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Let $R,\alpha >0$ and ${\mathcal{C}}_{\alpha }=\{(x,t)\in {\mathbb{C}}^2:|x|<R,|t|<\alpha (1-|x|/R)\}$. The displayed norm makes the holomorphic functions with finite norm a Banach space. Finiteness forces $u(x,0)=0$: for fixed $x$, the weight grows like $1/|t|$ near zero. A Cauchy sequence in this norm converges uniformly on every compact subset of ${\mathcal{C}}_{\alpha }$, including at $t=0$. Its limit is holomorphic by locally uniform convergence of holomorphic functions, and the pointwise norm bounds show convergence in the original norm. The shrinking domain compensates for loss of an $x$-derivative in a Cauchy-Kovalevskaya theorem argument.

For $X={\partial }_{\mu }\varphi {\partial }^{\mu }\varphi /2$ in signature $(+---)$, the derivative part of the quadratic action on a timelike background is $\frac{1}{2}(P_{,X}+2X{P}_{,XX})\delta {\dot{\varphi }}^2-\frac{1}{2}{P}_{,X}{a}^{-2}(\nabla \delta \varphi {)}^2$. Absence of a ghost requires $P_{,X}+2X{P}_{,XX}>0$; positive gradient energy requires $P_{,X}>0$. The squared sound speed is their ratio $P_{,X}/(P_{,X}+2X{P}_{,XX})$.

The normalized angular average is over the unit sphere, with solid angle measure $d\Omega$. Rotational symmetry gives $⟨e^i{⟩}_{\Omega }=0$, $⟨e^i{e}^j{⟩}_{\Omega }={\delta }^{ij}/3$, and $⟨e^i{e}^j{e}^k{e}^{\ell }{⟩}_{\Omega }=({\delta }^{ij}{\delta }^{k\ell }+{\delta }^{ik}{\delta }^{j\ell }+{\delta }^{i\ell }{\delta }^{jk})/15$. The constants follow by contracting indices and using $e^i{e}_i=1$. These identities extract photon angular temperature moments. For normalized spherical harmonics, $\int Y_{\ell m}^{\ast }{Y}_{{\ell }^{\prime }{m}^{\prime }}d\Omega ={\delta }_{\ell {\ell }^{\prime }}{\delta }_{m{m}^{\prime }}$; their angular average instead has an extra factor $1/(4\pi )$.

A deterministic social choice function maps profiles of preference orders to single alternatives. Unrestricted domain permits every profile in ${\mathcal{R}}^n$; onto means every alternative lies in the range. A unanimous rule selects an alternative whenever every agent ranks it first.

For $u\in W^{1,q}((0,1{)}^2)$ with $q>1$, almost every coordinate-line slice has an absolutely continuous representative and is Hölder continuous with exponent $1-1/q$. Slice bounds need not be uniform. Global Hölder continuity follows from Morrey's inequality only when $q>2$, with exponent $1-2/q$; unbounded examples exist at and below $q=2$.

For $d<p<\infty$, every Sobolev space element $u\in W^{1,p}({\mathbb{R}}^d)$ has a Hölder continuous function representative of exponent $1-d/p$, with the displayed bound. It also satisfies $\Vert u{\Vert }_{\infty }\le C_{d,p}(\Vert u{\Vert }_p+\Vert \nabla u{\Vert }_p)$. Averaging the fundamental theorem of calculus along a line segment over a ball givesThe Holder inequality bounds this by $C_{d,p}{r}^{1-d/p}\Vert \nabla u{\Vert }_p$, because $(d-1)p^{\prime }<d$. To compare two ball averages, translate a ball along the segment between their centres and use the same fundamental theorem of calculus along a line segment. These bounds give the displayed estimate. Density of smooth functions in a Sobolev space supplies the representative for nonsmooth $u$. For $p=\infty$, the corresponding conclusion is Lipschitz continuity.

For an integer $k\ge 0$ and sufficiently regular bounded functions, $\Vert fg{\Vert }_{H^k}\le C_k(\Vert f{\Vert }_{\infty }\Vert g{\Vert }_{H^k}+\Vert g{\Vert }_{\infty }\Vert f{\Vert }_{H^k})$. The Leibniz rule and derivative interpolation give the estimate. In a differentiated semilinear wave equation, this separates a controlled low-order coefficient from an arbitrarily high Sobolev norm, allowing propagation of smoothness after a fixed base norm has been controlled.

Choosing a stopping time to maximize an expected reward. For an adapted integrable reward process over a finite discrete horizon, the Snell envelope computes the conditional optimal reward by comparing immediate exercise with continuation.

A pullback of a smooth function composes a function on the target with a smooth map between manifolds, producing a function on the source. It needs no inverse map. Acting on the pulled-back function defines the differential of a smooth map intrinsically as a map of derivations.

A pushforward of a curve composes a parametrized curve in the source with a smooth map between manifolds. Its tangent is the image of the original tangent under the differential of a smooth map. A constant image curve has zero tangent, so a general smooth map need not preserve regularity of curves.

Over a perfect field, a point of a variety is smooth when its local ring is a regular local ring. For an irreducible variety of dimension $n$ at a closed point, this is equivalent to its Zariski tangent space having dimension $n$. The Jacobian criterion expresses the equality as a matrix-rank condition.

For nonzero finite-dimensional spaces $V,W$ of sections of two line bundles on a smooth integral curve over an algebraically closed field, their product span in sections of the tensor product satisfies this bound. At one point choose bases with strictly increasing orders of vanishing. Multiplying every first-space basis vector by the first second-space vector, and then the last first-space vector by the remaining second-space vectors, produces distinct increasing orders and hence independent sections. This proves the dimension inequality needed for the Clifford inequality for curves.

Let positive transverse displacement point along a uniform load per unit length $w$. The filament bending modulus $B$ gives energy ${\int }_0^L[B(h_{xx}{)}^2/2-wh]dx$. The variational derivative is $B{h}_{xxxx}-w$, so resistive-force theory gives ${\zeta }_{\perp }{h}_t=-B{h}_{xxxx}+w$. Clamping imposes $h(0)=h_x(0)=0$, while a free tip imposes $h_{xx}(L)=h_{xxx}(L)=0$. The steady solution isIts tip displacement is $w{L}^4/(8B)$. For a submerged cylindrical filament under Newtonian gravity, buoyancy gives $w=({\rho }_f-\rho )\pi a^{2}g$. This linear approximation requires small slope and negligible inertia.

Mapping out of the object preserves sequential colimits. A map factors through a finite stage, and two finite-stage maps agreeing in the colimit agree at a later stage. Relative sequential smallness restricts the diagrams to a stated class of morphisms.

A composite built by successive pushouts of coproducts of prescribed generating maps. In nonnegative chain complexes, attaching a sphere-to-disk generator adds a free element with a specified existing cycle as boundary. The resulting inclusion has a degreewise free cokernel.

For an axisymmetric slender viscous thread, the Young–Laplace equation and radial extensional viscous stress give $p=p_e+\gamma /a-\mu w_z$. The axial liquid stress relative to ambient pressure is $3\mu w_z-\gamma /a$. Its cross-sectional integral plus the circumferential surface tension force $2\pi \gamma a$ gives the displayed tensile force. Thus the net capillary contribution is $\pi \gamma a$, and axial force balance with downward gravity is $T_z+\pi \rho g{a}^2=0$.

Uniform axial strain rate $E$ in an annular Newtonian fluid gives radial velocity $u_r=-Er/2+C/r$. For radii $R_i,R_e$, area parameter $A=R_e^2-R_i^2$, and constant gas pressures,The opposite signs of inner and outer curvature in the Young–Laplace equation give these expressions. The kinematic boundary condition implies $\dot{A}=-EA$ and

For a straight rod $\mathbf{X}=s\mathbf{p}$, $-L<s<L$, leading slender-body theory uses $\mathbf{f}=C(I-\mathbf{pp}/2)[\Delta \mathbf{U}+s(\Delta \mathbf{\Omega }\times \mathbf{p}-E\mathbf{p})]$. Integration gives $\mathbf{F}=2CL(I-\mathbf{pp}/2)\Delta \mathbf{U}$ and $\mathbf{G}=(2C{L}^3/3)[(I-\mathbf{pp})\Delta \mathbf{\Omega }-\mathbf{p}\times E\mathbf{p}]$. Axial spin is outside this centreline approximation.

For a smooth Skyrme model field $U:{\mathbb{R}}^3\to SU(2)$ with $U(\infty )=1$, compactification gives a map $S^3\to S^3$. With $L_i=U^{†}{\partial }_{i}U$, its degree of a map between oriented manifolds is $B=-(24{\pi }^2{)}^{-1}\int {ϵ}_{ijk}\mathrm{tr}(L_i{L}_j{L}_k)d^{3}x$, with sign chosen so the standard decreasing hedgehog profile has $B=1$. The integrand is the pullback of the normalized volume form on $SU(2)$. Hence it is integer-valued and unchanged by smooth finite-energy homotopies. In the rational map approximation for Skyrmions, a degree-$N$ angular map and profile $f(0)=\pi ,f(\infty )=0$ give $B=-(2N/\pi )\int f^{\prime }{\sin }^{2}fdr=N$. This is a topological conservation law, not the Noether charge of isospin.

The standard low-energy $B=2$ Skyrmion in the ordinary Skyrme model has a toroidal baryon-density shape, with axial density symmetry $D_{\infty h}$. Its proper combined field symmetries have an $O(2)$-type stabilizer. The angular ansatz $R(z)=z^2$ in the rational map approximation for Skyrmions explains the axial relation $R(e^{i\theta }z)=e^{2i\theta }R(z)$: spatial rotation through $\theta$ is accompanied by isorotation through $2\theta$. The angular density vanishes at the two axial directions, yielding the toroidal hole. This describes the familiar minimal branch and a useful approximate field; it is not an assertion that every degree-two field is toroidal.