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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.

A nonprincipal ultrafilter on the natural numbers gives a Banach limit by taking the ultralimit of the Cesàro means of each bounded sequence. Compactness of a closed bounded interval, or disk for complex sequences, supplies the limit. Finite intersections in the filter give linearity; nonnegative averages give positivity; constants give norm one. The Cesàro theorem for convergent sequences gives agreement with ordinary limits because the ultrafilter contains every cofinite set. For the left shift on bounded sequences, the two means differ by $(x_{n+1}-x_1)/n$, which tends to zero, proving shift invariance.

The displayed sublinear functional on the real l-infinity sequence space agrees with the ordinary limit on convergent sequences. A dominated extension by the Hahn-Banach theorem therefore defines a generalized limit bounded between the lower and upper limits of the Cesàro means. For the left shift on bounded sequences $S$, the averages of $Sx-x$ equal $(x_{N+1}-x_1)/N$, which tend to zero. Domination applied to both signs gives $L(Sx-x)=0$, hence shift invariance. Positivity, agreement on constant sequences and the operator norm one follow from the bounds, so $L$ is a Banach limit.

For a nonempty ordered coordinate set $A$, write $A^{-}=[\min A-1]$ and $A^{+}=[\max A-1]\setminus A$. An exact $k$-cover satisfies ${\sum }_{A}H(X_A\mid X_{A^{+}})\le kH(X)\le {\sum }_{A}H(X_A\mid X_{A^{-}})$. A short proof uses the chain rule for information entropy contributions $h_i=H(X_i\mid X_{[i-1]})$: each conditional subset entropy bounds ${\sum }_{i\in A}{h}_i$ on the appropriate side, by conditioning reduces entropy. Weighting gives the lower inequality for a fractional packing and the upper inequality for a fractional cover. See Madiman and Tetali, "Information Inequalities for Joint Distributions".