For the degree-one mod-two generator, ${\mathrm{Sq}}^{1}x=x^2$ and higher squares of $x$ vanish by instability. The Cartan formula gives the displayed binomial identity. Naturality applies it to relative classes of a stunted real projective space.

Collapse the indicated standard skeleton to a point. Besides its basepoint, the quotient has one cell in dimensions $k+1$ through $m$. Its integral cellular boundary is two in even dimensions and zero in odd dimensions. Steenrod operations on its relative cohomology are inherited from real projective space.

This is the CW complex obtained as the increasing union of finite Real projective spaces under their standard inclusions. It has one cell in each nonnegative dimension. Its cellular chain complex has boundary multiplication by two in positive even dimensions and zero in odd dimensions. Consequently its integral cohomology is $\mathbb{Z}$ in degree zero, $\mathbb{Z}/2$ in positive even degrees and zero in odd degrees. Its mod-two cohomology is ${\mathbb{Z}}_2$ in every nonnegative degree.

For inviscid plane Couette flow $U=z$ between impermeable walls at $z=\pm 1$, the nonzero real streamwise wavenumber $\alpha$ gives no nontrivial smooth discrete vertical-velocity modes. Instead, the vorticity variable $q=(D^2-{\alpha }^2)w$ obeys $({\partial }_t+i\alpha z)q=0$. Localized vorticity sheets give generalized eigenfunctions with real frequencies ${\omega }_{\xi }=\alpha \xi$. Their closure forms a neutral continuous spectrum.

A random variable is discrete when some finite or countable set $D\subseteq \mathbb{R}$ has $\mathbb{P}(X\in D)=1$. Its probability mass function is $p_X(x)=\mathbb{P}(X=x)$; these nonnegative masses sum to one. An event determined by the value has probability $\mathbb{P}(X\in A)={\sum }_{x\in A\cap D}{p}_X(x)$. Values outside $D$ may still occur on a zero-probability event without changing this definition.

Countable support need not consist of integers: a random variable constantly equal to $3/2$ is discrete. Integer-indexed tail-sum formulas therefore require a separate integer-valued hypothesis. For the dice sum in random variable, the expected value is ${\sum }_{s=2}^{12}s{p}_S(s)=7$ and the variance is ${\sum }_{s=2}^{12}(s-7{)}^2{p}_S(s)=35/6$; the same answer follows by adding the variances of its independent summands.

For finitely coloured $(r+1)$-element subsets, choose successive least points and use the inductive $r$-set theorem to thin the remaining reservoir after each choice. Every tuple with that least point then has one assigned colour. The infinite pigeonhole principle retains infinitely many least points with the same assigned colour, giving an infinite homogeneous set. The $r=1$ base case is the same pigeonhole principle. This proves the Ramsey theorem for r-sets for all finite $r$.

Take increasing even $u_i$ for which every ordered four-index expression $u_i+u_j+2u_k+2u_l$ has one colour. The Ramsey theorem for r-sets supplies them. Set $x_i=u_{2i-1}+u_{2i}$ and $y_i=(u_1+u_2)/2+2u_{i+2}$. Then both $x_i+2x_j$ and $y_i+y_j$ use the same ordered coefficient pattern $(1,1,2,2)$, so their union is monochromatic. The shared half-prefix makes the unweighted pair family possible without asking the two families to use the same generating sequence.

For integer transverse string modes, $N_R={\sum }_{k>0}({\alpha }_{-k}\cdot {\alpha }_k+k{d}_{-k}\cdot d_k)$ with vacuum-annihilating normal ordering. Each nonzero-mode creation operator raises the level by $k$; fermionic zero modes commute with it. The open-string mass condition is $M^2=2\pi T{N}_R$, with zero Ramond intercept.

The zero-mode anticommutator $\{d_0^I,d_0^J{\}}_{+}={\delta }^{IJ}$ becomes the Euclidean Clifford algebra after ${\gamma }^I=\sqrt{2}{d}_0^I$. It acts within the oscillator-vacuum space. Eight transverse Gamma matrices give a minimal complex representation of dimension sixteen, split into two eight-dimensional chirality sectors.

A worldsheet operator that changes fermion boundary conditions to the Ramond sector and creates a target spinor state. For $D$ real matter fermions its conformal weight is $D/16$. At $D=10$, combining it with the appropriate superghost factor gives a dimension-one spacetime-supersymmetry current.

The integral of a source over a directed line, weighted by exponential absorption between each source point and a detector. For line coordinates $x=p\mathbf{n}+s\mathbf{d}$ and a detector at the negative end, it is $\int f(p,s)\exp [-{\int }_{-\infty }^s\mu (p,r)dr]ds$. The integrating factor of a transport equation derives the weight. Reversing the detector direction changes the weight for spatially varying attenuation, so the two directed measurements need not coincide. Setting $\mu =0$ recovers the Radon transform.

The radiative transfer source function is emission per unit extinction. If $j_{\nu }$ is emission per unit path length and $\rho {\kappa }_{\nu }$ is extinction per unit path length, $S_{\nu }=j_{\nu }/(\rho {\kappa }_{\nu })$ and the transfer equation is $\mu {\partial }_{{\tau }_{\nu }}{I}_{\nu }=I_{\nu }-S_{\nu }$. Under thermal absorption in local thermodynamic equilibrium it is the Planck function; for conservative coherent isotropic scattering it is the mean intensity.

Combining $F=c{\Omega }^{2}z/\kappa$ with viscous heating $dF/dz=(9/4)\mu {\Omega }^2$ gives a height-independent dynamic viscosity $\mu =4c/(9\kappa )$ under constant opacity. This exact limiting relation differs from an approximate vertically averaged alpha disk closure.

Taking $\bar{E}=GMr/q^2>0$ in a radial orbit time transformation gives $r=(\bar{E}/GM)q^2$ and $\bar{\Phi }(q)=-E(\bar{E}/GM{)}^2{q}^2$. Bound Kepler orbits with $E<0$ map to a confining harmonic oscillator; positive energy maps to an inverted oscillator. Full angular reconstruction halves the original orbital angle.

With $a=(1-\lambda )GM$, $h=\lambda E>0$ and $\bar{E}>0$, set $\bar{E}{q}^2=h{r}^2+ar$. If $a\ge 0$, its positive-radius root is $r=\sqrt{\bar{E}/h}(\sqrt{q^2+b^2}-b)$, with $b=a/(2\sqrt{h\bar{E}})$. The transformed Newtonian gravitational potential is an additive constant plus $-G\bar{M}/(b+\sqrt{q^2+b^2})$, the spherical isochrone model. The mass sign and branch conditions must be checked.

A transformed radial equation containing $L^2/(2q^2)$ becomes a full planar central-force orbit by choosing $d\bar{\theta }/d\tau =L/q^2$. For $d\tau /dt=(q/r)dq/dr$ and old angular velocity $L/r^2$, this gives $d\bar{\theta }/d\theta =(r/q)dq/dr$. Keeping the old angle generally fails to preserve the stated transformed angular momentum.

An outgoing radial null geodesic of the Vaidya metric prescribed as $r=r_0(v)$ determines the mass along that curve by $M(v)=r_0(v)(1-2r_0^{\prime }(v))/2$. This inverse relation comes from the radial null condition. It determines a function of advanced time wherever the prescribed curve is differentiable and has positive radius.

For $\Phi =-GM/r+\alpha {\omega }^2{r}^2/2$, nearly circular motion has ${\Omega }^2=GM/r^3+\alpha {\omega }^2$ and ${\kappa }^2=GM/r^3+4\alpha {\omega }^2$. Their generally nonrational ratio makes small-eccentricity orbits precess even if $\alpha$ was added slowly. This illustrates adiabatic actions do not preserve orbital closure.

For a nonnegative, radially nonincreasing spherical mass density, ${\Omega }^2=GM(r)/r^3$ and ${\kappa }^2={\Omega }^2+4\pi G\rho (r)$. Since $\rho (r)$ does not exceed the mean enclosed density, the radial epicyclic frequency lies between $\Omega$ and $2\Omega$. General axisymmetric disks need not satisfy this bound.

A quotient topological space is a quotient set equipped with the quotient topology. For a concrete gluing, give the closed interval $I=[0,1]$ its usual subspace topology, identify $0\sim 1$, and leave every interior point in its own singleton equivalence class. The resulting quotient space is a circle: the two endpoints become one point, while interior points remain distinct.

The exact continuous map to the unit circle in the plane isHere sine and cosine give $p(0)=p(1)=(1,0)$, and $p(s)=p(t)$ holds exactly when $s=t$ or $\{s,t\}=\{0,1\}$. Thus $p$ induces a bijection $\bar{p}:I/\sim \to S^1$. ^[ref] The universal property of the quotient topology makes $\bar{p}$ continuous. The quotient of a compact space is compact, and the unit circle is Hausdorff, so the compact-to-Hausdorff continuous bijection theorem makes $\bar{p}$ a homeomorphism. This verifies the topology, not merely the point identifications.

To inspect an open neighbourhood of the seam, choose $0<\epsilon <1/2$ and let $U_{\epsilon }=p([0,\epsilon )\cup (1-\epsilon ,1])$. Its full preimage isBoth pieces are open in the subspace topology on $I$, so $U_{\epsilon }$ is open in the circle. In contrast, $A=[0,\epsilon )$ is open in $I$, butis not open in $I$: it contains $1$ without any neighbourhood of $1$. Hence $p(A)$ is not open in the circle. A quotient map can therefore fail to be an open map.

For a continuous function $f:I\to \mathbb{R}$, the only gluing condition is $f(0)=f(1)$. For example, $f(t)=\cos (2\pi t)$ defines a continuous function on the circle, whereas $f(t)=t$ does not. This lets us use an interval as a coordinate for a periodic function while making the seam invisible. More general quotient spaces encode glued geometric pieces in the same way: describe continuous maps before gluing and check that their values agree on every equivalence class.

The Hausdorff property need not survive a quotient topology. In the non-Hausdorff quotient of the real line by rational translation, $x\sim y$ means $x-y\in \mathbb{Q}$. A nonempty open preimage $W$ must contain all translates by rational numbers of any open interval inside it. The density of the rational numbers then forces $W=\mathbb{R}$: for any $y$, some rational number $r$ has $y-r$ in that interval. Thus this quotient space has the indiscrete topology. The irrationality of the square root of two gives distinct equivalence classes for $0$ and $\sqrt{2}$, so this quotient space is not Hausdorff. Unlike the endpoint-gluing circle, this gluing identifies a dense subset with each point.