Dirichlet boundary data prescribe the boundary values of a solution as a function $\phi$ on the boundary of its domain. Continuity on a polygon requires agreement of the values at corners, but continuity alone does not ensure solvability if the equation loses uniform ellipticity at a boundary side.

For a closed inclusion $i:X\hookrightarrow Y$, direct image sheaf formation is exact: stalks on $X$ are unchanged and stalks outside $X$ are zero. It also preserves flasque sheaves. Pushing forward a flasque resolution thus gives a resolution computing cohomology on $Y$, and its complex of global sections is identical to the original one on $X$. This proves the displayed isomorphism in every nonnegative degree.

Tensoring sections on $f^{-1}V$ gives a balanced pairing of the two direct image sheaves on $V$. The universal property of the tensor product of modules and sheafification produce the displayed morphism. Composing with the pullback-direct-image adjunction unit gives the projection formula for sheaves. The comparison itself need not be an isomorphism.

For an open inclusion $j:U\hookrightarrow X$, this sheaf has sections $\mathcal{F}(V\cap U)$ on an open set $V$. Restriction defines $\mathcal{F}\to {}_U\mathcal{F}$. Its stalks outside $U$ may be nonzero, so it differs from extension by zero.

For $m^2>0$ and $n>d/2$, Schwinger parameterization and a Gaussian integral give $I_{d,n}(m)=\int d^{d}k(2\pi {)}^{-d}(k^2+m^2{)}^{-n}=\Gamma (n-d/2)(m^2{)}^{d/2-n}/[(4\pi {)}^{d/2}\Gamma (n)]$. Its meromorphic continuation supplies dimensional regularization poles.

Solutal diffusivity is the diffusion coefficient governing transport of a dissolved constituent down its concentration gradient. For uniform $D$ and a stationary solvent, the diffusion equation is $C_t=D{\nabla }^{2}C$. It need not equal the thermal diffusivity.

A spatial resolution element is one independently resolved angular or spatial scale under a specified resolution criterion. For an ideal telescope of diameter $D$, its angular scale is of order $\lambda /D$ and its angular area of order $(\lambda /D{)}^2$. A photodetector detector pixel or spaxel can be smaller than a resolution element without adding independent spatial information.

For $f\in C^2[0,1]$ and $\Vert f^{\delta }-f{\Vert }_2\le \delta$, two translated half-interval integrals have overlap multiplicity at most two. The inequality $|a-b{|}^2\le 2|a{|}^2+2|b{|}^2$ gives $\Vert D_h\Vert \le \sqrt{6}/h$. Averaging $f^{\prime }$ over the relevant interval and applying $|f^{\prime }(x+t)-f^{\prime }(x)|\le \Vert f^{\prime \prime }{\Vert }_{\infty }|t|$ gives the second term. For this to estimate a Moore–Penrose inverse of an operator, also require $f(0)=0$; otherwise the derivative is still approximated but $K^{†}f$ is not defined.

If $DU-UD=I$, commuting every $D$ past every $U$ gives $(D+U{)}^{\ell }={\sum }_{i+j+2m=\ell }\ell !U^i{D}^j/(2^{m}i!j!m!)$. The recurrence follows from $D{U}^i=U^{i}D+i{U}^{i-1}$. Applied to the empty diagram in the Young lattice, it counts oscillating tableaux.

For a vector field $X$ on $M$, its pushforward by a smooth map between manifolds $f:M\to N$ is the vector field along a map defined by $(f_{\ast }X{)}_p=d{f}_p(X_p)$. Acting on a function $h$ on $N$, it satisfies $(f_{\ast }X{)}_p(h)=X_p(h\circ f)$. For a diffeomorphism this defines a vector field on $N$ by evaluation at $p=f^{-1}(q)$. For a general map there need not be a single vector at each image point: $f(x)=x^2$, $X={\partial }_x$ gives $2x{\partial }_y$, with opposite values over $y>0$.

The pullback of a covariant tensor composes each of its vector arguments with the differential of a smooth map. It takes a type $(0,s)$ tensor on the target to one on the source. Antisymmetry is not required; the pullback of a differential form is the alternating special case. If the target vectors are projected onto the image tangent space first, evaluating the pullback is unchanged.

The pushforward of a contravariant tensor applies the differential of a smooth map to each vector factor of a type $(r,0)$ tensor. It is defined pointwise at a specified source point. A field generally gives a field along the map, not necessarily a unique tensor field on the image when several points have the same image. A diffeomorphism allows the usual transport of arbitrary mixed tensors through its inverse as well.

The different exponent at a prime ideal $\mathfrak{P}$ is its exponent in the prime ideal factorization of the different ideal. It equals the local different exponent of the corresponding extension of completions. The different exponent and tame ramification theorem detects unramified, tame, and wild behavior without assuming a Galois extension.

With ${\delta }_{h}f(x)=[f(x+h{e}_{\ell })-f(x)]/h$, a change of variables givesThe compact support of $f$ and its translates must remain inside the integration domain, or one must integrate over the whole space with convergent integrals. In particular ${\delta }_{-h}g(x)=[g(x)-g(x-h{e}_{\ell })]/h$: the denominator's sign matters. This is the difference quotient analogue of integration by parts.

The diagonal lemma can be carried out uniformly: from a one-variable formula code, compute a sentence code whose assertion is equivalent to the original formula applied to that sentence code. The effective substitution operation is what turns a semantic diagonal argument into a productive function.

An infinite game of perfect information with an open set of winning plays for I in the product topology on a discrete move set has a winning strategy in an infinite game for one player. Construct a winning-position attractor in an infinite game by transfinite recursion; ranks give I a terminating descent strategy inside it, and its complement gives II a strategy avoiding every winning prefix. The move set need not be countable, in the usual set theory with axiom of choice.

To evaluate a layered product on $A_x|{\psi }_0⟩$, scan the bond orthogonal projections from right to left. A projection disjoint from the current support commutes through the accumulated operator and disappears against the common ground state. Retained projections expand the support by at most one site per layer. Their product $L$ gives a local operator $L{A}_x$ with both $K^r{A}_x|{\psi }_0⟩=L{A}_x|{\psi }_0⟩$ and $⟨{\psi }_0|L{A}_x=⟨{\psi }_0|A_x$.

The derived algebra is the linear span of all Lie brackets $[x,y]$ in a Lie algebra. The Jacobi identity makes it an ideal of a Lie algebra. Its quotient is the largest abelian Lie algebra quotient of $\mathfrak{g}$.

The derivations of a Lie algebra form a Lie subalgebra of $\mathrm{End}(\mathfrak{g})$ with the commutator bracket. The identity $[D,\mathrm{ad}x]=\mathrm{ad}(Dx)$ makes the inner derivations an ideal of a Lie algebra. For a complex semisimple Lie algebra, the Weyl complete reducibility theorem and the vanishing center of a Lie algebra imply $\mathrm{Der}(\mathfrak{g})=\mathrm{ad}\mathfrak{g}$.

A Yarkovsky removal law $t_y=BD+C/D$, with positive constants $B,C$ and body diameter $D$, removes intermediate sizes fastest. Define $D_0=\sqrt{C/B}$, the diameter of minimum Yarkovsky removal time. Suppose an initially collision-dominated Dohnanyi collisional cascade has differential number distribution $n(D)=K{D}^{-7/2}$. Let $D_{+}\gg D_0$ be its large-diameter crossing of collision and removal times. At this crossing $t_c(D_{+})\simeq B{D}_{+}$; for the same fixed collision geometry and strength, $t_c(D)\propto D^{1/2}/K$.

Use a local steady model in which fragment injection into the depleted interval has number spectrum $S_N(D)\propto D^{-7/2}$. Removal-dominated fragment size distribution gives $n=S_N{t}_y$. Matching to the large-size collisional cascade at $D_{+}$ and taking $D\ll D_0$ givesThe steeper distribution can meet a second collision-dominated population $n_{-}(D)=K_{-}{D}^{-7/2}$ at a smaller diameter $D_{-}$. Continuity requiresThe reduced projectile population lengthens the collisional-cascade relaxation time by $K/K_{-}$. Therefore equality of removal and collision times at $D_{-}$ isCombining the last two equations provesThus the low-size collisional cascade recovers the differential-number slope $-7/2$ at a reduced normalization. Its boundary cannot be found by keeping the original projectile population in the collision rate. These are asymptotic matching relations for the stated injection and local-collision closures; a full fragment redistribution function can change transition coefficients and smooth the joins.