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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.

When a second-order equation has a repeated indicial root and only one independent Frobenius series, a second solution has the local form

Under $\zeta =1/z$, the point $z=\infty$ is regular singular forexactly when $zP(z)$ and $z^{2}Q(z)$ are holomorphic functions of $1/z$ near $1/z=0$.

The disjoint union of $r$ triangles is $2$-regular on $3r$ vertices and has matching number $r$, attaining the regular-graph lower bound $n/3$.

For a regular curve, $s(t)={\int }_{t_0}^t|{\alpha }^{\prime }(r)|dr$ has positive derivative and hence a smooth inverse. Reparametrization by $s$ gives unit speed.