The plane containing a central-force Kepler orbit, perpendicular to its conserved specific angular momentum. Its orientation can be specified by an orbital inclination and an orbital node. Perturbing forces can change that orientation.

For an algebraic group action, a group orbit has dimensionwhere $G_x$ is the stabilizer subgroup. The orbit map has fibres that are cosets of $G_x$, so the fiber dimension theorem gives the formula.

For the base change action on quiver representations, the stabilizer subgroup is the automorphism group of a quiver representation. It is a Zariski-open set in the endomorphism ring of that quiver representation, so their dimensions agree. The Ringel form then identifies orbit codimension with the dimension of the self-extension group. This open-unit-group argument applies to this representation-classification setting; a general vector stabilizer subgroup need not have that form.

The general linear group over a finite field acting diagonally on projective triples has five orbits of a group action: all equal, three different positions for the repeated coordinate when exactly two are equal, and all distinct. The corresponding representative stabilizer subgroups are upper triangular, diagonal, and scalar matrices. Their orders are $p(p-1{)}^2$, $(p-1{)}^2$, and $p-1$, respectively; the orbit sizes are $p+1$, $p(p+1)$, and $p(p+1)(p-1)$. Transitivity follows from the action being sharply three-transitive on a projective line.

A natural transformation $\tau :F\Rightarrow G$ satisfying $G_2{\tau }_{X\otimes Y}=({\tau }_X\otimes {\tau }_Y)F_2$ and $G_0{\tau }_I=F_0$.

The Euclidean logarithmic norm is the largest eigenvalue of a matrix's Hermitian part. Differentiating $\Vert y{\Vert }_2^2$ along $y^{\prime }=Ay$ gives $\Vert e^{tA}{\Vert }_2\le e^{t{\mu }_2(A)}$. It is the least exponent for such a bound with prefactor one, as a first-order expansion at zero shows. It equals the spectral abscissa for a normal matrix, but need not do so otherwise. A scalar rational stability function does not generally inherit a bound by its value at this real number.

The ordered tree-level four-tachyon open string amplitude is proportional to $\Gamma (-1-{\alpha }^{\prime }s)\Gamma (-1-{\alpha }^{\prime }t)/\Gamma (-2-{\alpha }^{\prime }(s+t))$. Its Mandelstam variables locate the mass levels, and pole residues reveal spin content.

Free endpoint variations set the open string flux $e{T}^2{X}_m^{\prime }+u{P}_m$ to zero. In a boundary-adapted temporal gauge, freedom to vary time forces $u=0$ at the end. Nonzero lapse then gives $X^{\prime m}=0$, a Neumann boundary condition.

The Hawaiian earring is the union of a countable family of circles tangent at one point, with their radii tending to zero, endowed with its planar subspace topology. Equivalently it is the one-point compactification of a countable disjoint union of real lines. Every neighbourhood of the common point contains all but finitely many whole circles. This compact topological space is not locally contractible there and has different singular homology from the infinite CW complex wedge of circles.

The temporal-temporal component of the photon stress-energy tensor in a local orthonormal tetrad. For an isotropic one-particle distribution function expressed using comoving energy $ϵ=aE$, it is ${\bar{\rho }}_{\gamma }=4\pi a^{-4}\int dϵ{ϵ}^3\bar{f}(ϵ)$. A thermal photon temperature perturbation gives a first-order directional energy perturbation four times its fractional temperature change. Boundary conditions on the distribution justify this factor through integration by parts.

Compare the same total number of finite-box modes in the soliton and vacuum sectors before removing the ultraviolet cutoff. A localized bound state replaces a continuum mode, and its contribution must be retained. For the Sine-Gordon kink fluctuation operator and the odd phase branch $\delta (k)=2\arctan (1/k)$, the bound translation mode replaces the free $k=0$ oscillator. Thus the complete oscillator correction is $-m/2-(m/2)\int dk\delta (k)k/[2\pi \sqrt{k^2+1}]$ in this convention. The continuum integral alone misses the finite $-m/2$ contribution.

Expanding an action about a classical stationary field and retaining its second variation gives a Gaussian path integral. Its nonzero fluctuation modes determine a functional determinant and a one-loop soliton mass correction. Symmetry-generated zero modes in field theory must be removed from that determinant and integrated as collective coordinates.

For a nonzero positive-weight eigenform at a good prime, the two oldforms $f(\tau ),f(p\tau )$ are independent. At level multiplied by $p$, the Hecke operator is the bad-prime operator, with matrix $\left(\begin{array}{cc}\lambda & 1\\ -\chi (p)p^{k-1}& 0\end{array}\right)$. Distinct roots $\alpha ,\beta$ of its characteristic polynomial give eigenforms $f-\beta f(p\tau )$ and $f-\alpha f(p\tau )$. Weight-zero constants are an exception to the independence assertion.

A polarization obeys $p\cdot ϵ=0$ and $p^2=(a-1)/{\alpha }^{\prime }$. If $a>1$, spacelike momentum allows a negative-norm timelike polarization. Thus ghost absence requires $a\le 1$. At $a=1$ the null string state identification $ϵ\sim ϵ+\lambda p$ leaves $D-2$ polarizations.

At $a=1$, let $k=\sqrt{2{\alpha }^{\prime }}p$, $k^2=-2$. The physical scalar $[{\alpha }_{-1}\cdot {\alpha }_{-1}+(D+4)(k\cdot {\alpha }_{-1}{)}^2/10+(D-1)k\cdot {\alpha }_{-2}/5]|p⟩$ has norm $2(D-1)(26-D)/25$. It is positive below 26, null at 26, and a string ghost state above 26. Matching the ordinary transverse spectrum removes this extra scalar at 26.

For $f:(0,\infty )\to \mathbb{R}$, its odd extension to the real line isThe zero value is forced by oddness, whether or not the extension is continuous. If $f$ has a finite right limit at zero, the extension is continuous there exactly when that limit is zero. Thus extending $f(x)=1$ gives a jump, while extending $f(x)=x$ gives the smooth function $\tilde{f}(x)=x$.

More generally, suppose $f$ has continuous right derivatives through order $k$ at zero. Its odd extension is $C^k$ exactly when every even-order endpoint derivative through order $k$, including $f(0)$, is zero. Differentiating the reflected formula shows that the left order-$j$ limit is $(-1{)}^{j+1}{f}^{(j)}(0+)$. In the radial reduction of the three-dimensional wave equation, the factor $r\varphi (t,r)$ has a smooth odd extension when $\varphi$ is a smooth radial field at the origin.

For $[a,a^{†}]=1$, the occupation number operator $a^{†}a$ has eigenvalues $0,1,2,\dots$ on the bosonic oscillator Fock space.

For $\{f,f^{†}{\}}_{+}=1$ and $f^2=0$, the occupation number operator $f^{†}f$ has eigenvalues zero and one. This gives the single-mode exclusion principle.

For a normalized state with finite second moment of a quantum observable $Q$, its uncertainty is the standard deviation of the measurement outcome: $(\Delta Q{)}^2=⟨Q^2⟩-⟨Q{⟩}^2$. This equals $\Vert (Q-⟨Q⟩)\psi {\Vert }^2$. The Robertson uncertainty principle bounds products of uncertainties using the expectation of the commutator.

The two-vector tensor product splits into a trace singlet, symmetric traceless tensors and antisymmetric tensors. For an O(N)-invariant S-matrix, the corresponding eigenvalues are $s_0=N{S}_1+S_2+S_3$, $s_{+}=S_2+S_3$ and $s_{-}=S_2-S_3$. For $N\ge 2$, analytic unitarity requires $s_a(\theta )s_a(-\theta )=1$. With Hermitian analyticity of a two-particle S-matrix, each channel has unit modulus on the real rapidity axis.

For bounded self-adjoint $T$, put $B=\beta I-T\ge 0$ with $\beta =\sup W(T)$. Unit vectors with $⟨B{u}_n,u_n⟩\to 0$ satisfy $\Vert B{u}_n{\Vert }^2\le \Vert B\Vert ⟨B{u}_n,u_n⟩$ by positive-form Cauchy-Schwarz inequality. Thus $\beta$ is an approximate eigenvalue. Applying the same argument to $T-\alpha I$ proves the result at $\alpha =\inf W(T)$.