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For a bulk gauge field, the leading boundary value of its time component is the dual chemical potential, while the normalizable radial coefficient determines the charge density.

Fefferman--Graham coordinates put an asymptotically anti-de Sitter metric in the form $d{s}^2=L^2[d{Z}^2/Z^2+g_{ij}(Z,x)d{x}^{i}d{x}^j/Z^2]$.

The Baumslag-Solitar group is

Suppose an urn starts with $m$ green balls and $m+2$ red balls. Drawing green removes one green ball, while drawing red adds one green and one red ball. The difference remains two, and $h(g)=1/(g+1)$ is harmonic for the green-ball chain. Stopping $h$ on hitting $0$ or $N$ and then letting $N$ tend to infinity gives termination probability

A bounded operator $T$ on a separable Hilbert space is trace class when $|T{|}^{1/2}$ is Hilbert--Schmidt. Its trace norm is

The Higgs boson is the neutral radial excitation of the Higgs field about its vacuum expectation value.

In the single-doublet Standard Model this physical excitation is a spin zero excitation of a scalar field. Its mass depends on the Higgs potential, so the theory's gauge symmetry alone does not specify where to search. The Higgs boson discovery in 2012 tested the existence of a new particle resonance; establishing its detailed identity requires testing its interactions and spin as well.

In the Higgs mechanism, gauge fields associated with broken generators acquire longitudinal polarizations and masses by absorbing Goldstone modes. Gauge fields of the unbroken subgroup remain massless.

The phrase “absorbing” describes a rearrangement of degrees of freedom, not a collision in which a gauge boson swallows another particle. The gauge symmetry remains a redundancy of the description. A chosen vacuum expectation value leaves a smaller subgroup manifest, while gauge-invariant observables describe the physical masses.

A one-field calculation isolates the mechanism. In an Abelian Higgs model, write a complex scalar field locally as $\Phi =(v+H)e^{i\theta }/\sqrt{2}$, use $D_{\mu }={\partial }_{\mu }-ie{A}_{\mu }$, and define $B_{\mu }=A_{\mu }-{\partial }_{\mu }\theta /e$. ThenThe second term contains a Proca field mass $m_B=ev$ and its interactions with the radial Higgs boson. A local gauge transformation removes $\theta$ from the scalar field but leaves its information in the longitudinal vector polarization. The electroweak symmetry breaking calculation applies this idea to three broken directions.

If a well-order is the union of suborders of types $\xi$ and $\eta$, its type is at most $\xi \#\eta$.

For a basis $(v_i)$, the matrix $A$ of a Hermitian form has entries $A_{ij}=H(v_i,v_j)$. With the convention that $H$ is linear in its first argument, coordinate columns $x,y$ satisfy $H(x,y)=x^{T}A\overline{y}$.

A function $d\in {\omega }^{\omega }$ dominates $h\in {\omega }^{\omega }$ when $d(n)>h(n)$ for all but finitely many $n$. A dominating real over a ground model dominates every such function belonging to that model.

With transverse and axial moments $I,J$, the Hamiltonian is

For $e_k=x^{\ast }-x_k$,If $A$ is diagonal with entries ${\lambda }_i$, a coordinate permutation turns this matrix into blocks

For $F(x)=x^{T}Ax/2-b^{T}x$ and $r_k=b-A{x}_k$,Thus $r_k$ is a polynomial in $A$ applied to the initial residual and lies in the corresponding Krylov subspace.

For $u_t=\kappa u_{xx}$ on the real line,has total mass one and converges to the delta distribution as $t$ decreases to zero. The solution is $u(t)=K_t\ast u_0$.

The nonlinear equationis the potential form of the viscous Burgers equation. The substitution $w=e^u$ converts it to the heat equation $w_t=w_{xx}$.

For $T_t=\kappa T_{xx}$ on $x>0$, with $T(x,0)=0$ and $T(0,t)=\sin \omega t$,Consequently the transform of $I(t)={\int }_0^{\infty }T(x,t)dx$ is

For a fixed-composition closed system, the heat capacity at fixed volume is $C_V=(\partial U/\partial T{)}_V$, with internal energy $U$. Under the quasistatic process assumptions in heat capacity, only pressure-volume work is permitted, so $dV=0$ gives $\delta Q=dU=C_{V}dT$. A rigid vessel therefore needs $Q=\Delta U$, or $Q=C_V\Delta T$ when $C_V$ is constant. Its per-unit-mass counterpart is specific heat capacity at constant volume.

The map $x\mapsto \{x\}$ is an isometry into the hyperspace of nonempty closed bounded subsets. Its image is closed because a Hausdorff limit of singletons has diameter zero and is therefore a singleton.

The Bona--Masso family is $({\partial }_t-{\beta }^i{\partial }_i)\alpha =-{\alpha }^{2}f(\alpha )K$. Harmonic slicing is the choice $f=1$.

Every nonconstant harmonic function $u:\mathbb{C}\to \mathbb{R}$ is unbounded above and below by the harmonic Liouville theorem. The intermediate value theorem then gives $u(\mathbb{C})=\mathbb{R}$.