A pure birth process on ${\mathbb{N}}_0$ jumps only from $n$ to $n+1$, at a strictly positive rate ${\lambda }_n$. Its successive holding times are independent exponential variables of rates ${\lambda }_n$.

With $m$ groups of $q$ trials and common marginal failure probability $p$, each count $X_j$ has expectation $qp$, so the displayed estimate is unbiased without assuming independence within groups. independent individual trials give variance $p(1-p)/(mq)$. If only groups are independent, estimate the variance instead by $s_X^2/(m{q}^2)$, retaining within-group dependence. Equal marginal probabilities alone do not justify a binomial confidence interval.

A contingent claim paying one at maturity if a specified price path has remained below an upper barrier, and zero otherwise. This survival payoff differs from a terminal digital put option, which tests only the terminal price. In the dividend-free Black-Scholes model, its value is the discount factor times the risk-neutral probability of avoiding the barrier, obtainable from the linear-boundary Brownian first-passage distribution.

A cash-at-maturity upper-barrier digital option pays a fixed amount if the stock touches an upper barrier by maturity. In the Black-Scholes model, write $a=\log (c/S_0)/\sigma$ and $b=-(r-{\sigma }^2/2)/\sigma$. Its unit-payoff price is $e^{-rT}$ times the linear-boundary Brownian first-passage distribution at $T$. This differs from paying at the hitting time, when the discount factor is random, and from an ordinary terminal-price digital option.

The polynomial identity $(X+1)g(X)g^{\mathrm{rev}}(X)=X^{23}+1$, where $g^{\mathrm{rev}}(X)=X^{11}g(X^{-1})$, impliesEvery coefficient is the binary inner product of the coefficient vector of $g$ with one cyclic shift. Thus every such inner product is one. Appending a parity bit to the cyclic shifts gives mutually orthogonal weight-eight vectors, so orthogonal generators of doubly even codes applies.

// Target: algebra.bigb

A root $\beta$ of the binary Golay generator polynomial has order $23$, and the Frobenius endomorphism shows that $\beta ,{\beta }^2,{\beta }^3,{\beta }^4$ are roots: $2^8\equiv 3(\mathrm{mod}23)$. The BCH bound gives minimum Hamming distance at least five. The circular autocorrelation of the binary Golay generator makes its parity extension doubly even, excluding original weights five and six. The generator itself has Hamming weight seven, proving that the distance is exactly seven.

// Target: algebra.bigb

The generator polynomial of a cyclic code defining the binary Golay code is $1+X+X^5+X^6+X^7+X^9+X^{11}$.

The extended binary Golay code is the parity extension of the length-$23$ binary Golay code. It is a self-dual code and a doubly even code, with minimum Hamming distance eight. Its radius-three Hamming balls are disjoint but do not cover its Hamming space; it is not a perfect code.

A single step has negative entropy production with probability $q$, independent of how long an observation lasts. A whole window is negative only if negative steps outnumber positive ones:For even $n$, the tie $K=n/2$ has zero entropy production and is excluded. At $p=3/4$, the negative-window probabilities for $n=1,9,21$ are approximately $0.25$, $0.04893$, and $0.00642$. Longer windows still contain negative increments: their expected number is $nq$. Their total is usually positive because the expected number of positive increments is $np$.

Only odd window lengths are plotted to avoid alternating tie effects. The left panel gives the exact negative-window probability and the upper bound $(2\sqrt{pq}{)}^n$; the right gives expected counts $np$ and $nq$. For steps of fixed duration $\delta t$, physical duration is $n\delta t$. These are counts and probabilities predicted by the model, not bead traces. The bound follows from $\mathrm{Pr}(S_n<0)\le ⟨e^{-S_n/2}⟩=(2\sqrt{pq}{)}^n$. Since $2\sqrt{pq}<1$ for $p\ne q$, negative totals are exponentially suppressed even though the negative-step fraction stays $q$.

If a Lie group has an Adjoint representation of a Lie group element with a real eigenvector of eigenvalue whose modulus differs from one, it has no bi-invariant Riemannian metric. Conjugation is a composition of left and right translations, hence would preserve the positive identity inner product. Applying it to that eigenvector gives $\Vert v\Vert =|\lambda |\Vert v\Vert$, a contradiction. The orientation-preserving affine group of the real line supplies a connected example: conjugation by dilation of factor two multiplies the infinitesimal translation by two.

A barrier on an infinite $A\subseteq \mathbb{N}$ is a family of nonempty finite increasing sequences, viewed as finite subsets, no one contained properly in another, such that every infinite subset of $A$ has a member of the family as an initial segment. Write $s◃t$ if there is a finite increasing sequence $u$ of which $s$ is an initial segment and of which $t$ is an initial segment after its first term is deleted. This shift relation underlies the good-array definition of better-quasi-ordering.

A continuous array takes infinite subsets of an infinite $A\subseteq \omega$, in the ordinary topology on infinite subsets, to a discrete preorder $Q$. It is bad if $F(X)\nleqq F(X\setminus \{\min X\})$ for every infinite $X$. A barrier in better-quasi-order theory array induces such a continuous array by evaluating its unique barrier initial segment. The continuous-array and barrier formulations of better-quasi-ordering are equivalent, using refinement of fronts to barriers in the reverse direction.

If $Q$ is a better-quasi-ordering, its full power set is better-quasi-ordered by the Hoare domination preorder. A bad continuous array of subsets would allow a choice of $q_X\in A_X$ not below any member of $A_{X^{-}}$, where $X^{-}$ deletes the least element. Choose this witness as a fixed function of the two locally constant subsets. Then $X\mapsto q_X$ is a bad continuous array in better-quasi-order theory into $Q$, a contradiction. Empty source subsets cannot occur in a bad array because they precede every subset.

For three quark generations, their corresponding leptons, and one complex Higgs doublet, use $Q=T_3+Y$ and $D_{\mu }\supset i{g}_{Y}Y{B}_{\mu }$. In $\mu d{g}_i/d\mu =b_i{g}_i^3/(16{\pi }^2)$ the coefficients are the displayed values. Per generation, the fermionic Dynkin index sums are two for both non-Abelian factors and the sum of squared hypercharges is $10/3$. The Higgs index for $SU(2)$ and its sum of squared hypercharges are each $1/2$. Gauge terms are $-22/3,-11$ for $SU(2),SU(3)$, respectively. For the grand-unified normalization $g_1=\sqrt{5/3}{g}_Y$, the Abelian coefficient instead is $41/10$.

In $d=4-ϵ$, the quartic scalar coupling has engineering dimension $ϵ$. Factoring out ${\mu }^{ϵ}$ makes the renormalized running coupling dimensionless. Differentiating its bare expression at fixed bare parameters generates the linear term $-ϵ\lambda$, even when the quantum contribution to the renormalization-group beta function begins quadratically. The simple-pole formulas for minimal-subtraction renormalization group functions determine the remaining contribution from the simple pole of $F$.

For a four-dimensional dimensionless gauge coupling, the one-loop correction is cubic in $g$. With $\alpha =g^2/(4\pi )$, the chain rule and $d/d\ln {\mu }^2=(1/2)d/d\ln \mu$ give the displayed conversion. In QCD, $c=-(11N-2n_f)/(48{\pi }^2)$, hence $d{\alpha }_s/d\ln {\mu }^2=-(11N-2n_f){\alpha }_s^2/(12\pi )+O({\alpha }_s^3)$. Other types of coupling or several interacting couplings can have different leading expansions; the squared-scale convention must not be mixed with the unsquared one.

For a field in a tensor-product representation of a product gauge group, its contribution to the beta function of factor $i$ includes the dimensions of all other factors. Those indices label independent copies for that factor. For an Abelian factor, replace its Dynkin index by the squared charge. Thus a quark weak doublet contributes three copies to the weak beta function and two copies to the colour beta function; hypercharge sums include all six components. Chirality conjugation changes a representation and the sign of an Abelian charge, but not its quadratic index.

Beta equivalence is semidecidable: enumerate finite conversion proofs. It is not decidable. A proposed total numeral-valued classifier of two nonempty disjoint beta equivalence classes can be reversed by a Church Boolean selector; a self-quoting fixed point of this reversal then belongs to the opposite class from the classifier output.

A uniform prior distribution on the success chance makes the number of successes in any fixed number $n$ of Bernoulli trials uniform on $0,\dots ,n$, because $\left(\genfrac{}{}{0ex}{}{n}{y}\right)\mathrm{B}(y+1,n-y+1)=1/(n+1)$. This is a statement about counts, not a uniform distribution over the $2^n$ ordered outcome sequences. It also makes predictions reversible between equally sized batches: their joint probability distribution is symmetric and their marginal count distributions are the same uniform distribution.

For Brownian motion started at radius $x>0$, the displayed process is a continuous local martingale. It is a bounded true martingale when $0<a\le x$, but a strict local martingale when $a=0$ or $a>x$. In the latter case the boundary is outside the starting point, so stopping does not remove the singularity at the origin. Brownian scaling gives $\mathbb{E}\Vert B_t{\Vert }^{2-d}=O(t^{-(d-2)/2})$; when $a>x$, the stopped expectation tends to $a^{2-d}<x^{2-d}$.