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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.

For a regular category $\mathcal{C}$, the category $\mathbf{Rel}(\mathcal{C})$ has the objects of $\mathcal{C}$ and subobjects of $A\times B$ as relations $A⇝B$. Composition takes the image of the pullback expressing existential quantification over the middle object.

For $\mathcal{C}=\mathbf{Set}$ this is the category of relations (sets); the general construction also applies to other regular categories.

At a perfectly rigid plane, every component of the displacement field vanishes.

For incident SV angle $\theta$, tangential phase matching gives $\sin \varphi =(c_P/c_S)\sin \theta$ for the reflected P-wave. It is evanescent when $\sin \theta >c_S/c_P$.

For $0\le r<d$, the standard binary bilinear form givesIndeed, the product of representing polynomials has polynomial degree at most $d-1$. Each square-free monomial in that product omits at least one of the $d$ variables and therefore takes the value $1$ an even number of times on ${\mathbb{F}}_2^d$. Summing the product over all evaluation points gives zero in the field with two elements, proving orthogonality. The dimensions of the two Reed-Muller codes add to $2^d$, by the symmetry $\left(\genfrac{}{}{0ex}{}{d}{j}\right)=\left(\genfrac{}{}{0ex}{}{d}{d-j}\right)$, so containment is equality.

At $r=d$, the Reed-Muller code is the full vector space and its dual code is zero. The same displayed identity holds if $\mathrm{RM}(d,-1)$ denotes the zero code, as in the boundary convention for Reed-Muller codes.

For length-$n$ binary linear codes $C_2\subseteq C_1$, their bar product isIt has length $2n$ and dimension $\dim C_1+\dim C_2$.

For a biased random walk on the nonnegative integers with upward probability $p>q=1-p>0$, the hitting probability of zero from $i$ is $(q/p{)}^i$. Indeed, first-step analysis gives $h_i=p{h}_{i+1}+q{h}_{i-1}$ for $i\ge 1$, with $h_0=1$. Its characteristic roots are $1$ and $\beta =q/p$, so any solution is $A+(1-A){\beta }^i$. Nonnegativity as $i\to \infty$ forces $A\ge 0$ and hence a value at least ${\beta }^i$. This candidate itself satisfies the recurrence. The theorem on the minimal nonnegative solution for hitting probabilities therefore identifies it with $h_i$. A walk on all of $\mathbb{Z}$ has the same probability until it first reaches zero.

A state is recurrent exactly when the sum over time of its return probabilities diverges.

A Recurrent state is null recurrent when its expected return time is infinite.

A Recurrent state is positive recurrent when its expected return time is finite.