For an adjoint vacuum expectation value proportional to $\mathrm{diag}(1,1,-2)$, the centralizer is $(SU(2)\times U(1))/{\mathbb{Z}}_2$. Four gauge bosons remain massless and four acquire a common mass. This differs from Fundamental-Higgs breaking of SU(3) to SU(2).

The neutral vector mass term is $\frac{1}{2}{m}_Z^2{Z}_{\mu }{Z}^{\mu }(1+H/v{)}^2$. Expanding it gives the cubic $HZZ$ and quartic $HHZZ$ interactions in unitary gauge. Differentiating the action with respect to identical fields gives Feynman rules $2i{m}_Z^2{g}_{\mu \nu }/v$ and $2i{m}_Z^2{g}_{\mu \nu }/v^2$. A single $Z$ does not couple to two copies of the real neutral radial Higgs boson through an elementary vertex.

If measured far-field patterns determine a density $g$ and its Herglotz wave function equals a prescribed spherical wave on the obstacle boundary, that trace identity can locate the boundary. A star-shaped axisymmetric obstacle can be represented by one radial function $h(\theta )$. Numerically, expand $g$ and $h$ in suitable bases, match the far-field constraints, and minimize the complex boundary residual at angular collocation points, using regularization of an inverse problem for noise and small singular values. A single scalar pairing constraint alone does not determine an arbitrary density.

A theory in a language with constants has the Henkin witness property when every existential sentence $\exists x\varphi (x)$ belonging to it has an instance $\varphi (c)$ belonging to it for some constant $c$. This property proves the existential step of the truth lemma for a Henkin term model.

For a complete consistent theory with the Henkin witness property, the term model has closed terms modulo provable equality as elements. Functions are evaluated by forming terms, and atomic relations hold exactly when the corresponding atomic sentences belong to the theory. Provable equality makes these interpretations well defined.

When head translation is weakly resisted compared with the flagellum but head rotation is strongly resisted, $A_0\ll A$ and $D_0\gg D$. Then $U\simeq -B\omega /A=\omega b\sin \varphi \cos \varphi /(1+{\cos }^2\varphi )$. Maximizing this expression gives the displayed pitch and speed for the chosen handedness and positive $\omega$.

The Heegaard surface is the common positive boundary of the two pieces in a Heegaard splitting.

A compression body is obtained from a product $F\times I$ by attaching one-handles to $F\times \{1\}$, allowing three-ball components as well. Its negative boundary is the retained copy of $F$ and its positive boundary is the other boundary component. A handlebody has empty negative boundary.

If increasing events $A_1,\dots ,A_k$ have a common probability, applying the Harris-FKG inequality to their decreasing complements gives $\mathbb{P}(A_1)\ge 1-(1-\mathbb{P}({\bigcup }_i{A}_i){)}^{1/k}$.

The two scalar coefficients in the unpolarized electromagnetic hadronic tensor. Extra polarization or parity-violating currents can require further coefficients.

For an initially bare cone with accumulation $a\simeq A$ near its apex, the early outer thickness is $At$. The radial gravity flux becomes comparable to accumulation over distance $x_a\sim D{A}^2{t}^3$. Setting $h=AtF(\eta )$ and $\eta =x/(D{A}^2{t}^3)$ givesRegular apex flux requires $F\sim (\eta /2{)}^{1/3}$ near zero. The boundary layer therefore approaches the local steady thickness $(Ax/(2D){)}^{1/3}$ while farther regions are still filling.

After ice accumulation and ice ablation cease, the similarity solution conserving $M=\int xhdx$ isIt is dry outside this interval. At its terminus the thickness $h_N$ is nonzero: the Rankine-Hugoniot condition gives ${\dot{x}}_N=q_N/h_N=D{h}_N^2=x_N/(5\tau )$. This is an entropy solution of the gravity-only kinematic wave model and a long-time approximation for general finite-volume initial profiles. Local pressure-gradient effects smooth the idealized discontinuous front.

For ice accumulation $a=A(1-x/\ell )$, regular zero flux at the apex givesThe steady zero-thickness terminus is at $3\ell /2$, below the snowline $x=\ell$. Accumulation above the snowline exactly balances ice ablation below it.

Left differentiation with respect to an odd Grassmann variable is the odd linear operation ${\partial }_{\theta }^L\theta =1$, with zero derivative on every other generator. For a homogeneous element $a$ of parity $|a|$, it obeys the graded product ruleFor independent odd generators, ${\partial }_{\bar{\theta }}^L(\theta \bar{\theta })=-\theta$. This sign comes from moving the derivative through the first odd factor in the Grassmann algebra. It fixes chirality in the chiral-superfield component expansion; an ordinary commuting-variable product rule would give the wrong sign.

The bracket on a graded Lie algebra is a bilinear operation satisfying graded antisymmetry and the graded Jacobi identity. For homogeneous elements of degrees $r,s$, $[u,v]=-(-1{)}^{rs}[v,u]$. A Gerstenhaber bracket uses this convention on the shifted degrees $r=|u|-1$ and $s=|v|-1$.

On a graded algebra, a left derivation of degree $r$ satisfies the displayed rule for homogeneous $a$. A right derivation of degree $r$ satisfies $R(ab)=aR(b)+(-1{)}^{r|b|}R(a)b$. For odd $r$, these reduce to the familiar parity sign rules. The left Gerstenhaber bracket makes $[u,-]$ a left derivation of degree $|u|-1$ on Hochschild cohomology; this rule holds for the induced cohomology operations, not necessarily for their cochain representatives.

A Gerstenhaber algebra has an associative product making it a graded commutative algebra and a degree-minus-one graded Lie bracket. In the left convention, $[u,vw]=[u,v]w+(-1{)}^{(|u|-1)|v|}v[u,w]$. The shifted degrees $|u|-1$ determine the signs of antisymmetry and the Jacobi identity. Hochschild cohomology and the exterior algebra of polynomial derivations are examples.

An element of a single graded component $A_n$ of a graded algebra is homogeneous of degree $n$. The degree sign rules for an exterior product or a graded Lie bracket apply first to homogeneous elements, then extend by linearity.

A graded algebra is a graded commutative algebra if $uv=(-1{)}^{pq}vu$ for homogeneous elements of a graded algebra of degrees $p,q$. In characteristic not equal to two, every odd-degree element then squares to zero. The exterior algebra and Hochschild cohomology with the Hochschild cup product are examples.

For a finite list $(a_0,\dots ,a_{r-1})$ of nonnegative integers, encode its length with the exponent of two and its entries with the following prime numbers: $2^r{\prod }_{j<r}{p}_{j+2}^{a_j+1}$, where $p_i$ is the $i$th prime number. Unique prime factorization recovers both the length and entries. Requiring positive exponents on exactly the prescribed consecutive primes also makes validity decidable. Nesting list descriptors therefore gives a Gödel encoding of deterministic finite automata and other finite syntactic objects.