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A continuous function on a compact interval is uniformly continuous. A partition with sufficiently small mesh then makes the oscillation on every subinterval small, so its upper and lower Darboux sums can be made arbitrarily close.

With no cosmological term, the two-dimensional Einstein field equations reduce identically on the geometric side to $G_{\alpha \beta }=0$. They impose $T_{\alpha \beta }=0$ and supply no local metric dynamics; the Einstein-Hilbert action is topological.

The Cheeger constant of a finite reversible chain is the least stationary boundary flow divided by the smaller stationary mass of a set and its complement.

For a reversible transition matrix $P$ with invariant distribution $\pi$,

This chain energy is a discrete example of a Dirichlet form.

Detailed balance is the identity pi_i P_ij = pi_j P_ji for every pair of states.

Every retract of a contractible space is contractible. If $i:A\hookrightarrow X$ is the inclusion, $r:X\to A$ is a retraction, and $H$ contracts $X$, thencontracts $A$.

For a source point $y(s)$ and field event $(x,t)$, the retarded time solves

The invariant residual-life mass at i is proportional to the probability that a lifetime exceeds i.

With bounded integer lifetimes, residual lifetime decreases deterministically and resets randomly at zero.

For non-arithmetic inter-renewal times with finite positive mean $\mu$, the excess $E(t)=S_{N(t)+1}-t$ converges in distribution to $E_{\infty }$ with

For $\lambda \ne \pm 1$, the rotation acts by $\mathrm{diag}(\lambda ,{\lambda }^{-1})$ and the reflection interchanges the two eigenlines.

If a central subgroup is isomorphic to $S^1$, its weight spaces are invariant under the entire group, because every representing operator commutes with the central circle action.

At each node of a binomial tree, a claim with next-period values $V_u,V_d$ has valuewhere $q$ is the local risk-neutral probability in a binomial market. For an American claim, replace this continuation value by the maximum of continuation and immediate exercise value.

A running coupling is a renormalized coupling regarded as a function of the renormalization scale. Its scale derivative is its beta function.

At every deterministic $t>0$, the containing interval stochastically dominates an unselected interarrival time:Conditioning on the first interarrival gives a renewal equation; subtracting the constant tail probability leaves a renewal equation with a nonnegative forcing term.

For exponential interarrival times of rate $\lambda$, the age and residual life at time $t$ converge jointly to independent rate-$\lambda$ exponential variables. Their sum therefore converges to the gamma distribution with density ${\lambda }^{2}x{e}^{-\lambda x}$, the size-biased exponential law.

For a nonnegative random variable $X$ with $0<\mathbb{E}X<\infty$, its size-biased version $\hat{X}$ is defined byfor bounded measurable $h$. If $X$ has density $f$, then $\hat{X}$ has density $xf(x)/\mathbb{E}X$.

Dimensional regularization analytically continues loop integrals from an integer spacetime dimension to $d=d_0-ϵ$. Ultraviolet logarithms then appear as poles in $1/ϵ$.

Cutoff regularization restricts loop momenta to $|\ell |<\Lambda$. The ultraviolet cutoff $\Lambda$ makes individual integrals finite while displaying power and logarithmic ultraviolet divergences explicitly.

Early stopping halts iterative fitting near the best validation performance, limiting adaptation to training noise.