Fix an $R$-bimodule $M$. A unital ground-ring-linear split extension with square-zero kernel $M$ can be identified with $R\oplus M$, with multiplication $(a,u)(b,v)=(ab,av+ub+f(a,b))$. Associativity is exactly $\delta f=0$. Changing a unital linear section by $g:R\to M$ changes $f$ by $\delta g$. Thus equivalence classes of these split extensions are $H{H}^2(R,M)$. Over a field every such vector-space extension splits; over a general ground ring a splitting hypothesis is essential for this ordinary cochain classification.

Over a field, the bimodule resolution $0\to k[x]\otimes k[x]\stackrel{x\otimes 1-1\otimes x}{\to }k[x]\otimes k[x]\to k[x]\to 0$ is exact: multiplication identifies the quotient by $x\otimes 1-1\otimes x$, and that element is a non-zero-divisor in the two-variable polynomial ring. Applying the bimodule Hom functor gives zero differential, proving the displayed Hochschild cohomology. Degree-one classes are the derivations $f(x){\partial }_x$.

The ordinary Hochschild cochain complex is defined using a chosen commutative ground ring $k$. The bar complex is exact via insertion of $1$, but its terms are projective over the enveloping algebra when $R$ is projective over $k$, in particular when $k$ is a field. Without that hypothesis the ordinary cochain groups compute relative Hochschild cohomology and cannot automatically be identified with absolute Ext. Likewise, ordinary second cohomology classifies ground-ring-linearly split square-zero extensions, not arbitrary nonsplit ring extensions.

// Destination: module-theory.bigb

For $Y_t=-vt+\sqrt{2D}{B}_t$ with $v,D,\ell >0$ and standard Brownian motion, the one-time occupation Probability is $\frac{1}{2}\mathrm{erfc}[(\ell +vt)/\sqrt{4Dt}]$. It is maximized at $t=\ell /v$, giving $\frac{1}{2}\mathrm{erfc}(\sqrt{v\ell /D})$. The actual hitting probability is $e^{-v\ell /D}$. To derive it, solve $D{h}^{\prime \prime }-v{h}^{\prime }=0$ on $(-A,\ell )$ with $h(-A)=0$, $h(\ell )=1$, obtaining $h(0)=(1-e^{-vA/D})/(e^{v\ell /D}-e^{-vA/D})$, and let $A\to \infty$. A path that reached the level and returned contributes to the latter event but not to a given one-time tail.

Two Hilbert space modules are unitarily equivalent if each embeds isometrically as a closed submodule of the other. For isometric intertwiners $u:H_1\to H_2$ and $v:H_2\to H_1$, let $D=H_1\ominus v{H}_2$ and $K={⨁}_{n\ge 0}(vu{)}^{n}D$. These summands are orthogonal. The operator acting as $u$ on $K$ and as $v^{\ast }$ on $K^{\perp }$ is the required unitary onto $H_2$.

The gauge-covariant kinetic term of the Higgs doublet gives $M_W^2(1+H/v{)}^2{W}_{\mu }^{+}{W}^{-\mu }+M_Z^2(1+H/v{)}^2{Z}_{\mu }{Z}^{\mu }/2$. Differentiation with respect to the external fields yields Feynman rules $2i{M}_V^2{g}_{\mu \nu }/v$ for one Higgs boson and two matching massive electroweak gauge bosons, and $2i{M}_V^2{g}_{\mu \nu }/v^2$ for two Higgs bosons and the same pair. Identical neutral fields supply combinatorial factors. The neutral radial Higgs boson has no corresponding elementary coupling to photons at tree level.

At tree level and energies much greater than a gauge-vector mass, an amplitude with longitudinal external gauge bosons equals the corresponding amplitude with their would-be Goldstone bosons, up to conventional external-state phases and relative $O(m_V/E)$ corrections for a nonzero leading amplitude. In a renormalizable gauge, the broken-symmetry identities relate $k_{\mu }{\mathcal{M}}^{\mu }$ to $m_V{\mathcal{M}}_{\chi }$; inserting ${\epsilon }_L^{\mu }=k^{\mu }/m_V+O(m_V/E)$ explains the result. Loop-level versions include field-normalization factors. This is different from an equivalence theorem for field redefinitions.

In unitary gauge, ${\varphi }^{†}\varphi -v^2/2=vH+H^2/2$, so this Higgs potential gives $M_H^2=2\kappa v^2$, a cubic coefficient $\kappa v$ and a quartic coefficient $\kappa /4$. The corresponding Feynman rules are $-3i{M}_H^2/v$ and $-3i{M}_H^2/v^2$. Writing the potential with an additional factor $1/2$ changes the relation between its named coupling and the Higgs boson mass; it does not change these mass-based rules.

For $V=\frac{1}{2}\lambda ({\varphi }^{†}\varphi -v^2/2{)}^2$ and the canonically normalized radial field, the expansion is $V=\frac{1}{2}\lambda v^2{H}^2+\frac{1}{2}\lambda v{H}^3+\frac{1}{8}\lambda H^4$. Comparing the quadratic term with $m_H^2{H}^2/2$ gives the displayed mass. Replacing the prefactor $\lambda /2$ by $\lambda$ changes the answer to $2\lambda v^2$, so the potential's normalization must be retained.

Higgsstrahlung produces a Higgs boson together with a massive electroweak gauge boson. At an electron-positron collider the process $e^{+}{e}^{-}\to Z^{\ast }\to ZH$ uses the Higgs boson coupling to Z bosons. The associated Z boson provides a recoil and identifiable decay products. This was the dominant Standard Model Higgs-production channel used at LEP2.

In a Heyting algebra, $a\to b$ is the greatest $x$ with $x\wedge a\le b$. Thus $x\le a\to b$ if and only if $x\wedge a\le b$. Every frame has this operation, obtained by taking the join of all such $x$. The open nucleus attached to $a$ is $x\mapsto a\to x$.

For $F\in C^2({\mathbb{R}}^n)$ with bounded Hessian matrix, Taylor's formula along the line from zero to $p$ gives $|F(p)|\le |F(0)|+|DF(0)||p|+\frac{1}{2}\Vert D^{2}F{\Vert }_{\infty }|p{|}^2$. Its gradient obeys $|DF(p)|\le |DF(0)|+\Vert D^{2}F{\Vert }_{\infty }|p|$. Consequently ${\int }_{\Omega }F(Du)$ is finite for $u\in H^1(\Omega )$ when $\Omega$ has finite measure, and $DF(Du)\in L^2$.

A nondegenerate Hermitian form over a quadratic finite field has an orthonormal basis. If all diagonal values vanished, evaluating $B(v+aw,v+aw)$ would give $\mathrm{Tr}(\bar{a}B(v,w))=0$ for every $a$, contradicting quadratic finite-field norm and trace surjectivity and nondegeneracy. Thus some $v$ has nonzero norm; rescaling using the field norm makes it one. Its orthogonal complement for a sesquilinear form is nondegenerate, so induction completes the basis.

In a two-dimensional nondegenerate Hermitian form over a quadratic finite field, choose an orthonormal basis $v,w$ and $a$ with $a\bar{a}=-1$. Then $e=v+aw$ is an isotropic vector. Choose $g$ with $B(e,g)=1$, and choose $c$ with $c+\bar{c}=B(g,g)$. The vector $f=g-ce$ satisfies $B(e,f)=1$ and $B(f,f)=0$. Thus $e,f$ give the displayed matrix, including in characteristic two. Iterating on nondegenerate orthogonal complement for a sesquilinear form leaves only a unit-norm line in odd dimension.

For $|t|<1$, this identity sums the Hermite function expansion into a Gaussian. For $0<t<1$, one proof takes the Bargmann-Fock space inner product of the generating functions ${\pi }^{-1/4}\exp (-x^2/2+\sqrt{2}xz-z^2/2)$ at $\sqrt{t}z$. Writing $z=u+iv$, the two resulting real Gaussian integrals have quadratic coefficients $1+t$ and $1-t$ and linear coefficients $\sqrt{2t}(x+y)$ and $i\sqrt{2t}(x-y)$. Evaluation gives the formula. Setting $t=e^{-2s}$ and multiplying by $e^{-s}$ gives the harmonic oscillator transition kernel of $e^{-s(-d^2/d{x}^2+x^2)}$.

This real-parameter calculation extends to complex $|t|<1$ by analytic continuation, with the square root chosen analytic and positive at $t=0$. Restricting the Gaussian-integral argument first to $0<t<1$ avoids treating its complex coefficients as real ones.

Use the unitary Fourier transform with phase $e^{-i\lambda x}$. It commutes with $-d^2/d{x}^2+x^2$, because differentiation transforms to multiplication by $i\lambda$ and multiplication by $x$ transforms to $id/d\lambda$. The Hermite differential equation has a unique polynomial solution of each degree up to scale, and multiplying by the Gaussian gives a Hermite function. Transforming a polynomial times this Gaussian again gives a polynomial times the Gaussian of the same degree; comparison of leading coefficients gives the eigenvalue $(-i{)}^n$.

In the displayed formula $H_n$ denotes the physicists' Hermite polynomial; the normalized Hermite function is a constant multiple of $H_n(x)e^{-x^2/2}$. Thus the normalized Hermite function has the same Fourier transform eigenvalue. This distinction prevents confusing the polynomial notation with the normalized function notation used in the parent article.

For a hereditary hypergraph property of $r$-uniform hypergraphs, ${\mathcal{P}}_n$ is the family on the labelled vertex set $[n]$. For $n\ge r$, its normalized speed lies between zero and two and is nonincreasing. Choose a uniformly random member on $n+1$ vertices and record its edge indicators. Each edge occurs in $n+1-r$ vertex-deletion restrictions. The Shearer inequality bounds its information entropy by the sum of the restriction entropies divided by $n+1-r$. Each restriction has at most $|{\mathcal{P}}_n|$ possible values. Division by $\left(\genfrac{}{}{0ex}{}{n+1}{r}\right)$ gives $\log d_{n+1}\le \log d_n$. Empty families are handled directly by heredity.

Every infinite-dimensional Banach space contains either an unconditional basic sequence or a hereditarily indecomposable Banach space. For a space with a basis and no unconditional basic sequence, apply the block Ramsey theorem to finite sequences whose odd and even spans contain nearby unit vectors. A diagonal sequence of successively sharper games gives a subspace in which every pair of infinite-dimensional subspaces has angle zero.

A set has countable-subset closure when every countable subset of it is one of its elements, including the empty subset. In ZFC, the hereditarily countable sets form the least such set. First realize them as the separated subset of $V_{{\omega }_1}$ with countable transitive closure. Countable unions show closure; well-founded induction proves leastness because each of these sets is a countable collection of earlier members.

A countable well-founded relation that is an extensional relation codes a transitive set through the Mostowski collapse theorem. Adding a distinguished point selects one decoded set. Every hereditarily countable set has such a code by enumerating its transitive membership ancestry. There are only continuum many codes. In ZFC, selecting least codes in a well-order gives an injection of the hereditarily countable sets into a set of size $2^{{\aleph }_0}$.