For a down-set $D$ in the Boolean lattice, every member $A$ has all its $|A|$ immediate lower neighbors in $D$. Counting an internal edge at its upper endpoint gives $e(D)={\sum }_{A\in D}|A|$. The degree identity for the edge boundary of the hypercube graph then gives the displayed formula.

Two degree-raising differentials square to zero and anticommute. If an initial convention uses commuting differentials, introduce a sign in one of them before forming the total cochain complex.

For a nonnegative martingale, $T_R=\inf \{n:M_n\ge R\}$ satisfies $\mathbb{P}(T_R<\infty )\le \mathbb{E}{M}_0/R$. Stop at $T_R\wedge n$ and use nonnegativity before passing to the increasing event. A limit of thresholds from below also gives the bound for $\{{\sup }_n{M}_n\ge R\}$ even when the supremum is not attained.

A monotone binary allocation is made truthful by charging its winning threshold, truncated below at the lowest allowed value, and charging zero to losers. The displayed integral formula sets the lowest type's utility to zero. To check it, hold the other reports fixed: a type above the threshold benefits from winning at that price, while a type below it benefits from losing. At a threshold, either deterministic tie choice is compatible with indifference.

For a zero-starting continuous local martingale $X$, the stochastic exponential satisfiesThe Holder inequality and the supermartingale property of the first exponential bound its stopped $p$th moment by $(\mathbb{E}{e}^{C(p,q)X_T}{)}^{(q-1)/q}$. The subtraction in the coefficient is $\sqrt{pq}-1$, not a square root of $pq-1$.

For a convergent continuous local martingale and $r>1$, factor $\mathcal{E}(X)$ into $\mathcal{E}(rX{)}^{1/r^2}$ and an ordinary exponential. Apply the Holder inequality and then the concave Jensen inequality to the fractional power $2/(r+1)$. The displayed lower bound is informative when the terminal exponential moment is finite.

For a strictly positive continuous semimartingale $Z$, its stochastic logarithm is $\int Z^{-1}dZ$. When $Z$ is a continuous local martingale starting at one, this is a local martingale $L$ and the Itô formula gives $Z=\exp (L-⟨L⟩/2)$. Unlike the ordinary logarithm, it has no compensating finite-variation term.

For a bounded measurable coefficient matrix $A$, an $H^1$ weak solution of $\mathrm{div}(A\nabla u)=0$ satisfies $\int A\nabla u\cdot \nabla \phi =0$ for every compactly supported smooth test function. By density this identity extends to compactly supported $H^1$ tests. If $A$ is symmetric and $\lambda I\le A\le \Lambda I$, Caccioppoli inequalities follow without differentiating $A$.

If $Lu=\mathrm{div}F+g$ and $L$ contains $\mathrm{div}(bu)+du$, then $L(u-k)=\mathrm{div}(F-kb)+(g-kd)$. Thus shifting a weak solution to make it nonnegative changes the forcing unless the operator annihilates constants. This bookkeeping is essential in applying the Weak Harnack inequality to upper and lower oscillations.

On an interval, the weak equation $(a{u}^{\prime }{)}^{\prime }=0$ makes $a{u}^{\prime }$ constant almost everywhere. If $\lambda \le a\le \Lambda$, the derivative energy on concentric subintervals satisfies $E(r)\le (\Lambda /\lambda {)}^2(r/R)E(R)$. This supplies dyadic energy decay even though a one-dimensional annulus is disconnected.

The aspect ratio of an astrophysical disk is its vertical thickness scale divided by cylindrical radius. A thin disk has $H/r\ll 1$. In gas-supported vertical hydrostatic equilibrium with harmonic external gravity $-{\Omega }_z^{2}z$, $H/r\sim c_s/(r{\Omega }_z)$, where ${\Omega }_z$ is the vertical epicyclic frequency and $c_s$ is the sound speed. In a centrally dominated Keplerian disk, ${\Omega }_z=\Omega$, so this compares sound speed with orbital speed. If disk self-gravity supplies the vertical restoring force, the relation $H\sim c_s/\Omega$ requires replacement. A polytropic free-surface height and an isothermal scale height differ by model-dependent constants even though their aspect ratios have the same basic role.

An ordered sample from $n$ equally likely values is nondecreasing exactly when it is the unique sorted representative of its multiplicities. Stars and bars counts the nonnegative multiplicities summing to $r$; repeated values are included automatically.

The Discrete-time multi-state diagnosis model tracks the delay between infection and diagnosis. Let $h_i$ be the expected number of new infections at step $i$, with $h_i=0$ for $i<0$. An infection at step $i$ has one early opportunity for diagnosis at step $i+1$, with Probability $a$. If not diagnosed early, it has a diagnosis opportunity with Probability $b$ at each step $i+2,i+3,\dots$, conditional on still being undiagnosed. Assume $0\le a\le 1$ and $0<b\le 1$, with the same transition probabilities for all cohorts.

The expected early and late diagnosis counts at step $k$ arewhere an empty sum is zero. Each term in the second expression is a cohort size multiplied by its Probability of avoiding early diagnosis, surviving the earlier late opportunities, and being diagnosed now. The waiting time in the late state therefore has a positive-support geometric distribution.

For a single cohort with $h_0=100$, $a=0.2$, $b=0.5$, and no later infections, the expected counts are $e_1=20$, $n_2=40$, $n_3=20$, and $n_4=10$. The ratio $e_k/(e_k+n_k)$, when its denominator is positive, is the early fraction of the expected diagnosis intensity. It is not generally the expected value of the observed random fraction: taking a ratio and taking an expected value do not commute. Initial infections before step zero require additional cohort terms, so the time origin must be stated.

Relative to losing against a particular rival, the gain from winning now is the current value minus discounted continuation utility. In an elimination contest this is $A_i=v_i-\delta J_i(S\setminus \{j\},r-1)$. When the relevant losing baseline is fixed, current effort incentives reduce to a two-player complete-information all-pay equilibrium. In general, the baseline can depend on which rival wins; that dependence must be verified before using a single effective prize.

For $n\ge 3$ and a bounded smooth domain, a Dirichlet Laplacian eigenfunction with eigenvalue $\lambda$ satisfies the displayed estimate. A signed power test for a Laplacian eigenfunction and the zero-boundary Sobolev inequality give a Moser iteration with exponents $p_j=2[n/(n-2){]}^j$. The geometric series ${\sum }_{j}1/p_j=n/4$ gives the exact power of $\lambda$, while ${\sum }_j(\log p_j)/p_j<\infty$ controls the remaining constant. The constant depends only on dimension, because the zero-boundary Sobolev inequality follows by extension by zero to the whole space.

After the Dirichlet gauge transform for constant drift, put $q=\sqrt{p+a^2}$. The Dirichlet Green function of $p-{\partial }_x^2+a^2$ isThe original unweighted resolvent kernel is $e^{-ax}{R}_q(x,y)e^{ay}$. Its poles are $p=-a^2-(n\pi /L{)}^2$. The kernel is even in $q$, so the apparent square-root branch point is removable.

Zero Dirichlet form of the Kac collision operator forces invariance under every pair rotation; strong continuity upgrades almost every angle to every angle. Coordinate-plane rotations generate the special orthogonal group, whose action is transitive on each sphere for $N\ge 2$. Averaging over normalized Haar measure therefore identifies the invariant $L^2$ functions with radial functions. Conversely radial functions are fixed by every pair rotation.

For a smooth map between Riemannian manifolds, its Dirichlet energy is half the integral of the squared norm of its differential, using the source and target metrics. When the source is two-dimensional, a conformal change of its metric leaves this energy unchanged. For a J-holomorphic curve into a symplectic manifold with a compatible almost complex structure, the Energy identity for a J-holomorphic curve identifies it with the pulled-back symplectic area. The norm of a differential here is the full tensor norm, summing its squared values on an orthonormal source frame.

Convolution of this kernel with Dirichlet boundary data supplies the boundary forcing for $u_t=u_{xx}+\alpha u_x$ on $x>0$. It is $-(2{\partial }_x+\alpha )H(x+\alpha t,t)$. For $\alpha \ge 0$, its total time mass is $e^{-\alpha x}$, tending to one as $x\downarrow 0$, and its mass away from zero time tends to zero. Thus the boundary value is recovered through an approximate identity, not by pointwise substitution $x=0$ in the kernel.

In a fixed-temperature convection problem, the boundary temperature is prescribed, and its perturbation satisfies a thermal Dirichlet boundary condition. A homogeneous condition means the boundary follows the prescribed conductive background temperature. This does not necessarily mean a vertical wall has a spatially constant temperature; that additional assumption can conflict with a background vertical gradient.