The finite-condition Lévy collapse $\mathrm{Lv}(\kappa )$ consists of finite partial functions $p$ with $\mathrm{dom}p\subseteq \kappa \times \omega$ and $p(\alpha ,n)<\alpha$, ordered by reverse inclusion. The generic union gives a surjection $\omega \to \alpha$ for every infinite $\alpha <\kappa$. If $\kappa$ is regular and uncountable, the Delta-system lemma at a regular uncountable cardinal proves the $\kappa$-chain condition, so the extension makes $\kappa ={\aleph }_1$.

The punctured circular coneis geodesically incomplete because a generator reaches the missing vertex in finite time. It is an inextendible embedded surface: any proper connected embedded extension would have to add the vertex, but its tangent planes have different limits as $\theta$ varies. Nevertheless its Gaussian curvature of a cone away from its vertex is identically zero.

An interest rate measures the growth of value across time. In discrete time, a one-period investment at rate $r_t$ grows by the factor $1+r_t$.

A zero-coupon bond makes one payment at maturity and no earlier coupon payments. A unit-face-value bond maturing at $T$ has terminal value $P_T^T=1$.

For a quantum harmonic oscillator perturbed by $\lambda X^4$, write $X=\sqrt{\hslash /(2m\omega )}(A+A^{†})$. Sincefirst-order nondegenerate perturbation theory gives

An abelian extension is a Galois extension whose Galois group is an abelian group.

The map $x\mapsto x^q$ is an automorphism of ${\mathbb{F}}_{q^n}$ fixing ${\mathbb{F}}_q$. Its powers are all the ${\mathbb{F}}_q$-automorphisms, and its order is $n$.

Given noisy evaluations at $x_0+h{Z}_i$, where the $Z_i$ are independent Rademacher random variables, the estimatoraverages one-sided finite differences from both directions. If $|g^{\prime \prime }|\le M$ and the noise variance is ${\sigma }^2$, its mean squared error is at most $h^2{M}^2/4+{\sigma }^2/(N{h}^2)$.

For a genus-one fibered knot, the fiber is a once-punctured torus and its monodromy acts on first homology by a matrix $M\in S{L}_2(\mathbb{Z})$. Up to a Laurent unit, the one-variable Alexander polynomial is

The short-time limit rescales a Euclidean time circle toward zero size. Supersymmetry cancels paired nonzero-mode determinants, leaving an integral over constant or symmetry-fixed zero modes.

The Berezin integral is differentiation with respect to a Grassmann variable: $\int d\theta 1=0$ and $\int d\theta \theta =1$. A finite-dimensional fermionic Gaussian integral gives a determinant or Pfaffian.

The fermion number operator counts occupied fermionic modes. Its parity $(-1{)}^F$ is positive on even exterior degree and negative on odd exterior degree.

Fermionic creation and annihilation operators satisfy $\{a_i,a_j^{†}\}={\delta }_{ij}$ and vanishing equal-type anticommutators. These relations make each mode either empty or singly occupied.

For each fixed positive integer $t$,The standard proof counts $t$-element subsets of graph neighbourhoods in a bipartite subgraph. If one side has degrees $d_y$, the absence of $K_{t,t}$ givesconvexity then bounds the number of edges.

For a timelike geodesic moving in the $x$ direction with $t(0)=x(0)=0$ and $\dot{x}(0)=v_0>0$, spatial translation symmetry gives $e^{2Ht}\dot{x}=v_0$. Proper-time normalization then gives

For $A\subseteq [n]$, the coordinate projection $S_A$ of $S\subseteq {\mathbb{R}}^n$ consists of the coordinate tuples $(x_i{)}_{i\in A}$ obtained from points $x\in S$. Its measure is denoted $|S_A|$.

An axis-parallel box in ${\mathbb{R}}^n$ is a Cartesian product $I_1\times \cdots \times I_n$ of bounded intervals. Its volume is the product of its side lengths, and each coordinate projection is the product of the corresponding intervals.

For an integrable adapted discounted price process $X$ in finite discrete time, the following are equivalent under a probability measure $Q$: $X$ is a $Q$-martingale; every bounded predictable self-financing strategy has a martingale discounted value; and every such strategy has expected terminal value equal to its initial value. To recover the martingale property from the expectation identity, trade one asset only during one period on an arbitrary event in the preceding sigma-algebra.

An equatorial inertia--gravity wave is a high-frequency equatorially trapped mode restored jointly by buoyancy and the Coriolis force. Its dispersion relation has eastward and westward branches lying outside the low-frequency equatorial Rossby wave branch.

A circular null geodesic remains at constant areal radius. It lies at a stationary point of the null radial effective potential and may be stable or unstable according to the sign of the second derivative there.