For finite sets in an abelian group and a bipartite graph $\Gamma \subseteq A\times B$, the restricted sumset is $\{a+b:(a,b)\in \Gamma \}$. It retains only sums corresponding to graph edges.

A surjective submersion between Riemannian manifolds is Riemannian when its derivative is a linear isometry from the orthogonal complement of its kernel onto the target tangent space. Its vertical space is the kernel and its horizontal space is the orthogonal complement.

The weighted operator $L\xi =W^{-1}[-(P{\xi }^{\prime }{)}^{\prime }+Q\xi ]$, with real coefficients and $P,W>0$ in the interior, defines a Sturm-Liouville problem. Its realization as a self-adjoint differential operator requires a domain whose endpoint conditions annihilate the boundary form $[P(\overline{{\eta }^{\prime }}\xi -\overline{\eta }{\xi }^{\prime })]$. Its Rayleigh quotient uses the weight $W$ in the denominator.

On a spacelike constant-target-time slice, integrate the induced spatial line element to obtain the geometric string length. It depends on the embedding and is distinct from the fundamental string length ${\ell }_s=\sqrt{{\alpha }^{\prime }}$. For transverse local motion, energy additionally includes the local Lorentz factor; the geometric length alone does not determine the total energy.

A stress-energy superpotential is an antisymmetric tensor $B^{\rho \mu \nu }=-B^{\mu \rho \nu }$ whose divergence supplies a stress-energy tensor improvement. In the Maxwell field, the potential-dependent expression $B^{\rho \mu }{}_{\nu }=F^{\mu \rho }{A}_{\nu }$ produces the difference between canonical and gauge-invariant stress-energy tensors on shell. Antisymmetry ensures identically conserved improvement terms.

If real matrices satisfy $A^T=A\le 0$ and $B^T=-B$, then $e^{tA}$ is a contraction and $e^{tB}$ is an orthogonal matrix for $t\ge 0$. Hence $\Vert e^{kA/2}{e}^{kB}{e}^{kA/2}{\Vert }_2\le 1$ for every $k\ge 0$. The bound is independent of commutativity and gives unconditional stability for a centered convection-diffusion semidiscretization with homogeneous Dirichlet boundary conditions. It does not itself bound the mesh-dependent commutators that enter a time-error estimate.

The Brinkman equation augments Stokes flow with a local drag against a background matrix of velocity ${\mathbf{u}}_m$. The inverse screening length $\alpha$ has units of inverse length. With equal viscous coefficients the permeability is ${\alpha }^{-2}$. Taking the curl produces a screened vorticity equation.

For log-price dynamics $dX=-{\sigma }^{2}dt/2+\sigma d{W}^X$ and volatility dynamics $d\sigma =A(\sigma )dt+B(\sigma )d{W}^{\sigma }$, with correlation $\rho$, the exponential payoff ansatz $U=e^{\theta X}V$ reduces the backward partial differential equation to the displayed one-dimensional equation. The terminal value is $V(T,\sigma )=1$. Correlation changes the volatility drift, while the log-price drift supplies the negative $\theta$ term.

A stochastic process in which individuals independently produce offspring according to specified distributions. It models growing populations and gives an approximation to sparse percolation clusters when collisions between branches can be neglected. On a lattice, loops and overlapping branches prevent this approximation from being an exact identity.

On a finite distributive configuration lattice, probability weights satisfying ${\mu }_1(\omega \vee \eta ){\mu }_2(\omega \wedge \eta )\ge {\mu }_1(\omega ){\mu }_2(\eta )$ for all pairs give stochastic domination ${\mu }_1\ge {\mu }_2$. The result holds for nonnegative weights as well: apply the four functions theorem to ${\mu }_1{1}_{A^c},{\mu }_2{1}_A,{\mu }_1{1}_A,{\mu }_2{1}_{A^c}$ for an increasing event $A$, obtaining ${\mu }_1(A^c){\mu }_2(A)\le {\mu }_1(A){\mu }_2(A^c)$.

In singular stochastic control, a control is a finite-variation process rather than only a rate integrated against time. Its increments can have atoms or a part singular with respect to time. The Hamilton-Jacobi-Bellman equation becomes a variational inequality with a constraint on directional derivatives of the value function.

A narrow spectral feature records enhanced emission or absorption near a transition frequency. In a stellar atmosphere, its strength depends on atomic populations, opacity, velocity fields and the source function. Lines therefore diagnose composition as well as the physical atmosphere in which they form.

A heated layer above a photosphere can have an outward temperature increase. Strong lines formed there may show emission where an outward-cooling photosphere would produce absorption. The energy maintaining the temperature inversion is additional atmospheric heating; the exact height and transfer regime vary between stars.

When radiative rates compete with or dominate collisions, level populations need statistical equilibrium coupled to radiative transfer, rather than thermal population laws at the local temperature. Radiation from other depths can make the local radiation field differ from the Planck function. This changes ionization and spectral-line strengths, so a geometrically thin atmosphere may still require non-LTE abundance modelling.

A stellar abundance can be expressed as an elemental or isotopic mass fraction, or as a number ratio to hydrogen. Inferring it from a spectral line requires a model for level populations, opacity, radiation and velocity fields. Agreement between suitable ionization stages and weak photospheric lines tests whether the adopted stellar atmosphere is adequate.

For equal thermal properties and bounded travelling profiles on two deep half-spaces, the solid temperature is $T_m$ and the liquid profile is exponential. The Stefan condition requires $L=c_p\Delta T$. This balances the latent heat of solidification with warming of the initially supercooled liquid.

A crystal dendrite is a branched crystal-growth morphology. Diffusive access to undercooling or supersaturation favours protruding tips, while interfacial capillarity regularizes fine scales and crystal interface anisotropy selects directions. Dendritic growth of a snowflake is a vapour-growth example.

A dominated statistical path is differentiable in quadratic mean at $P$ if its square-root probability density function has an L2 space derivative $(1/2)g\sqrt{p}$, with $g\in L^2(P)$. The score function is $g$. This formulation controls mass near zeros of $p$ as well as the derivative on the support of $P$; a pointwise log-density derivative is insufficient by itself.

A statistical tangent set is the collection of score functions attained by a specified family of differentiable-in-quadratic-mean paths through $P$. The choice of paths is part of the definition. Its elements lie in the mean-zero L2 space, but the set need not already be a closed vector subspace.

Outcome propensities can vary between individuals because of their measured covariates. A linear regression can describe this variation by allowing its conditional mean to depend on those predictors.