A collisionless stellar system is one for which discrete gravitational encounters change its distribution negligibly over the time under study, usually $T\ll t_{\mathrm{rel}}$. Stars can orbit and respond collectively to a varying smooth gravitational potential even in this regime; collisionless does not mean force-free or stable against collective gravitational instabilities. Its smooth distribution follows the Collisionless Boltzmann equation.

For a mass $m$ passing at relative speed $u$ and impact parameter $b$, integration of $Gmb/(b^2+u^2{t}^2{)}^{3/2}$ gives the transverse kick $2Gm/(bu)$. Weak deflection requires $b\gg Gm/u^2$, up to the equal-mass order-one factor. This scale supplies a small-impact cutoff for the weak-scattering diffusion approximation, not a physical exclusion of stronger encounters.

In an unsoftened plane with number surface density $n_2$ and relative speed $u$, an encounter strip has area $2udtdb$. Thus the velocity diffusion rate is $8G^2{m}^2{n}_2/u\int db/b^2$, dominated by the small-impact cutoff. Setting this cutoff to $Gm/u^2$ and requiring a change of order $u^2$ gives $t_{\mathrm{rel}}\sim u/(Gm{n}_2)$. With $u=\lambda v$, $n_2\sim N/R^2$ and $v^2\sim GNm/R$, this is of order $\lambda R/v$, independent of $N$ at fixed total mass. The point-particle, zero-thickness limit is singular; fixed softening or nonzero thickness changes it. Numerical coefficients are not controlled by extending weak kicks down to order-one deflections.

For a disk with thickness $z_0$, random relative speeds $u=\lambda v$ and small-impact cutoff $b_0\ll z_0\ll R$, the leading three-dimensional logarithmic encounter term has upper cutoff of order $z_0$. At fixed mass and radius, its number volume density exceeds the spherical density by order $R/z_0$. Hence $t_{\mathrm{disk}}/t_{\mathrm{sphere}}\sim {\lambda }^3(z_0/R)\log (R/b_0)/\log (z_0/b_0)$ when the same cutoff is used for comparison. Different speed-dependent cutoffs must be distinguished in numerical applications. Distant planar encounters contribute a convergent, non-logarithmic correction. This scaling ceases to apply if thickness is comparable to the cutoff.

For the complete $\ell$-uniform hypergraph $K_r^{\ell }$, with $r\ge \ell \ge 2$, the bound is $\pi (K_r^{\ell })\le 1-1/\left(\genfrac{}{}{0ex}{}{r-1}{\ell -1}\right)$. In the complementary covering formulation, every $r$-set containing an edge forces asymptotic edge density at least $1/\left(\genfrac{}{}{0ex}{}{r-1}{\ell -1}\right)$. It follows from clique-extension incidence inequalities and Cauchy-Schwarz inequality estimates on link degrees.

For $v\ne 1$, $K_T(v)$ differs from $1_{v>1}$ by $O(v^c\min (1,(T|\log v|{)}^{-1}))$. Close the contour to the left for $v>1$ and to the right for $v<1$; horizontal sides give the logarithmic denominator and the nearby transition is bounded. At $v=1$, direct integration gives $K_T(1)={\pi }^{-1}\arctan (T/c)=1/2+O(c/T)$. This explains the half-weight endpoint in the truncated Perron formula. The constants may be taken uniform for $1\le c\le 2$ and $T\ge 2$.

For $c=1+1/\log x$, $1\le y\le x$ and $2\le T\le x$, apply the truncated Perron formula with Von Mangoldt function coefficients at both endpoints on the same vertical line. The difference kernel is $((x+y{)}^s-x^s)/s={\int }_x^{x+y}{u}^{s-1}du$, of modulus $O(y)$. The usual near-integer error bound contributes $O(x{\log }^{2}x/T)$. The logarithmic derivative on this line is finite because of the Euler product.

For $f=r{M}^2/2+u{M}^4/4+v{M}^6/6-hM$ with $v>0$, the ordinary critical edge of a tricritical wing satisfies $f^{(1)}=f^{(2)}=f^{(3)}=0$. For $u<0$, $M_c^2=-3u/(10v)$, $r_c=9u^2/(20v)$ and $h_c=6u^2{M}_c/(25v)$. Its fourth derivative is $-12u>0$. The two signs of $M_c$ give the two wings.

Take two nonnegative coexisting minima $a\le b$, put $s=a+b$ and $p=ab$, and fix $v>0$. For the sextic Landau free energy with quadratic, quartic, sextic and linear terms, equal stationary minimum values occur atAt these coefficients,Thus both are global minima. The case $a=0$ gives the zero-field three-phase line, including the negative minimum; the limit $a=b$ gives the tricritical wing critical edge. Reflecting $M,h$ gives the negative-field wing.

For $V=r{m}^2/2+u{m}^4/4+v{m}^6/6-hm$ with $v>0$, the displayed line has three global minima, $m=0$ and $m=\pm \sqrt{-3u/(4v)}$. The factorization $V=(v/6)m^2(m^2+3u/(4v){)}^2$ proves their coexistence. Two nonzero-field tricritical wings and the zero-field ordered coexistence sheet meet along this line. Its order-parameter discontinuity tends to zero as it terminates at the tricritical point.

At fixed leading power of a periodic planar filament, the family $a\sin (mkx+\Omega t)$ has propulsive force of a periodic planar filament increasing with spatial harmonic $m$. The formal quadratic objective is unbounded without a spatial cutoff; large harmonics eventually invalidate the small-slope model. With a finite admissible Fourier series band, force divided by power is a weighted average of the signed ratios $-({\xi }_{\perp }-{\xi }_{\parallel })k_j/({\xi }_{\perp }{\Omega }_j)$. An allowed extremal ratio is attained by a travelling wave mode, or by a superposition with one common phase speed.

The dimensionless accretion factor $(1-q\delta {)}^{q-1/\delta }$ tends to $e^q$ as $\delta \to 0$. For $q>0$, it tends to $1$ as $\delta \uparrow 1/q$, although the sonic point moves to the origin. Writing the latter logarithm as $-qh\ln h/(1-h)$ makes the finite limit transparent. These limits connect transonic spherical accretion rate in a power-law potential to the isothermal equation of state and the limiting critical adiabatic exponent.

Linear steady transport on a translating sphere gives $B=2A{C}_{0}a/(k{a}^2+2D_s)$. The components of its unit normal vector are degree-one spherical harmonics. The condition $2|A|Ua/(k{a}^2+2D_s)\ll 1$ ensures the concentration perturbation is small relative to $C_0$.

The dipolar normal traction is balanced by buoyant hydrostatic pressure and the concentration-dependent normal capillary stress, in addition to the constant gas pressure and equilibrium Laplace pressure. A degree-one shape perturbation translates a sphere rather than deforming it; the balance fixes the terminal speed. A uniform gas pressure alone cannot balance this mode.

In a finite unweighted connected graph, take distinct graph vertices $a\ne z$ and orient the unique graph path from $a$ to $z$ in a uniform spanning tree and assign signed unit current to its edges. Its mean over trees is the electrical unit flow from $a$ to $z$. The Kirchhoff node law follows by averaging graph path divergences. For the graph cycle law, let ${\mathcal{F}}_{az}$ be the two-component spanning forests separating the terminals and $A_F$ the component containing $a$. Deleting a tree-path edge and conversely adjoining an edge across a forest cut are inverse operations. Hence the mean signed current on an edge $u\to v$ isThis is the gradient of $h(u)=|\mathcal{T}(G){|}^{-1}{\sum }_F{1}_{A_F}(u)$, proving the graph cycle law. In particular $R_{\mathrm{eff}}(a,z)=|{\mathcal{F}}_{az}|/|\mathcal{T}(G)|$. For an oriented existing edge $e=(a,z)$, that edge is used by the tree graph path exactly when it belongs to the tree, and then has positive orientation. Thus its mean current is $\mathbb{P}(e\in T)$, giving the edge-inclusion formula for a uniform spanning tree by Ohm's law.

For any orthonormal basis and Hermitian operator, ${\sum }_i|⟨{\psi }_i|X|{\psi }_i⟩|\le \Vert X{\Vert }_1$. In the trace-norm variational principle for Hermitian operators, choose the diagonal $T$ whose entries are the signs of these diagonal entries. This bounds the total distinguishability visible in one fixed measurement basis.

For a nonzero fractional ideal $\mathfrak{b}$ of a number field, its trace-dual lattice is the fractional idealThe equality follows from the definition of the inverse different and invertibility of fractional ideals. Under the positive metric on the Minkowski embedding of a number field, the Euclidean dual lattice of $j(\mathfrak{b})$ is $\overline{j({\mathfrak{b}}^{\vee })}$: coordinatewise complex conjugation converts the positive inner product into the trace pairing. This assertion does not require that conjugation preserve the number field itself.

For an absolutely one-homogeneous functional $J=\mathrm{TV}$ on a real Hilbert space, a subgradient $q$ at $u$ is characterized by $⟨q,v⟩\le J(v)$ for every $v$ and $⟨q,u⟩=J(u)$. A bounded vector field with $|z|\le 1$ and $q=\mathrm{div}z$ proves the inequality through the dual definition, provided cutoff approximation is valid in the ambient space. Equality is called a calibration. It certifies a global minimizer of total variation denoising, rather than only a minimizer within a chosen ansatz.

For a quadratic data term and divergence maps $A_1,A_2$, the dual constraint $A_{1}v\in D$ can be split through ${\delta }_D(A_{1}v)={\sup }_w[⟨w,A_{1}v⟩-{\delta }_D^{\ast }(w)]$. For a product of Euclidean row balls of radius $\beta$, its support function is ${\delta }_D^{\ast }(w)=\beta {\sum }_i\Vert w_i{\Vert }_2$. The resulting saddle coupling is $K(u,w)=A_2^{\ast }u-A_1^{\ast }w$. The Chambolle–Pock algorithm then uses a row-ball projection, a quadratic proximal operator and radial soft thresholding, with no projection onto an intersection involving a divergence operator.

For a torsion-free abelian group $G$, its rational divisible hull is $G{\otimes }_{\mathbb{Z}}\mathbb{Q}$. It consists of fractions $g/n$, with $n>0$ and $g/n=h/m$ exactly when $mg=nh$. The natural map $G\to G\otimes \mathbb{Q}$ is injective. Every homomorphism from $G$ into a torsion-free divisible Abelian group extends uniquely through this hull.