On unit-capacity links $1,\dots ,I$, route zero uses every link and route $i$ uses only link $i$. Let $N={\sum }_{r=0}^I{n}_r$ and $M={\sum }_{i=1}^I{n}_i$. Under proportional fairness, every active local route has $n_0{x}_0+n_i{x}_i=1$. Maximizing $n_0\log w_0+M\log (1-w_0)$ for $w_0=n_0{x}_0$ gives $w_0=n_0/N$ and the displayed per-flow rates on active routes. At the empty state departures vanish, and rates of absent flows may be set to zero.

For the proportionally fair allocation on a linear flow network, independent Poisson flow arrivals at rates ${\nu }_r$ and exponential document sizes of rates ${\mu }_r$ give a continuous-time Markov chain with departures ${\mu }_r{n}_r{x}_r(n)$. Put ${\rho }_r={\nu }_r/{\mu }_r$. Under ${\rho }_0+{\rho }_i<1$ for each local route, the normalizing constant is $Z=(1-{\rho }_0{)}^{I-1}/{\prod }_i(1-{\rho }_0-{\rho }_i)$. The binomial coefficient weight has adjacent-state ratios equal to the aggregate service rates, proving detailed balance. For more general network topologies, proportional fairness need not give this reversible product form.

Under the hypotheses of the limit form of the special adjoint functor theorem, intersect all subobjects of $C$ supporting $x:B\to GC$. Limit preservation gives a smallest supporting $(C_0,x_0)$. If $G(u)x_0=G(v)x_0$ for maps $u,v:C_0\to Q_i$, their equalizer supports $x_0$, so minimality makes it invertible and $u=v$. Thus the indicated map of hom-sets is injective. The evaluation embedding into cogenerator products embeds $C_0$ in a product of cogenerators indexed by the realized subsets of the fixed sets $\mathcal{D}(B,G{Q}_i)$. There are only a set of such products, a set of their subobjects, and a set of maps from $B$ into their images under $G$. These data give a weakly initial set in $(B\downarrow G)$. Using realized subsets avoids assuming maps into every cogenerator exist.

Fixing nonzero longitudinal oscillators leaves the center-of-mass symplectic term, transverse oscillator terms, and a zero-mode multiplier imposing the mass-shell condition. On a patch with $p^{+}\ne 0$, further choosing $x^{+}$ as time solves that constraint for $p^{-}=[p_{\perp }^2+(N-a)/{\alpha }^{\prime }]/(2p^{+})$. This distinguishes partial oscillator gauge fixing from the subsequent time choice.

For a test particle interior to a distant eccentric planet, write $\alpha =a/a_p\ll 1$. The leading Laplace coefficients are $b_{3/2}^1=3\alpha$ and $b_{3/2}^2=15{\alpha }^2/4$. Thus its forced eccentricity is $Z_f=5\alpha e_p/4$ and its secular frequency is $A=(3/4)n_p(m_p/M){\alpha }^{3/2}$. More distant inner particles have larger forced orbital eccentricities and faster secular precession, producing a growing gradient of complex eccentricity.

Exterior multiplication $h_j(u)=e_j\wedge u$ is a chain homotopy between multiplication by $f_j$ and zero on a Koszul complex. A linear combination of these homotopies is a contracting homotopy when the generators span the unit ideal.

If $T$ is a normal splitting Suslin tree and $P$ is Knaster, then $P\times T$ is CCC, so $P$ forces that the ground tree is still CCC. Countable levels and splitting persist. A new cofinal branch would give an uncountable antichain by splitting, so none is added. This permits adding many Cohen reals while retaining a Suslin tree, and distinguishes Knaster preservation from an unjustified preservation claim for every CCC forcing.

The square is a rank-two Killing tensor. It is symmetric because it pairs the covectors $Y_{a\cdot }$ and $Y_{b\cdot }$ using the metric. Along every affinely parametrized geodesic, its contraction $K(v,v)$ is the squared norm of the carried by parallel transport covector $Y(v,\cdot )$. Metric compatibility makes that norm constant, proving the Killing tensor equation. On a Riemannian manifold this tensor is positive semidefinite.

For an $L^2$ kernel and an independent sample from a density $f$, the centred kernel density estimator hasindependence removes cross terms and Tonelli theorem integrates the pointwise variance. Young's convolution inequality ensures $K_h\ast f\in L^2$ even without $f\in L^2$. The Cauchy-Schwarz inequality yields the expected-norm bound $\mathbb{E}\Vert {\hat{f}}_{n,h}-K_h\ast f{\Vert }_2\le \Vert K{\Vert }_2/\sqrt{nh}$.

Two integrations of the steady Keplerian viscous diffusion equation give this general transported quantity. $A$ fixes the mass accretion rate, while $B$ fixes the additive angular-momentum flux or inner torque. A zero-torque inner boundary condition gives $B=-A\sqrt{r_{\mathrm{in}}}$; a nonaccreting constant-torque disk has $A=0$.

For a steady zero-mass-flux background and constant viscous transport response exponent, this change removes the first derivative in the radial diffusion operator. If ${\Sigma }_0\propto r^{-p}$, the kinematic viscosity scales as $r^{p-1/2}$, so the transformed diffusivity scales as $x^{2p-3}$. It is constant at $p=3/2$.

In any cyclic ordering on $n$ points, an antichain contains at most $n$ nonempty proper cyclic intervals. The intervals with one prescribed final position are nested, so at most one belongs to the antichain. Summing over final positions proves the bound. Averaging it over cyclic orderings proves the LYM inequality; the empty set and full set must be handled separately.

For infinite regular cardinals $\kappa <\lambda$, a $\lambda$-tree whose levels all have size less than $\kappa$ has a cofinal branch. At levels of cofinality $\kappa$, bound all heights distinguishing distinct predecessor chains. The Fodor lemma makes this bound constant on a stationary set; one predecessor at that height occurs stationarily often. The resulting predecessor chains agree below their common heights and unite into a branch through every level. Distinct limit-level nodes with identical predecessor chains do not affect this argument.

For a separated angular net, two translated unit rectangles of width $\delta$ intersect in area at most a constant times ${\delta }^2/(\delta +\alpha )$, where $\alpha$ is their unoriented angular distance. Each row of their intersection matrix therefore has sum at most $C\delta \log (2/\delta )$. Applying the Schur test to the adjoint averaging operator gives the displayed bound. A wider-tube comparison extends it from the net to all directions. Since $\sqrt{\log (2/\delta )}{\lesssim }_{\epsilon }{\delta }^{-\epsilon }$, this proves the planar version of the Kakeya maximal conjecture.

Use the Kauffman bracket assigning the empty diagram value $1$ and a disjoint circle factor $-A^2-A^{-2}$. Correct by the writhe of a link diagram and substitute $t=A^{-4}$. Relative to the Jones polynomial normalized to be $1$ on the unknot, this unreduced invariant has an additional factor $-(t^{1/2}+t^{-1/2})$. For a nonempty $l$-component link, its value at $t=1$ is $(-2{)}^l$.

For a differentiable scale family with finite positive Fisher information, the score is $-[1+u(\log f{)}^{\prime }(u)]/\sigma$. Its squared expectation is a constant times ${\sigma }^{-2}$, yielding the displayed Jeffreys prior. This is an improper prior on the entire positive half-line.

Choose a topological generator $\gamma$ of the Galois group of a Zp-extension and set $T=\gamma -1$. The completed group algebra becomes the formal power series ring ${\mathbb{Z}}_p[[T]]$, a two-dimensional complete regular local integral domain with maximal ideal $(p,T)$. Its finite-layer quotient is $\Lambda /((1+T{)}^{p^n}-1)\cong {\mathbb{Z}}_p[\Gamma /{\Gamma }^{p^n}]$.

The scalar stochastic differential equation $d{X}_t=\tanh X_{t}dt+d{W}_t$, $X_0=x$, has a global strong stochastic solution and pathwise uniqueness because its coefficients are globally Lipschitz continuous. The Itô formula gives the normalized positive martingaleBoundedness of $\tanh$ verifies Novikov's condition. Under $dQ=Z_{T}dP$, the Girsanov theorem makes $X_t-x$ a Brownian motion through time $T$. The factor $\cosh x$ ensures $Z_0=1$.

The transition density under the original measure isThis is a Doob h-transform with $h=\cosh$. Expanding the hyperbolic cosine gives a mixture of $N(x+t,t)$ and $N(x-t,t)$ with weights $e^x/(2\cosh x)$ and $e^{-x}/(2\cosh x)$, showing directly that the density integrates to one.

The neutral one-dimensional Wright–Fisher diffusion records a proportion in $[0,1]$, with absorbing endpoints, zero drift and generator $Lf(x)=x(1-x)f^{\prime \prime }(x)/2$. Its conditional variance rate is largest near one-half and vanishes at fixation. Mutation or selection variants add drift and must be specified separately.

For the wealth-variable Legendre dual $J(z)={\sup }_{w\ge 0}[V(w)-zw]$, a finite slope $z_{\ast }=V^{\prime }(0+)$ at an absorbing zero-wealth boundary makes $J$ identically zero for $z\ge z_{\ast }$. Matching gives $J(z_{\ast })=J^{\prime }(z_{\ast })=0$. It does not generally give $J^{\prime \prime }(z_{\ast }^{-})=0$: killing at ruin permits the dual curvature to jump.