A gapless mode associated with a canonically paired pair of broken symmetry generators. In an isotropic ferromagnet, the nonzero expectation of $[S^x,S^y]=i{S}^z$ makes two transverse coordinates one dynamical pair, giving one quadratic magnon branch. Broken-generator counting without this pairing would incorrectly predict two independent modes. This contrasts with the linear phase mode of a superfluid.

For a resting shallow-water strip $-L<x<0$ adjacent to an impermeable coast, conserved linear potential vorticity gives $a^2{\eta }_{xx}-\eta =-{\eta }_{\mathrm{initial}}$. Decay offshore, matching and conserved volume give $\eta ={\eta }_0{e}^{x/a}\sinh (L/a)$ outside the strip and $\eta ={\eta }_0[1-e^{-L/a}\cosh (x/a)]$ inside it. The adjusted velocity is $v=g{\eta }_x/f$, with zero tangential wall velocity inherited from initial rest and alongshore uniformity.

For the stated critical geometric offspring law, the conditional upper tail is exactly $[n/(n+1){]}^{\lfloor nx\rfloor }$ for $x\ge 0$. Its logarithm tends to $-x$, proving an exponential distribution limit of rate one without a transform continuity theorem. The unconditional mass at zero tends to one, so survival conditioning changes the relevant limiting population scale.

For a geodesic variation of unit timelike geodesics, proper time normalization holds across the entire variation: $g(T,T)=-1$. Metric compatibility and ${\nabla }_{T}S={\nabla }_{S}T$ then give $dg(S,T)/d\tau =\frac{1}{2}S(g(T,T))=0$. Consequently an initially orthogonal deviation vector stays orthogonal. The common normalization is essential: an arbitrary family of affinely parametrized curves with different tangent norms does not imply this conclusion.

In constant sectional curvature $\kappa$, a deviation vector orthogonal to a unit timelike geodesic obeys $D^{2}S=\kappa S$, where $D={\nabla }_T$. Therefore $K=g(DS,DS)-\kappa g(S,S)$ is constant, since its derivative is $2g(DS,D^{2}S)-2\kappa g(S,DS)=0$. Its squared length $f=g(S,S)$ satisfies $\ddot{f}=2K+4\kappa f$. In four spacetime dimensions, $\kappa =R/12$.

A generalized Bell state is one member of a generalized Bell basis. Its reduced density matrices are both $I/n$, so it is a maximally entangled state. For $n=2$ these are the four Bell states.

The Bell basis is the two-qubit case of the generalized Bell basis. Its four vectors are the Bell states, and a Bell-basis measurement projects onto them.

A monomial matrix has exactly one nonzero entry in each row and column. It factors as an invertible diagonal matrix times a permutation matrix. Such matrices permute the coordinate axes and normalize the group of invertible diagonal matrices; over very small fields that normalizer can be larger if the diagonal subgroup fails to distinguish the axes.

For weight $w(x)=x$ on $[0,1]$, the two-node Gaussian quadrature nodes are roots of $x^2-6x/5+3/10$, the monic quadratic orthogonal to all linear polynomials. Its weights come from integrating the Lagrange interpolation cardinal functions. Orthogonality gives exactness on cubic polynomials and positive weights.

On $[0,T]$ with vanishing endpoint fluctuations, the sine-mode eigenvalues are $(\pi n/T{)}^2-{\omega }^2$. Divide by the free eigenvalues and use the sine infinite product. The ratio determines the harmonic oscillator transition kernel once the free configuration-space path integral is normalized. Zeros correspond to caustics where the fluctuation operator has a zero mode.

For a quadratic potential energy, shifting a path to its classical solution leaves an exactly quadratic fluctuation action with homogeneous Dirichlet boundary conditions. The first variation vanishes by the Euler-Lagrange equation. The Gaussian path integral prefactor is an inverse square root of a regulated functional determinant and does not depend on the endpoints. At conjugate times the ordinary formula must instead be read by distributional continuation.

The Fourier transform of a Gaussian and the convolution theorem give this multiplier with the angular-frequency convention $e^{-ix\cdot \xi }$. In cycles-per-length frequencies $k$, it is $e^{-2{\pi }^2{\sigma }^2|k{|}^2}$. Each coordinate has variance ${\sigma }^2$, and higher frequencies receive stronger attenuation.

For the Maxwell field, the variation $\delta A_{\nu }=\alpha F_{0\nu }=\alpha ({\partial }_0{A}_{\nu }-{\partial }_{\nu }{A}_0)$ is time translation combined with a gauge transformation. With constant $\alpha$, $\delta F_{\mu \nu }=\alpha {\partial }_0{F}_{\mu \nu }$, and the Lagrangian changes by $\alpha {\partial }_0\mathcal{L}$. Its Noether current is directly the time component of the gauge-invariant Maxwell stress-energy tensor, providing the physical energy density without a potential-dependent improvement step.

For vector coupling to a Dirac fermion, write the Hermitian color connection ${\mathcal{A}}_{\mu }=A_{\mu }^a{T}^a$. Charge conjugation reverses the vector current and transposes its color matrix, requiring ${\mathcal{A}}_{\mu }^C=-{\mathcal{A}}_{\mu }^T$. Combining with parity gives ${\mathcal{A}}_{\mu }^{CP}(x)=-P_{\mu }{}^{\nu }{\mathcal{A}}_{\nu }^T(x_P)$, where $P=\mathrm{diag}(1,-1,-1,-1)$. The transpose acts only on color indices; intrinsic fermion phases cancel in the bilinear.

For a Galton-Watson process with one ancestor, the branching extinction probability is the probability that some generation is empty:The equality uses the absorbing state at zero. The complementary branching survival probability $1-q$ is the probability that every generation is nonempty. The Galton-Watson extinction fixed point computes $q$ from the offspring distribution.

For $p_0=p_1=1/4$, $p_2=1/2$, one ancestor has $q=1/2$. Starting instead from $m$ ancestors with independent descendant families givesAll $m$ families must die out. With three ancestors, eventual extinction has probability $1/8$ and survival has probability $7/8$; by generation two, extinction has probability $(11/32{)}^3=1331/32768\approx 0.040619$. Extra ancestors change the extinction probability without changing the offspring distribution. ^[ref]

For finite offspring mean $\mu$, the extinction classification is:

The critical exception is $p_1=1$: every individual has exactly one child, $F(s)=s$, and $Z_n=1$ forever from one ancestor. Thus $\mu =1$ but $q=0$; a blanket claim that every critical Galton-Watson process becomes extinct would be false. These cases concern the Galton-Watson process with its common offspring distribution and independence assumptions. ^[ref]

This probability answers whether any lineage persists, which the offspring mean alone cannot quantify in the supercritical case. A branching process conditioned on extinction uses the same fixed point to reweight reproduction and describe families known eventually to die out.

For a finite Galois module $M$ unramified outside a finite set $S$, define $H_S^1(K,M)$ as the classes in Galois cohomology whose restriction to inertia is zero at every finite prime outside $S$. This means zero as a cohomology class, rather than zero for every possible representative. Passing to a finite Galois extension on which $M$ is constant reduces the finiteness problem to characters with bounded ramification and a finite restriction kernel.

For a Finite Galois extension $L/K$ and a finite-dimensional $L$-space $V$ with a semilinear action of its Galois group, the map $L{\otimes }_K{V}^G\to V$ is an isomorphism. The vectors ${\sum }_{\sigma }\sigma (a)\sigma (v)$ are invariant and span $V$ by the Artin independence theorem. Choosing an invariant $L$-basis then identifies the invariant space with its $K$-span. The proof does not divide by the extension degree.

The mass-to-light ratio compares the mass assigned to an astronomical system with its luminosity in a specified band, often in solar units. A stellar value uses only stellar mass and depends on the stellar population, age, composition and band. A dynamical value can include gas and dark matter, so it need not equal the stellar value. In a halo-to-galaxy mapping, $L=M_{\ast }/{{\rm Y}}_{\ast }$ requires the stellar ratio and not the total dark matter halo ratio.

Every finitely presented group is the fundamental group of a closed connected smooth manifold in dimension at least four. Build a compact zero/one/two-handle manifold $W$ with the generators and relators of the presentation. Disjoint attaching loops can be embedded in its boundary of dimension at least three. The Seifert-van Kampen theorem gives ${\pi }_1(W)=T$, and the dual relative handle structure makes ${\pi }_1(\partial W)\to T$ surjective. Doubling $W$ along its boundary gives the amalgam of two copies of $T$ along that same surjection, which is again $T$. This construction underlies binary families of nonhomeomorphic Sunada quotients.

A quadratic functional on a real vector space combines a quadratic form $B(v,v)$, a linear functional $\ell (v)$ and a constant. If $B$ is a symmetric bounded bilinear form, its first variation is $DI(v)[w]=2B(v,w)-2\ell (w)$. For a strictly positive $B$, an existing stationary point is the unique minimizer by the quadratic variational principle for a symmetric positive operator.