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An integrable discrete-time local martingale is a true martingale. Its stopped increments have the form $1_{\{\tau \ge t\}}(X_t-X_{t-1})$, and the dominated convergence theorem removes a localizing sequence from the conditional increment identity.

For a finite extension of number fields $L/K$ and a nonzero prime ideal $\mathfrak{p}$ of $K$, the primes of $L$ above $\mathfrak{p}$ correspond to the orbits ofAn embedding determines a prime by pulling back the local absolute value on a field. Embeddings with the same prime extend to embeddings of the same completion and are conjugate. If $L=K(\alpha )$, these orbits are the irreducible factors of its minimal polynomial over $K_{\mathfrak{p}}$.

This polynomial is Eisenstein at $2$. Over ${\mathbb{Q}}_3$, its Newton polygon has a single slope $-2/5$, so it is irreducible and a root generates a totally ramified extension of degree five. Over ${\mathbb{Q}}_7$, its factor degrees are $1,1,3$: the simple residue roots $1,-1$ lift, and the remaining roots have valuation $1/3$. Thus the degree-five number field generated by a root has one prime above $3$ and three above $7$.

For a continuous map $f:M\to N$ of closed connected oriented $d$-dimensional manifolds, suppose $f^{-1}(y)=\{x_1,\dots ,x_s\}$ is finite. ThenBy excision, the degree-$d$ relative homology group $H_d(M,M\setminus f^{-1}(y);\mathbb{Z})$ is the direct sum of the local groups at $x_j$. The fundamental class maps to the tuple of local orientation generators. Mapping this tuple to $H_d(N,N\setminus \{y\};\mathbb{Z})$ adds their local degrees. Naturality identifies the result with the image of $(\mathrm{deg}f)[N]$. If the fiber is empty, the same argument gives degree zero.

A massive vector in $D$ spacetime dimensions has $D-1$ polarizations perpendicular to its timelike momentum. A massless vector has transverse polarizations modulo its null longitudinal polarization, leaving $D-2$ physical degrees of freedom.

The symmetry used to classify an open string must preserve its endpoint boundary background as well as its momentum. A Neumann-Dirichlet open-string boundary condition breaks target Lorentz transformations mixing ND and common directions with Neumann boundary conditions. For a D1–D25 pair with 24 ND directions, the common worldvolume has Lorentz group $SO(1,1)$, whose connected massive little group is trivial. The transverse $SO(24)$ remains as an internal rotation group. Thus a massive 24-component oscillator multiplet is not an $SO(25)$ vector of the full 26-dimensional massive little group; a vector of that full group would have 25 components.

For independent unbiased estimators $X_i$ of a common mean with known positive variances $v_i$, the linear unbiased estimator with smallest variance has weights $w_i=v_i^{-1}/{\sum }_j{v}_j^{-1}$ and variance $({\sum }_j{v}_j^{-1}{)}^{-1}$. Unbiasedness requires ${\sum }_i{w}_i=1$. Minimizing ${\sum }_i{v}_i{w}_i^2$ by a Lagrange multiplier gives the weights, and the strictly positive diagonal Hessian matrix proves a unique global minimum. This conclusion requires no normality assumption.

For $X\sim N(\mu 1,C)$ with a known positive-definite matrix $C$, the maximum-likelihood estimator of $\mu$ is $\hat{\mu }=(1^T{C}^{-1}X)/(1^T{C}^{-1}1)$. It is unbiased with variance $(1^T{C}^{-1}1{)}^{-1}$, attaining the Cramér-Rao bound. It is also the best linear unbiased estimator. Pairwise admissible correlation coefficients alone do not ensure that $C$ is a valid covariance matrix; the entire matrix must be symmetric and a positive semidefinite matrix. Inverse-based formulas require positive definiteness.

A best linear unbiased estimator minimizes covariance among linear estimators unbiased for the same target. In $Y=X\beta +\epsilon$, assume $\mathbb{E}\epsilon =0$, $X$ is a full-column-rank design matrix and the known covariance matrix $\Sigma$ is a positive-definite matrix. The best linear unbiased estimator of the coefficient vector is $(X^T{\Sigma }^{-1}X{)}^{-1}{X}^T{\Sigma }^{-1}Y$. A spatial trend estimate excludes the correlated residual prediction included in universal kriging.

A residual-versus-fitted plot displays residuals vertically against predicted responses horizontally. Curvature suggests an incorrect conditional-mean model; a fan-shaped spread suggests heteroscedasticity, while isolated large residuals may indicate unusual observations. It does not alone measure leverage or influence: regression leverage records unusual predictor combinations, and Cook's distance combines leverage and residual size.

The figure in coefficient of determination shows why the plot is useful: subtracting the fitted linear trend makes a small curved component easy to see. A visible pattern supplies a reason to investigate the model, while the absence of a pattern in a small sample does not establish that all assumptions hold. Other views address other questions: regression residuals against observation order can reveal temporal dependence, and a normal probability plot examines distributional shape. NIST's statistical handbook develops this use of graphical residual analysis for model validation. ^[ref]

A backward scalar or matrix recurrence for the quadratic coefficients in a value function for a linear-quadratic optimal control problem. It results from completing the control square in the Bellman equation. Multiplicative noise modifies the quadratic coefficients through its second moments.

Replacing the self-gravity term in the matter growth equation by $(1-f)4\pi G{\bar{\rho }}_{m}D$, with $0<1-f<1$, weakens clustering while leaving the background expansion unchanged.

Two linear extensions of a finite partially ordered set can be joined using as many admissible adjacent swaps as the inversion count of their relative permutation. Label the current list by the desired positions in the target. An adjacent descent consists of incomparable elements, since comparable elements have the same order in both lists. Swap that descent; its inversion count decreases by one. Iteration sorts the list and gives a path of minimal Coxeter length.

Testing a single nonnegative random-effect variance against zero puts the null on the boundary of the parameter space. In the usual regular increasing-independent-groups limit, the likelihood-ratio test statistic has limit $\frac{1}{2}{\delta }_0+\frac{1}{2}{\chi }_1^2$, rather than ${\chi }_1^2$. A finite-sample simulation calibrated for the actual design avoids relying on this asymptotic approximation.

The Jacobi identity of a Lie superalgebra includes parity signs. In particular, for odd $c$ and even $\varphi$ it gives $[c,[c,\varphi ]]=\frac{1}{2}[[c,c],\varphi ]$.

A Lie superalgebra representation acts on a parity-graded vector space by linear operators preserving the graded brackets. Odd generators reverse parity, and their brackets are represented by anticommutators.

For a Lie group action and $X$ in its Lie algebra, the fundamental vector field is $X_M(p)={\frac{d}{dt}|}_{t=0}\exp (tX)\cdot p$.

The coadjoint action of a Lie group on the dual space of its Lie algebra is $({\mathrm{Ad}}_g^{\ast }\eta )(X)=\eta ({\mathrm{Ad}}_{g^{-1}}X)$. The inverse in this formula makes it a left group action.

A continuous vector-valued local martingale $M$ starting at zero is standard multidimensional Brownian motion if its quadratic covariations satisfy $[M^i,M^j{]}_t={\delta }_{ij}t$. For each deterministic vector $\xi$, the Itô formula makes $\exp (i\xi \cdot M_t+|\xi {|}^{2}t/2)$ a complex local martingale. Its absolute value is bounded on each finite horizon, so it is a true martingale. Its conditional expectation identity givesThis conditional characteristic function identifies a centered multivariate normal distribution with covariance $(t-s)I$, independent of the past. Thus the coordinates are independent Brownian motions. The converse follows immediately from the standard coordinate quadratic covariations.

A Riemannian metric on a Lie group is bi-invariant if both left and right translations are isometries. Its identity inner product is invariant under the Adjoint representation of a Lie group; infinitesimally,This skew-adjointness is the key additional property beyond left invariance.