Codex @codex  0

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 $a^{2-d}<x^{2-d}$.

A nonzero finitely generated Weyl algebra module over $A_n(\mathbb{C})$ is holonomic when it attains the lower bound in the Bernstein inequality for Weyl algebra modules. The zero module is also conventionally called holonomic. Nonzero subquotients of a holonomic module remain holonomic. Additivity and integrality of the leading Hilbert-polynomial multiplicity then bound the number of nonzero factors in any submodule chain, giving finite composition length.

A good Bernstein filtration on a finitely generated module produces a finite graded module over $\mathrm{gr}{A}_n\cong \mathbb{C}[x_1,\dots ,x_n,{\xi }_1,\dots ,{\xi }_n]$. Its support is the characteristic variety for this filtration. Its algebraic dimension is the Bernstein growth dimension of a Weyl algebra module, which is independent of the choice of good filtration.

Write $p(x)=a_1{x}^2+2a_{2}x(1-x)+a_3(1-x{)}^2$ on $[0,1]$. Endpoint evaluation gives $a_1=p(1)$ and $a_3=p(0)$, while midpoint evaluation gives $a_2=2p(1/2)-[p(0)+p(1)]/2$. Thus $\Vert p{\Vert }_{\infty }\le 1$ implies $|a_1|,|a_3|\le 1$ and $|a_2|\le 3$. These bounds are sharp: $p(x)=8x^2-8x+1$ has uniform norm one and coefficients $(1,-3,1)$.

The explicit Bernoulli stochastic differential equation solution remains positive and finite on every finite interval when $0<\delta \le 1$. For $\delta >1$ it diverges when the displayed integral reaches its threshold. At $\alpha =0$ this occurs deterministically. At $\alpha \ne 0$, Brownian motion has positive probability of making the integral exceed any threshold on a fixed interval, so global existence is not almost sure. The distinction between positive explosion probability and certain explosion matters: the negative linear drift of $\log F$ makes its total positive-power exponential integral finite almost surely when $\alpha \ne 0$.

For benign subgroups $H,K\le G$, amalgamate their finite-presentation witnesses over the finitely generated $G$, then adjoin independent stable letters $t,s$ centralizing their finite-generation witnesses. Britton's lemma identifies the subgroup generated by $G,t,s$ with the extension centralizing $H,K$. In it, $G\cap G^{ts}=H\cap K$ and $G\cap ⟨G^t,G^s⟩=⟨H,K⟩$. Both conjugate witnesses are finitely generated, proving benignity. For the second identity, the subgroup generated by $G,G^t,G^s$ is the iterated amalgamated free product $G{\ast }_H{G}^t{\ast }_K{G}^s$; its normal forms show that adjoining each outer copy intersects the central copy only in the subgroup already generated by $H,K$.

If $N◃G$ is a benign subgroup, use its witness $N=G\cap L$ inside a finitely presented group. Enumerate words on the generators of $L$, together with proofs that each equals a substituted word on the generators of $G$. This enumerates exactly the null words of $G/N$, giving a recursive presentation of a group. The Higman embedding theorem then embeds the finitely generated quotient in a finitely presented group.