective momentum dimensions, differentiating the free-energy density with respect to the squared mass $R$ gives a term proportional to $\log ({\Lambda }^2/R)$. Integration gives $F_S\propto R\log R$ after subtracting analytic terms. Its thermal power index is still 1, but a strictly homogeneous pure-power expression misses this logarithm. It is a Gaussian determinant resonance, distinct from interaction-generated logarithms at an upper critical dimension.