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A sequence $(x_n{)}_{n\ge 1}$ of real numbers has limit of a sequence $L\in \mathbb{R}$, written $x_n\to L$, precisely whenThe absolute value measures the error. The tolerance $\epsilon$ is chosen first; $N$ may depend on it, but must control every later term. Infinitely many accurate terms alone do not suffice. For example, $x_n=(-1{)}^n$ repeatedly equals one but also repeatedly differs from one by two. The definition and uniqueness of the limit of a sequence are discussed in MIT 18.100A, Lecture 6, definitions 13 and theorem 15.

For a worked error estimate, let $x_n=3+2/n$. Then $|x_n-3|=2/n$, so any positive integer $N>2/\epsilon$ works. With $\epsilon =0.01$, choose $N=201$; every later error is at most $2/201<0.01$. The strict inequality matters: $N=200$ would give equality at its first term. A limit of a sequence turns the informal instruction “take enough terms” into an explicit accuracy guarantee.

The limit of a sequence is unique: two distinct proposed answers cannot both be within arbitrarily small error of every late term, by the triangle inequality. Changing finitely many initial terms does not change it. For an extended real number, $x_n\to +\infty$ means $\forall M\in \mathbb{R}\exists N\forall n\ge N:x_n>M$; convergence to $-\infty$ similarly requires $x_n<M$. These describe escape rather than a finite answer. The extended-real Fekete lemma is an example where the limit of a sequence need not be finite.

A finite filtration of a cochain complex produces a convergent spectral sequence. The induced abutment filtration is the image of $H^n(F^{p}C)\to H^n(C)$. Finiteness in each degree suffices; a bounded range of cochain degrees is not required.

Uniform convergence in probability means that the supremum of the absolute discrepancy on the entire index set tends to zero in probability. It controls evaluation at random indices in that set, unlike pointwise convergence in probability. On a fixed compact index set it is a special case of uniform convergence on compacts in probability.

A nonnegative continuous multiplier $k$ defines a contraction semigroup on the space of continuous functions vanishing at infinity and on $L^2(\mu )$ by $P_{t}f=e^{-tk}f$. Strong continuity follows from uniform convergence on compact sets and small tails in $C_0$, or the dominated convergence theorem in $L^2$. The multiplier itself may be unbounded.

The one-dimensional semigroup associated with the standard Ornstein-Uhlenbeck process has invariant standard Gaussian measure and generator $Lf=f^{\prime \prime }-x{f}^{\prime }$. Its Mehler formula for the Ornstein-Uhlenbeck semigroup makes positivity and invariance explicit. The normalized Probabilists' Hermite polynomials diagonalize it, with $P_t{h}_n=e^{-nt}{h}_n$. It has spectral gap one and satisfies the sharp Gaussian Poincaré inequality.

On a compact metric space, a conservative Feller semigroup is a strongly continuous semigroup of positive unital operators on $C(X)$. The Riesz-Markov-Kakutani representation theorem associates a Markov kernel to each operator. On noncompact spaces the standard definition uses $C_0(X)$; conservativity then refers to the corresponding kernels having mass one, even when $1\notin C_0(X)$.

A contraction is a homotopy from the identity map of $X$ to a constant map. Its existence says that $X$ is a contractible space. For a tree, choosing a root and moving each point along its unique finite path to that root gives a contraction, with continuity understood in the weak topology of a CW complex.

A continuous martingale is a continuous-time martingale whose paths are continuous outside one null event. Continuity does not remove the integrability or conditional-expectation requirements in the martingale definition.

A càdlàg martingale is a continuous-time martingale represented by a càdlàg process: its paths are right-continuous and have finite left limits at positive times, on one common probability-one event. This path regularity permits approximation of stopping times from above.

A martingale whose values over its whole time interval form a uniformly integrable family has an integrable terminal limit and is closed by that limit. Conversely, conditional expectations of an integrable terminal variable form a uniformly integrable martingale, by uniform integrability of conditional expectations. For continuous martingales, one may equivalently use uniform integrability of the family stopped at finite stopping times.

The Rayleigh distribution is the probability distribution of the Euclidean norm of a two-dimensional random vector with independent zero-mean normal random variables of equal variance as components. If $X,Y\sim N(0,{\sigma }^2)$ are independent, thenHere $\sigma$ is the standard deviation of each component, not of the radius $R$. At $\sigma =1$ this gives the unit-scale probability density function $r{e}^{-r^2/2}$.^[ref]

This geometry turns two signed errors into one nonnegative distance. For example, suppose horizontal and vertical positioning errors are independent $N(0,4{\mathrm{cm}}^2)$ random variables. Their means both vanish, but the distance from the intended point is usually positive. The Rayleigh distribution with $\sigma =2\mathrm{cm}$ gives the probability of landing inside any centered disk. These are chosen model assumptions, rather than measurements of a particular instrument.

An apparently paradoxical feature is that the most likely location per unit area is the origin, while the radial mode is $\sigma$. A narrow annulus farther out has more area. Near zero that growing circumference outweighs the falling planar density; farther out the exponential function wins. Distinguishing densities measured against different elements of area or length resolves the paradox.

A bounded linear operator between normed spaces is contractive if its operator norm is at most one. This norm property is distinct from a strict contraction mapping, whose Lipschitz constant must be less than one.

A linear map $U:C(K,\mathbb{R})\to C(K,\mathbb{R})$ is positive when $f\ge 0$ implies $Uf\ge 0$. This is positivity for the pointwise order, not positivity of a quadratic form. On a compact space, $|Uf|\le U|f|\le \Vert f{\Vert }_{\infty }U1$, so $\Vert U\Vert =\Vert U1{\Vert }_{\infty }$. This makes positive maps especially useful for uniform approximation.

A contingent claim is replicated when an admissible self-financing portfolio has exactly its terminal payoff. In the absence of arbitrage, two replicating strategies for the same payoff have the same initial cost.

The payout $\sqrt{S_T}$ is a concave stock payoff. Its conditional expected value has a discount-like dependence on future variance. The square-root stock supermartingale explains the negative variance drift.

The image of a morphism of algebraic varieties is constructible. This replaces the generally false assertion that all morphisms have closed image. Group structure adds the fact that a constructible subgroup is closed.

The equality $L\cap V_{\alpha }=L_{\alpha }$ can fail even at $\alpha =\omega +1$. Inside the constructible universe, $L_{\omega }=V_{\omega }$ is countable, so its definable power set $L_{\omega +1}$ is countable. The constructible power set of $\omega$ is uncountable there by the Cantor theorem. A real outside $L_{\omega +1}$ remains a constructible subset of $\omega$, hence belongs to $V_{\omega +1}$. The cardinal comparison must be made inside $L$, since its reals can be externally countable.

The coefficients $C_{ij}{}^k(q,p)$ in $\{{\phi }_i,{\phi }_j\}=C_{ij}{}^k{\phi }_k$ are structure functions. A finite real span of first-class constraints is a Lie algebra when it closes with constant coefficients. General structure functions instead describe closure with phase-space-dependent coefficients, while still permitting canonical gauge transformations.

For a saline Stefan problem with common thermal properties, $ϵ=\sqrt{D/\kappa }\ll 1$, and fixed latent-to-sensible heat ratio, set $T_0=-m{C}_0$, ${\theta }_{-}=T_0-T_{-\infty }$ and ${\theta }_{+}=T_{\infty }-T_0>0$. A finite-$\lambda$ similarity solution has $T_i=(T_{\infty }+T_{-\infty })/2+O(ϵ)$. At the onset of constitutional supercooling, $\lambda =O(ϵ)$ and the error in this mean is $O({ϵ}^2)$. The salt-rejection function for a saline Stefan front then gives $m\lambda C_i=({\theta }_{-}-{\theta }_{+})/(2\sqrt{\pi })+O({ϵ}^2)$, whereas $\sqrt{Dt}{T}_{l,x}(a^{+})=ϵ{\theta }_{+}/\sqrt{\pi }+O({ϵ}^2)$. Comparing the actual temperature gradient to the liquidus gradient gives the displayed onset, with supercooling on the side of greater ${\theta }_{-}$. For a freezing front at $a\ge 0$ and $D<\kappa$, the ratio of these two gradients decreases into the liquid, so their interface comparison also decides whether a supercooled interval occurs anywhere ahead. The formula is an asymptotic criterion; finite-$ϵ$ onset requires the full Stefan condition and salt balance.

The sets $A_i=[m]\setminus \{i\}$ have every $t$-fold intersection of size $m-t$ and every $(t-2)$-fold intersection of size $m-t+2$. For $m\ge t+1$, their common intersection is empty and they attain $m=k+t-2$ in the constant t-wise intersection dichotomy.