For a bounded linear regularization operator $R_{\alpha }$ and noisy data with $\Vert f_{\delta }-f\Vert \le \delta$, the triangle inequality givesThe first term is noise amplification; the second is the approximation bias on exact data. Thus $\alpha (\delta )\to 0$ and $\delta \Vert R_{\alpha (\delta )}\Vert \to 0$ suffice for a convergent regularization of an inverse problem, provided $R_{\alpha }$ is consistent on exact data.

If $X_n\stackrel{d}{\to }c$ for fixed real $c$, convergence of the distribution functions at $c-\epsilon$ and $c+\epsilon$ shows $\mathbb{P}(|X_n-c|>\epsilon )\to 0$. Thus convergence in distribution to a deterministic limit implies convergence in probability to that limit, even though the implication fails for nonconstant limits on a specified probability space.

A sequence of random variables $(X_n)$ is bounded in probability when ${\lim }_{K\to \infty }{\sup }_n\mathbb{P}(|X_n|>K)=0$. A sequence converging in probability to a finite random variable has this property, and its product with a sequence tending to zero in probability also tends to zero in probability.

Convergence in L1 means that the expected value of the absolute error tends to zero. It implies convergence in probability and convergence of expected values.

If a smooth posterior density has vanishing boundary terms, its log-posterior gradient satisfies $\mathbb{E}[g]=0$ andThis is a posterior integration-by-parts identity; it concerns differentiation in the random parameter, distinct from the usual mean-zero score identity for sampling distributions. It makes $g$ a vector control variate. Finite moments and known coefficients preserve unbiasedness of the Monte Carlo estimator.

A continuous martingale $M$ is L2-bounded when ${\sup }_{t\ge 0}\mathbb{E}|M_t{|}^2<\infty$. The L2 martingale convergence theorem gives a terminal value $M_{\infty }$ with convergence in $L^2$ and almost sure convergence. The resulting conditional expectation identity is $M_t=\mathbb{E}[M_{\infty }\mid {\mathcal{F}}_t]$. Boundedness on each separate finite horizon is a weaker condition. The subspace with $M_0=0$ is commonly denoted ${\mathcal{M}}_0^2$.

A hypoexponential distribution is the distribution of a sum of independent variables with exponential distributions. For two positive unequal rates $\lambda ,\mu$, convolution of independent random variables gives the probability density functionIt is zero for negative $s$. Its equal-rate limit is the shape-two gamma distribution with rate $\lambda$.

A continuous map injective map $f:X\to Y$ that is a homeomorphism onto its image with the subspace topology. A continuous map injective map from a compact space into a Hausdorff space is a topological embedding.

The operator range is the image of a linear map $K$. It need not be closed in an infinite-dimensional Hilbert space; that failure makes the Moore–Penrose inverse of an operator unbounded on its domain.

A replicating strategy is a self-financing portfolio whose terminal wealth equals the prescribed payoff of a contingent claim. Admissibility specifies the permitted wealth bounds and integrability of its holdings.

A right-linear grammar has productions of the form $A\to wB$ or $A\to w$, where $A,B$ are nonterminals and $w$ is a terminal word, possibly empty. Each right-hand side has at most one nonterminal, at its right end. Such grammars generate exactly regular languages: regard nonterminals as states and expand each terminal word into a finite path to another state or an accepting endpoint. Conversely the transition graph of a deterministic finite automaton gives these productions directly.

For a transitive model $M$ of ZFC and $\alpha \in M\cap \mathrm{Ord}$, the internally computed constructible hierarchy agrees with the ambient level: $L_{\alpha }^M=L_{\alpha }$. Satisfaction for a fixed set structure is computed from the same finite formulas and the same domain in both universes. Transfinite recursion then proves agreement at successors and limits. If $\theta =M\cap \mathrm{Ord}$, it follows that $L_{\theta }\subseteq M$.

For the flux $W_{ij}=-M{\partial }_i{\mu }_j+N_{ij}$ of a vector order parameter, microscopic reversibility requires independent Gaussian white noise components with covarianceThe current changes sign under time reversal. The resulting path-probability ratio contributes $-(k_{B}T{)}^{-1}\int W_{ij}{\partial }_i{\mu }_j$ to its logarithm; integration by parts converts this to the free-energy loss from the conserved dynamics.

The positive order-parameter mobility $M$ relates the diffusive transport flux to the chemical potential gradient: $\mathbf{J}=-M\nabla \mu$. Combining this law with local conservation gives the Cahn--Hilliard equation when $\mu$ is the functional derivative of a composition free energy.

The rate at which angular momentum crosses a surface, including advection and stresses. With positive inward mass accretion rate, the net inward flux in a steady accretion disk is $\dot{M}h-\mathcal{G}$.

In an orthonormal frame, the spin connection is the Lorentz connection acting on spinors. With ${\gamma }^{AB}=\frac{1}{2}[{\gamma }^A,{\gamma }^B]$, its covariant derivative is ${\nabla }_{\mu }\psi ={\partial }_{\mu }\psi +\frac{1}{4}{\omega }_{\mu AB}{\gamma }^{AB}\psi$. The metric connection determines $\omega$ when torsion conditions are specified. A zweibein gives its two-dimensional form; omitting it on a curved worldsheet generally destroys covariance.

A connected Feynman diagram is one-particle reducible if cutting one internal propagator disconnects it. These diagrams occur in connected correlation functions but do not contribute to a quantum effective action defined by a Legendre transform. For a strict Wilsonian effective action, a bridge also cannot be a high-momentum propagator when the sum of external momenta on one side lies below the integrated shell.

An amputated connected correlation function is obtained by removing external propagators. Amputation using full propagators eliminates external self-energy decorations; internal exchange diagrams remain. It is distinct from a one-particle-irreducible correlation function. In a scalar theory with vanishing one-point function, at zero momentum the four-point quantity satisfies ${\mathcal{A}}_4=-{\Gamma }^{(4)}+3[{\Gamma }^{(3)}{]}^2/{\Gamma }^{(2)}$ with a convention in which connected diagram vertices are minus effective-action derivatives. For a constant scalar source, the Legendre relation $J={\Gamma }^{\prime }(\varphi )$ gives $G^{\prime \prime }=1/{\Gamma }^{\prime \prime }$ and $G^{\prime \prime \prime \prime }=-{\Gamma }^{\prime \prime \prime \prime }/({\Gamma }^{\prime \prime }{)}^4+3({\Gamma }^{\prime \prime \prime }{)}^2/({\Gamma }^{\prime \prime }{)}^5$. Dividing by the four full external propagators proves the formula.

In dimension $N$, the conformal rescaling of a Riemannian metric $\tilde{g}=e^{2\omega }g$ changes the Ricci scalar by the displayed formula. In two dimensions the gradient-square term vanishes; for a flat base metric, $\tilde{R}=-2e^{-2\omega }\Delta \omega$.

For independent and identically distributed random variables $X_j$ that are integrable random variables, with $S_n={\sum }_{j=1}^n{X}_j$, symmetry gives $\mathbb{E}[X_1\mid S_n,S_{n+1},\dots ]=S_n/n$. The decreasing sigma-algebras generated by these future sums permit application of the reverse martingale convergence theorem.