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Under the usual filtration conditions, a càdlàg submartingale of Class D on a finite horizon has a unique decomposition into a uniformly integrable martingale and a predictable integrable nondecreasing process starting at zero. A general continuous semimartingale decomposition instead allows an arbitrary continuous finite-variation drift. A convex function of a semimartingale need not be a submartingale when that drift has unrestricted sign.

For a submartingale with ${\sup }_n\mathbb{E}|X_n|=C<\infty$, its positive part is a submartingale by the conditional Jensen inequality. Thus the displayed conditional expectations increase in $p$. Their limits are finite almost surely and have expectations at most $C$, by the monotone convergence theorem. Conditional monotone convergence and the tower property of conditional expectation give $\mathbb{E}[M_{n+1}\mid {\mathcal{F}}_n]=M_n$. Moreover $M_n\ge X_n^{+}$. Consequently $Y_n=M_n-X_n$ is a nonnegative supermartingale, ${\sup }_n\mathbb{E}{Y}_n\le 2C$, and $X=M-Y$. The resulting martingale need not have uniform integrability.

For a sublimating surface with exponentially absorbed volumetric heating, use a frame moving into the solid at speed $V$. Its steady heat equation is $k{T}^{\prime \prime }+\rho c_{p}V{T}^{\prime }+q_0{e}^{-\lambda z}=0$. The surface Stefan condition is $k{T}^{\prime }(0)=\rho LV$. Integrating over the half-space gives the displayed energy-limited speed. The surface gradient is positive, but the far field is colder, so the temperature has a subsurface maximum where melting can begin.

For a locally linear ice temperature with positive gradient $G$ into the solid, a corrugated sublimation front receives a perturbation in heat flux proportional to its horizontal wavenumber $\alpha$. The Gibbs-Thomson effect suppresses short waves, giving the displayed quasi-static growth rate. It requires $V/(\kappa \alpha )\ll 1$ and $|\sigma |/(\kappa {\alpha }^2)\ll 1$, as well as a perturbation depth short enough for the linear background approximation. Unstable modes have $0<\alpha <\sqrt{G/\Gamma }$.

For a subset $W$ of the vertex set of a hypergraph $H$, the induced subhypergraph retains exactly the edges wholly contained in $W$. This is the restriction used by a hereditary hypergraph property; it differs from intersecting every edge with $W$, which can change uniformity.

For locally integrable real $u$ with nonnegative distributional derivative $\Delta u$, its ball means increase with radius. To see this, mollify: for a smooth subharmonic function the derivative of its spherical mean is the integral of $\Delta u$ over the ball divided by the sphere area. The ball means inherit monotonicity; local $L^1$ convergence passes this to $u$. Their decreasing-radius limit exists, may be minus infinity at exceptional points, and equals $u$ almost everywhere by the Lebesgue differentiation theorem. It is upper semicontinuous as an infimum of continuous ball means and satisfies the mean-value inequality. Arbitrary changes of a representative on a null set can destroy its pointwise maximum principle.

A locally integrable real function is superharmonic in the distributional sense if its Laplacian is nonpositive on every nonnegative test function. Equivalently its negative is a subharmonic function. The canonical lower semicontinuous representative satisfies the reverse ball mean inequality. A function both subharmonic and superharmonic has zero distributional Laplacian and a smooth harmonic function representative by the Weyl lemma.

For a countable family of subharmonic functions, a locally integrable pointwise supremum is subharmonic in the distributional derivative sense. Finite maxima preserve subharmonicity: approximate the maximum by the smooth convex coordinatewise increasing function $(s+t+\sqrt{(s-t{)}^2+{\delta }^2})/2$, apply the chain rule after mollification, then pass to a local L1 norm convergence limit. The maxima $g_N=\max (f_1,\dots ,f_N)$ increase to $g$ and are bounded in absolute value by $|f_1|+|g|$ on compact sets, so the dominated convergence theorem completes the proof. The hypothesis cannot be omitted: $f_j=j$ has infinite supremum. To obtain a pointwise subharmonic function, take the canonical representative of a subharmonic distribution.

On a relatively compact coordinate disk $V$ in a Riemann surface, replace a continuous subharmonic function $u$ inside $V$ by the Poisson integral of its boundary values, leaving it unchanged outside. The resulting function $L_{V}u$ is continuous and subharmonic, equals $u$ off $V$, is harmonic on $V$, and satisfies $L_{V}u\ge u$. The maximum principle for subharmonic functions proves the inequality. The pasting assertion follows from harmonic comparison: a harmonic comparison function dominating the pasted function on the boundary of a test domain first dominates $u$, then dominates the disk lift on its part inside $V$.

A harmonic majorant of a real function $u$ is a real harmonic function $h$ with $h\ge u$ pointwise. A least harmonic majorant lies below every other harmonic majorant. A continuous subharmonic function need not have one: $u(z)=(1-|z{|}^2{)}^{-1}$ on the unit disk has increasing circular means diverging to infinity, whereas any harmonic majorant would have the same finite mean on every centered circle.

Write a holomorphic function as $f=u+iv$. The Cauchy-Riemann equations make $u,v$ harmonic functions and give $u_x=v_y$, $u_y=-v_x$. The product rule therefore gives $\Delta (u^2+v^2)=2(|\nabla u{|}^2+|\nabla v{|}^2)=4|f^{\prime }{|}^2$. Thus its squared modulus is subharmonic and is harmonic only where $f^{\prime }$ vanishes.

For every holomorphic function and $p>0$, $|f{|}^p$ is subharmonic. Away from its zeros the displayed Laplacian calculation proves this. At the zeros, use the smooth approximations $(|f{|}^2+\epsilon {)}^{p/2}$, whose Laplacians are nonnegative, and let $\epsilon \downarrow 0$ in their mean inequalities. This permits Poisson integral comparison for fractional modulus powers even when a global analytic fractional power does not exist.

A proper subgroup with no strictly intermediate subgroup between it and the ambient group. In a transitive action a point stabilizer is maximal exactly when the action is primitive.

For a specified subgroup $H\le G$ and a chosen alphabet for $G$, the membership problem asks whether an input word represents an element of $H$. Solubility of the ordinary word problem for a group does not imply solubility of this problem, even for finitely generated subgroups: a Mihailova subgroup of a product of finite-rank free groups can have undecidable membership.

A subgroup is finitely generated when finitely many of its elements generate it as a group. Even inside a finitely generated group, subgroups need not be finitely generated. Finite generation of a witness subgroup, rather than of the original intersection, is what makes centralizing HNN extensions usable for a benign subgroup.

A real convex function locally bounded on an open convex set has a continuous supporting linear functional at every interior point. Apply the dominated Hahn-Banach theorem to its sublinear directional derivative $p_x$ to obtain $l\le p_x$. Increasing secant slopes imply $p_x(z)\le f(x+z)-f(x)$ whenever the endpoint is in the domain. A local Lipschitz continuity bound gives both $l(y)\le L\Vert y\Vert$ and $-l(y)=l(-y)\le L\Vert y\Vert$, proving continuity.

A stationary binary subdivision mask $a$ specifies a refinement operator $q_j={\sum }_i{a}_{j-2i}{p}_i$. Indices may be vectors for a surface. Coefficients restricted to a parity class give the stencil for one type of refined vertex.

// Target: analysis.bigb

Loop subdivision refines an interior triangular mesh by splitting each face into four triangles and updating control points with local affine combinations. For an old vertex $P$ with $n$ cyclic neighbors $P_1,\dots ,P_n$, the original vertex rule isFor an interior edge with endpoints $A,B$ and opposite triangle vertices $C,D$, the new edge vertex isOn the regular triangular lattice $n=6$, ${\beta }_6=1/16$, so $P^{\prime }=(10P+{\sum }_j{P}_j)/16$. The coefficient sums are one, giving affine equivariance of a geometric basis; their nonnegativity also gives a convex hull bound. These are interior rules: a boundary or a deliberately retained sharp crease needs its own specified stencil.

The scheme is approximating rather than interpolatory. If an old vertex has height one and its six regular neighbors have height zero, its new height is $10/16$, so refinement moves the old vertex. On regular patches the limit is a quartic box spline; extraordinary subdivision vertices require the local subdivision matrix and characteristic map of a subdivision surface analysis. The method is named for Charles Loop, whose 1987 thesis introduced the triangular scheme.^[ref][ref]

// Target: analysis.bigb

A regular quincunx subdivision step replaces a square lattice by the union of its old vertices and face centres. In lattice indices one may useThus two steps give ordinary binary refinement. If $A(z,w)$ is the one-step subdivision mask symbol, the two-step symbol is $A(z,w)A(zw,z/w)$. The two lattice parity classes separately determine the old-vertex and face-vertex stencils; their coefficient sums must each be one to reproduce constants.

An extraordinary subdivision vertex has a valence different from that of the regular refinement lattice. Regular box spline calculations do not alone establish smoothness at this vertex. The finite local subdivision matrix, its eigenvalues, and the associated characteristic map of a subdivision surface determine convergence and tangent behavior.