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Suppose an estimating equation $n^{-1}{\sum }_i\psi (X_i,\hat{\theta })=0$ has the expansionwhere $A=\mathbb{E}[\partial \psi (X,{\theta }_0)/\partial {\theta }^T]$ is nonsingular and the centered estimating function has covariance matrix $B$. A central limit theorem gives asymptotic covariance $A^{-1}B{A}^{-T}$ for the rescaled error, or approximately $n^{-1}{A}^{-1}B{A}^{-T}$ for $\hat{\theta }$.

The transpose is essential when $A$ is not symmetric; the simplified form $A^{-1}B{A}^{-1}$ applies only in the symmetric case. Plugging in consistent estimates gives a sandwich covariance estimate. Regression heteroskedasticity-consistent standard errors and cluster-level generalized estimating equations use instances of this construction.

If $X_n\stackrel{d}{\to }X$ and $Y_n\stackrel{\mathbb{P}}{\to }c$ for a deterministic constant $c$, then $(X_n,Y_n)\stackrel{d}{\to }(X,c)$. The continuous mapping theorem therefore givesand $X_n/Y_n\stackrel{d}{\to }X/c$ when $c\ne 0$. No independence between the two sequences is required. The deterministic limit matters: convergence of two separate nondegenerate marginal distributions does not determine their joint limit.

For example, if $\sqrt{n}(\hat{\theta }-\theta )\stackrel{d}{\to }N(0,v)$ with $v>0$ and $\hat{v}\stackrel{\mathbb{P}}{\to }v$, then $\sqrt{n}(\hat{\theta }-\theta )/\sqrt{\hat{v}}\stackrel{d}{\to }N(0,1)$ wherever the denominator is defined. This justifies substituting a consistent estimator of asymptotic variance when constructing a Wald confidence interval.

Forrepeated integration by parts gives

For fixed $a$ and real $y\to +\infty$, it is the upper incomplete gamma function that has the endpoint-tail expansionThe empty product is one. Repeated integration by parts gives each coefficient and an integral remainder. For a positive integer $a$, the product eventually vanishes and the expansion terminates. The lower function instead satisfies $\gamma (a,y)=\Gamma (a)-\Gamma (a,y)$ for $\mathrm{Re}a>0$; confusing the two functions would incorrectly make the lower function decay to zero.^[ref]

If an algebra is defined over a parameter ring $R$, a ring homomorphism $R\to k$ produces a specialized $k$-algebra by extension of scalars. Defining relations specialize by replacing every parameter by its image in $k$.

A finite-dimensional associative algebra is semisimple when its left regular module is a direct sum of simple modules. Equivalently, every left ideal is a direct summand. In particular, a semisimple algebra has no nonzero square-zero left ideal: if $L^2=0$ and a module projection $A\to L$ exists, its value $e$ at $1$ satisfies $e\in L$ and $e^2=e$, forcing $e=0$.

A left ideal $L$ of an associative algebra $A$ is a vector subspace satisfying $AL\subseteq L$. Equivalently, it is a submodule of the left regular $A$-module.

The group algebra $k[G]$ is the vector space with basis $G$ and multiplication obtained by extending the group law $k$-bilinearly.

A commutative Noetherian ring is Artinian if and only if every prime ideal is maximal. Equivalently, it is Artinian exactly when its Krull dimension is zero.

The Condorcet paradox is a cycle in collective pairwise preferences even though each individual voter's preference is transitive.

A set is in ${\Pi }_2$ when membership can be written with one universal block followed by one existential block over a computable predicate. Equivalently, in the word-function formulation, there is a partial computable $f$ such that

A set $A$ is in ${\Sigma }_1$ exactly when it is the domain of a partial computable function. Equivalently, there is a computable predicate $R$ such that

A ${\Sigma }_1$ formula is equivalent to $\exists \mathbf{y}\delta$ with $\delta$ bounded. A ${\Pi }_1$ formula is equivalent to $\forall \mathbf{y}\delta$ with $\delta$ bounded.

If $g$ is holomorphic and nowhere zero on $0<|z|<r$, the residue of its logarithmic derivative is an integer:where $C$ circles zero once counterclockwise. It is the winding number of $g(C)$ about zero. Removing $z^k$ makes every period of the remaining logarithmic derivative vanish, so $z^{-k}g$ has a holomorphic logarithm on the punctured disk.

This does not make that derivative removable at zero. The counterexample $g(z)=e^{1/z}$ has $g^{\prime }/g=-1/z^2$ and residue zero, but still has a double pole in its logarithmic derivative. Its holomorphic logarithm $1/z$ is defined on the punctured disk, not across the missing point.

If $g$ additionally extends meromorphically across zero, write $g=z^{k}h$ with $h$ holomorphic and nonzero there. Then $g^{\prime }/g=k/z+h^{\prime }/h$, and the derivative after removing $z^k$ really is removable. In that stronger setting, $k$ is the zero order, or the negative of the pole order.

If $f,g$ are holomorphic near the closure of a bounded domain and $|g|<|f|$ on its boundary, then $f$ and $f+g$ have the same number of interior zeros, counted with multiplicity. Apply the argument principle to the zero-free boundary homotopy $f+tg$.

A portfolio is self-financing when every change in its asset holdings is paid for entirely by selling or buying assets within the portfolio, with no external cash added or removed.

An anyon condenses at a boundary when its string operator may terminate there without leaving an excitation, so the boundary can absorb or emit that anyon.

Two particle types are mutual semions when taking either once around the other multiplies the state by $-1$.

If $\psi =z{e}^{-r^2/(2a^2)}$ in two dimensions, then $L_z\psi =\hslash \psi$ and $L_z{\psi }^{\ast }=-\hslash {\psi }^{\ast }$. Orthogonality therefore makes the angular-momentum expectation of $\alpha \psi +\beta {\psi }^{\ast }$ equal to $(|\alpha {|}^2-|\beta {|}^2)\hslash$ for normalized coefficients.

Areal velocity measures how fast a position vector from a chosen origin sweeps signed swept area. It connects a geometric picture of motion to angular momentum: a long position vector with slow transverse motion can sweep the same area per unit time as a short one with fast transverse motion. Constancy requires a dynamical condition, given by the torque criterion for constant areal velocity.

Choose a fixed origin $O$ in an inertial frame and a fixed plane through $O$, with constant unit vector $\mathbf{n}$ normal to it. For a differentiable trajectory in that plane, let $\mathbf{r}(t)$ be position relative to $O$ and $\mathbf{v}=\dot{\mathbf{r}}$ its velocity. Positive swept area follows the right-hand rule about $\mathbf{n}$. In positively oriented Cartesian coordinates in the plane, and in plane polar coordinates with angle $\theta$ in radians,Here the dot product selects the normal component of the cross product. Motion away from or towards $O$ alone contributes no area. The signed specific angular momentum component $h_n=\mathbf{n}\cdot (\mathbf{r}\times \mathbf{v})$ obeysfor a constant mass $m$. If $\mathbf{n}$ is chosen along a nonzero conserved specific angular momentum vector, then $h_n=h=|\mathbf{r}\times \mathbf{v}|$ and the familiar positive formula is $\dot{A}=h/2$. Reversing the orientation changes the sign; the instantaneous unsigned swept-area rate is $|\dot{A}|$.