If $R=({\partial }_t+{\Delta }_x)P$ is smooth and bounded up to time zero on a closed manifold, the Volterra convolution of kernels solves $Q+R+R\ast Q=0$ by $Q={\sum }_{k\ge 1}(-1{)}^k{R}^{\ast k}$. Bounds by $C^k\mathrm{vol}(M{)}^{k-1}{t}^{k-1}/(k-1)!$ prove convergence. Then $H=P+P\ast Q$ is the exact Riemannian heat kernel and has the same initial delta limit.

When a finite union is a compact H-hull, hitting that union before the real axis implies hitting at least one member before the real axis. The union bound followed by the defining limit proves subadditivity of harmonic capacity from infinity in the upper half-plane.

For $c=\sqrt{1-{\epsilon }^2}$, three vertical slits of height $\epsilon$ rooted at $-c,0,c$ form a compact H-hull of enclosing radius one. Subadditivity of harmonic hull capacity gives capacity at most $6\epsilon$, which can tend to zero. Thus a uniform lower bound requires additional geometric hypotheses.

For a connected slit joining $0$ to $x+iy$ with $x^2+y^2=1$, reflect across the vertical line through $x$. The two slits separate the vertical segment below the common tip and the intervening real boundary from infinity. Brownian reflection symmetry bounds the corresponding model hitting probability by twice the original hull-hitting probability. Mapping out the vertical segment gives total boundary-image length $1+y$ for its two banks together with the real interval between $0$ and $x$. The harmonic-measure asymptotic at infinity gives the displayed bound. This argument requires a separating connected barrier; radius gives no positive lower bound for disconnected harmonic hull capacity.

On bounded continuously differentiable functions with uniformly bounded total variation of a function, the map $T$ is Hadamard differentiable in the supremum norm with derivative ${\dot{T}}_{F,G}(a,b)={\int }_0^{u}b{F}^{\prime }+G(u)a(u)-G(0)a(0)-{\int }_0^u{G}^{\prime }a$. Use weak convergence of bounded-variation integrators for the first term and integration by parts for the remaining terms. No derivatives of the limiting continuous directions are required.

A site is a category equipped with a Grothendieck topology, specifying which sieves are covering. The sheaves on a site satisfy unique amalgamation of matching families on those sieves. A small site presents a Grothendieck topos.

For ghost term $-({\partial }^{\mu }\bar{c})\cdot D_{\mu }c$ and the current convention $\delta S=-\int ({\partial }_{\mu }\theta )j_G^{\mu }$, localize $\delta c=\theta c$, $\delta \bar{c}=-\theta \bar{c}$. The two differentiated parameters give this current. Its Noether charge grades ghosts by $+1$ and antighosts by $-1$, up to the corresponding overall generator normalization.

For an axial rotlet dipole parallel to a flat free surface, its reflected image gives the displayed normal vorticity at height $h$. A spherical torque-free body responds with angular velocity ${\omega }_z/2$ and an elongated body also responds to strain. At constant height, self-propulsion plus yaw gives a circular path; the signed moment sets its handedness.

For a real smooth cutoff $\varphi$ supported on $[-1/3,1/3]$ with $\varphi (0)=1$, define $F_X(n)=({\sum }_{d\mid n}\mu (d)\varphi (\log d/\log X){)}^2$. Its sum over any interval of length $X$ is $c_{\varphi }X/\log X+O_{\varphi }(X/{\log }^{2}X+X^{2/3})$, uniformly in the interval location. Here $c_{\varphi }$ is the derivative energy constant for a smooth sieve cutoff. Expanding the square yields an interval-counting error $O(X^{2/3})$ and an Euler product for a smoothed divisor-square correlation; its zeta function pole provides the main term. Every prime exceeding $X^{1/3}$ has weight one, giving the short-interval prime upper bound from a smooth divisor weight.

Let $N_t$ be the backlog in slotted ALOHA, and let stations use attempt probability $1/S_t$ with estimate $S_t\ge 1$. Update the estimate by $a$ after an idle slot, $b$ after a success and $c$ after a collision, clipping its value below at one. These feedback parameters therefore specify an actual controller, rather than only a drift formula.^[ref]

In the interior fluid model, write $s,n>0$ and $\kappa =n/s$. The idle and success probabilities approach $e^{-\kappa }$ and $\kappa e^{-\kappa }$. For $a=b=-A$, $c=2A/(e-2)$ and $A\ge 1$, their weighted mean givesHere $\nu$ is the mean number of arrivals per slot. Since $g(1)=0$ and $g^{\prime }(\kappa )=Ae\kappa e^{-\kappa }/(e-2)>0$, the estimate falls below contention ratio one and rises above it.

Use the ratio time change for a homogeneous fluid model. Its ratio drift is $h=\nu -\kappa e^{-\kappa }-\kappa g$. For $\kappa \ge 1$,because $g\ge 0$ and $Ae{\kappa }^2/(e-2)>\kappa -1$. If $\nu <e^{-1}$, then $h(1)<0$; continuity supplies a trapping upper bound below one. There $g$ is bounded strictly below zero, giving finite-time draining of a fluid model. If $\nu >e^{-1}$, instead $\dot{n}\ge \nu -e^{-1}>0$. These are fluid conclusions; stochastic stability requires a separate argument.

For line-bundle transition functions $g_{ij}=e^{2\pi i{f}_{ij}}$, the integer cocycle is $c=\delta f$. Connection forms satisfy $A_j-A_i=2\pi id{f}_{ij}$. Set $B=-A/(2\pi i)$ and $F=i\Theta /(2\pi )$. In the Čech-de Rham total complex, $F-c=D_{\mathrm{tot}}(B-f)$, so the normalized vector-bundle curvature represents the image of the integral First Chern class. Curvature cannot detect integral torsion.

For a continuous Itô diffusion with diffusion generator $\mathcal{L}$, let $u$ solve $\mathcal{L}u=\lambda u$ with $\lambda >0$, be bounded on an open domain and its boundary, and equal $g$ on the boundary. Stop at the first boundary hit $\tau$. The Itô formula makes $e^{-\lambda (t\wedge \tau )}u(X_{t\wedge \tau })$ a bounded martingale. Its limit is zero on $\{\tau =\infty \}$ and equals $e^{-\lambda \tau }g(X_{\tau })$ on $\{\tau <\infty \}$. ThusBoundedness justifies passage to the terminal expectation even if hitting is not almost surely finite.

A chiral superfield has a complex scalar. Minimizing its potential only on the real axis can miss supersymmetric vacua. For example $W^{\prime }(\phi )=1+{\phi }^2$ has positive potential $(1+x^2{)}^2$ for real $x$, yet its complex vacua $\phi =\pm i$ have zero energy. Supersymmetry breaking must therefore be tested in the full complex field space, not inferred from a positive minimum on an arbitrarily chosen real slice.

If each $m$ admits $p,q$ with $pmq=p$, then $B^A$ is decidable whenever the left M-set $B$ is. For equivariant $f,g$, equality of all values at $(1,a)$ implies equality at $(m,a)$: apply $p$ and use $f(pm,pa)=pm\cdot f(1,qa)$, then cancel the injective action of $p$ on $B$. Equality after the exponential action of $m$ then gives equal traces by cancelling its action on $B$.

For the free monoid on $x,y$, act on $\mathbb{N}$ by adding word length and on $\{0,1\}$ trivially. Both actions are injective. The equivariant function $f(w,n)$ indicating words of length strictly greater than $n$ ending in $x$ is nonzero, but $y\cdot f=0$. Thus the exponential of monoid sets is not decidable. Strict inequality ensures invariance when the original word is empty.

A unital quantum channel mixes its input spectrum with a doubly stochastic matrix, so its output spectrum satisfies the majorization relation with its input spectrum. The reverse inequality is generally false: a pure state can be sent to a maximally mixed state. The direction expresses the fact that unital noise can make eigenvalues more uniform. Equality of the two spectra is possible, for example under a unitary channel.

If two smooth plane conics over $\mathbb{C}$ meet at four distinct points, their tangent incidence curve is a smooth projective genus one curve. Projection to $C_2$ is a double cover branched at those four intersections. The tangent equation becomes a quadratic whose discriminant cuts out $C_1\cap C_2$; its four simple zeros make the cover smooth and connected. The Riemann-Hurwitz formula then gives genus one.

The Drude model relative permittivity is $1-{\omega }_p^2/[\omega (\omega +i/\tau )]$. At frequencies large compared with the collision rate but below ${\omega }_p$, its leading real part is negative and the physical spatial wave-number branch has positive imaginary part. The field therefore decays into the conductor; close to the threshold the full complex response must be retained.

Using the bundle-valued conjugate-linear Hodge star, the formal adjoint of the holomorphic bundle's Dolbeault operator is ${\bar{\partial }}_E^{\ast }=-{\ast }_{E^{\ast }}{\bar{\partial }}_{E^{\ast }}{\ast }_E$. It lowers antiholomorphic degree. Stokes applied to the coefficient-contracted form $\psi \wedge {\ast }_E\eta$ proves the integrated adjoint identity on a compact complex manifold.

For closed subspaces of a Hilbert space, set $T=P_{V^{\perp }}{|}_W$. The two positive cosines bound $T$ and its adjoint operator $P_W{|}_{V^{\perp }}$ below. The first bound gives a closed range and injectivity; the second makes the range dense by image-kernel orthogonality for an adjoint. Thus $T$ is a bounded bijection, and $w=T^{-1}{P}_{V^{\perp }}f$ yields the unique direct sum decomposition $f=w+v$. In equal finite dimensions, a lower bound on $T$ alone suffices by the rank-nullity theorem.