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The cylindrical KdV equation in this normalization is $q_t+q_{xxx}+q{q}_x+q/(3t)=0$, for $t\ne 0$. The additional time-dependent term distinguishes it from the ordinary KdV equation. A scaling-invariant solution has $q=t^{-2/3}F(x{t}^{-1/3})$. Substitution gives $F^{(3)}+F{F}^{\prime }-(\xi F{)}^{\prime }/3=0$. Integration reduces the similarity problem to $F^{\prime \prime }+F^2/2-\xi F/3=C$. The integration constant depends on supplementary conditions and cannot be set to zero without them. Use real cube roots separately on either sign of $t$.

One normalization of the first Hamiltonian structure uses the constant differential Poisson operator $J=-2{\partial }_x$. For functionals with variational derivatives $f,g$, its Poisson bracket is $\int fJgdx=-2\int f{g}^{\prime }dx$ under periodic or vanishing-boundary conventions. Thus the field bracket is $\{u(x),u(y)\}=-2{\partial }_x\delta (x-y)$. The factor and sign change with normalization; the constant first-order operator is the invariant structural feature.

Dehn surgery removes tubular neighborhoods of a link in a three-dimensional manifold and performs Dehn filling on their boundary tori with specified primitive meridian slopes. Dehn surgery on a framed link is the integral-slope special case.

For otherwise outbred opposite-sex first cousins, the probability that the mother's randomly transmitted X copy matches the father's X by descent depends on which of their parents are siblings. It is zero when the male cousin is related through his father, $1/8$ when the female cousin is related through her father and the male through his mother, and $3/16$ when their mothers are sisters. Daughters can be X-autozygous; sons have one X and a Y and are not homozygous for a paired X chromosome. Recombination and pseudoautosomal regions require a locus-based formulation instead of whole-chromosome blocks.

For $N$ individuals in monogamous families of exactly two offspring, let ${\Phi }_G$ be mean pairwise kinship coefficient, $F_G$ mean inbreeding and ${\Phi }_{m,G}$ mean kinship of mates. A random distinct offspring pair are siblings with probability $1/(N-1)$. Accounting separately for within-family and cross-family pairs gives the displayed recurrence, while $F_{G+1}={\Phi }_{m,G}$. Random mating makes ${\Phi }_{m,G}={\Phi }_G$. Sibling avoidance changes the mating-pair mean; it does not create unrelated founders or stop eventual drift to ancestral identity in a closed finite biological population.

The Vlasov equation describes collisionless evolution of a phase-space distribution function in a self-consistent force field: ${\partial }_{t}f+\mathbf{v}\cdot {\nabla }_{x}f+\mathbf{F}\cdot {\nabla }_{p}f=0$. The force can be electromagnetic or gravitational; the Collisionless dark-matter Vlasov equation is the cosmological gravitational specialization.

Thermal transpiration is gas transport driven by an imposed temperature difference in a rarefied connection. Gas can move from cold to hot even when the two end pressures start equal. If the reservoirs are closed, redistribution builds a counteracting pressure difference until the net transfer stops. This final stationary state is not global thermal equilibrium: the two temperatures remain different. It explains why a heated pressure sensor can read a different pressure from the vessel to which it is connected. The correction depends on gas, connection geometry and molecular interaction with its surfaces. ^[ref]

The simple molecular limit is beautiful: equal opposing particle rates need not mean equal pressures. Consider effusion through a thin hole connecting two locally equilibrated reservoirs of the same gas, at $T_h>T_c$. Equating their one-way rates givesAt equal pressure, the hotter reservoir has lower number density; its faster molecules do not fully compensate for that reduction in outgoing particle rate. With $T_c=300\mathrm{K}$ and $T_h=351\mathrm{K}$, the ideal zero-transfer ratio is $\sqrt{351/300}\simeq 1.0817$. The hydrodynamic, zero-flow limit instead has equal pressures when other forces are absent. These are limiting models, not an interpolation formula for a glass capillary. ^[ref] In a real tube, incomplete exchange of molecular energy with the wall can alter the limiting ratio; a Knudsen number does not specify that exchange. ^[ref]

Effusion is molecular passage through a small opening whose dimensions, including its thickness, are much smaller than the intermolecular mean free path. For an equilibrium ideal gas, the one-way particle rate through a thin hole of area $A$ is $\dot{N}=Ap/\sqrt{2\pi m{k}_{B}T}$. This weights molecules by their wall-normal velocity, rather than just counting all molecules in the reservoir. ^[ref]

For a Killing vector field $K$, its metric-dual differential one-form produces the electromagnetic field $F_{ab}=2{\nabla }_{[a}{K}_{b]}=2{\nabla }_a{K}_b$. The exterior derivative gives $dF=0$, while the second covariant derivative of a Killing vector gives ${\nabla }^a{F}_{ab}=-2R_{bd}{K}^d$. Thus this construction solves the source-free Maxwell equations in a vacuum spacetime with zero cosmological constant. The field is a test field: adding its stress-energy tensor would generally change the original metric tensor.

With metric signature $(+---)$, normalize a timelike Killing vector field by $U^a=K^a/N$, $N^2=K^a{K}_a$. The Killing equation gives $K^b{\nabla }_b{K}_a=-\frac{1}{2}{\nabla }_a{N}^2$, while $K^b{\nabla }_{b}N=0$. Thus the four-acceleration of its normalized flow is the displayed gradient. With metric signature $(-+++)$ the corresponding lower-index formula has the opposite sign.

On a connected Riemannian manifold, these data determine a Killing field uniquely, without a completeness assumption. The second covariant derivative of a Killing vector makes its restriction to every geodesic a Jacobi field. The difference of two fields with identical data has zero Jacobi initial values and vanishes in a convex normal neighborhood. The set on which both its value and covariant derivative vanish is closed by continuity and open by the same local argument, so connectedness makes it the whole manifold.

The Killing equation and the second covariant derivative of a Killing vector transport a Killing field and its antisymmetric derivative along any curve. Here $L_{ab}={\nabla }_b{\xi }_a$ and the curvature convention is the one in curvature-sign convention in Killing derivative identities. In a parallel frame the equations form a homogeneous linear system of ordinary differential equations. Initial value zero has unique zero continuation. Thus on a connected pseudo-Riemannian manifold, not just on a Riemannian manifold, the field is determined by its value and first derivative at one point.

On a geodesically complete Riemannian manifold, every Killing vector field is complete and integrates to a global one-parameter subgroup of its isometry group. On the incomplete interval $(0,1)$ with its Euclidean metric, ${\partial }_x$ satisfies the Killing equation, but translations leave the interval in finite time. The full isometry group is only the identity and reflection. Local solutions of the Killing equation therefore need not equal the Lie algebra of global isometries without completeness.

On a closed Riemannian manifold, a Killing vector field $X$ satisfies ${\nabla }^{\ast }\nabla X={\mathrm{Ric}}^{♯}X$ for the nonnegative rough Laplacian. This follows by differentiating ${\nabla }_i{X}_j+{\nabla }_j{X}_i=0$, contracting and commuting covariant derivatives; the contraction also gives $\mathrm{div}X=0$. Taking the inner product with $X$ and using integration by parts gives the displayed identity. If Ricci curvature is nonpositive, the nonnegative left side must vanish, so $X$ is a parallel vector field. In particular every Killing vector field on a closed flat Riemannian manifold is a parallel vector field.

Suppose the full Lie algebra of Killing fields has basis $T,\Phi$ and the flow of $\Phi$ has a nonzero period $P$. Write $[T,\Phi ]=aT+b\Phi$ with constant coefficients. The Adjoint representation of the periodic flow satisfies $e^{P\mathrm{ad}\Phi }=1$. Its real eigenvalues are $-a$ and zero, so $a=0$. The remaining adjoint operator is nilpotent and its exponential is $1+P\mathrm{ad}\Phi$, forcing $b=0$. This proves commutation from genuine periodic axial symmetry, without differentiating mere asymptotic component limits.

A gravitational magnetic monopole for a compact circle, whose fibre is nontrivially twisted over a surrounding two-sphere. With flat spectator directions its nonsingular local geometry is a Taub–NUT space. Reducing along the M-circle yields the D6-brane. The magnetic dual of the IIB circle vector is instead a five-brane in nine noncompact dimensions.

// Target: geometry-and-topology.bigb

A real five-dimensional Maxwell field on a circle of radius $r$ has a Fourier series with coefficients $A_M^{(n)}$, $A_M^{(-n)}=(A_M^{(n)}{)}^{\ast }$. Its mixed field-strength component is $F_{\mu 5}^{(n)}={\partial }_{\mu }{A}_5^{(n)}-in{A}_{\mu }^{(n)}/r$. The nonzero modes transform as $\delta A_{\mu }^{(n)}={\partial }_{\mu }{\Lambda }^{(n)}$, $\delta A_5^{(n)}=in{\Lambda }^{(n)}/r$. Thus $A_5^{(n)}$ is a Stueckelberg mechanism field and the gauge-invariant massive vector is $A_{\mu }^{(n)}-r{\partial }_{\mu }{A}_5^{(n)}/(in)$. Its mass is $|n|/r$ and it has three physical polarizations. The zero modes instead consist of a massless four-dimensional Maxwell field and a real massless scalar field; the latter cannot be removed by a periodic infinitesimal gauge transformation. A positive Fourier level and its negative level are conjugate, not independent real fields.

For a massless field on Minkowski space times a circle of circumference $R$, the compact delta distribution expands as $R^{-1}{\sum }_n{e}^{2\pi in\Delta w/R}$. The noncompact retarded Green function in each Fourier sector is that of a four-dimensional field with mass $m_n$. Equivalently, the time-frequency denominator is ${\mathbf{k}}^2+m_n^2-(\omega +i0{)}^2$, putting both poles below the real axis. At equal compact positions the sum has one zero mode and a doubly counted nonzero mass tower.

For the Poisson equation ${\nabla }_4^2\varphi =-4\pi Rq{\delta }^3(\mathbf{x}){\delta }_{2\pi R}(z)$ on ${\mathbb{R}}^3\times S^1$, expand the periodic delta as $(2\pi R{)}^{-1}{\sum }_n{e}^{inz/R}$. The $n$th Kaluza-Klein mode has radial potential $q{e}^{-|n|r/R}/(2\pi r)$, since $({\nabla }_3^2-m^2)e^{-mr}/r=-4\pi {\delta }^3(\mathbf{x})$. Summing the geometric series gives the displayed Green function. At equal compact coordinate it reduces to $q\coth (r/(2R))/(2\pi r)$. Thus the interaction energy changes from $q_1{q}_2/(2\pi r)$ at large separation to $q_1{q}_{2}R/(\pi r^2)$ at small separation, with normalization fixed by the stated Poisson equation.

For a periodic scalar on a circle of circumference $2\pi R$, its Fourier series has modes ${\psi }_n(x)e^{inz/R}$. In the fixed-radius metric $d{s}_5^2={\gamma }_{ij}d{x}^{i}d{x}^j+(dz+A_{i}d{x}^i{)}^2$, the five-dimensional Klein-Gordon equation becomes $(D_i{D}^i-M^2-n^2/R^2){\psi }_n=0$ with $D_i={\nabla }_i-in{A}_i/R$. Thus the geometric gauge charge is quantized in units $1/R$ in units $\hslash =1$, or the conserved compact momentum in units $\hslash /R$. A rescaling to a canonically normalized electromagnetic potential changes the corresponding numerical unit of physical electric charge.