For a differentiable link delay $D_j(y)$, the toll $T_j(y)=y{D}_j^{\prime }(y)$ charges a user for the additional delay imposed on the other users. Its perceived link cost is the derivative of $y{D}_j(y)$. The resulting Beckmann potential equals total travel delay. If total delay is convex, its Wardrop equilibria minimize that delay globally. Strict increase of $D_j$ alone does not imply this convexity; optimal-flow calibrated congestion tolling avoids that extra requirement.

Independent normal arm means with variances $v_0,v_1,v_2$ give treatment-contrast Wald test statistics whose correlation coefficient is $v_0/\sqrt{(v_0+v_1)(v_0+v_2)}$. The common control mean creates positive dependence between comparisons.

For an independent normal sample with known standard deviation $\sigma$, the signed Wald statistic for testing mean zero is $W=\sqrt{n}\bar{Y}/\sigma$. It has the standard normal distribution under the null hypothesis and distribution $N(\sqrt{n}\delta /\sigma ,1)$ under mean $\delta$. Its square has a chi-squared distribution; retaining the sign permits a one-sided test.

Ignoring accessible quark masses and strong corrections, the W boson flavor width is $\Gamma (W^{+}\to q{\bar{q}}^{\prime })=N_c{G}_F{M}_W^3|V_{q{q}^{\prime }}{|}^2/(6\pi \sqrt{2})$. The CKM matrix fixes the flavor factor, while the color multiplicity in a decay width supplies $N_c=3$. Only the up and charm rows are accessible for an on-shell W boson, giving a total hadronic width $G_F{M}_W^3/(\pi \sqrt{2})$ by row unitarity.

A quantum vortex has integer winding number of a condensate's complex argument. For atomic mass $m$, its superfluid velocity gives circulation $2\pi \hslash q/m$ for integer $q$. A straight vortex has a density-depleted core and, outside it, tangential velocity $q\hslash /(mr)$. Its long-range flow energy is proportional to $q^2\log (R/a)$, while its angular momentum has the sign of $q$.

For the periodic-grid recurrencethe exact von Neumann stability analysis range at fixed $\mu >0$ is $\mu \ge 3$. The Fourier symbol gives $A(\theta ){\xi }^2-4\xi +1=0$, $A=3e^{2i\theta }+8\mu {\sin }^2(\theta /2)$. The root near one obeys $|\xi {|}^2=1+(6-2\mu ){\theta }^2+O({\theta }^4)$, proving instability for $\mu <3$.

For $\mu \ge 3$, $\mathrm{Re}A-3=6(\cos \theta -1{)}^2+4(\mu -3)(1-\cos \theta )\ge 0$. Writing the modal recurrence as $3v^{n+2}-4v^{n+1}+v^n=-(A-3)v^{n+2}$, the BDF2 discrete energy identity proves nonincrease of $|v^{n+1}{|}^2+|2v^{n+1}-v^n{|}^2$. Summing modes proves stability uniformly even when $\mu$ varies within $[3,\infty )$.

This shifted scheme is not a consistent standard heat discretization under $k=\mu h^2$: dividing its extra spatial-shift defect by $2k/3$ produces $3h{u}_x/k+O(h^2/k)$. On a finite interval with Dirichlet boundary conditions it also requires an extra boundary closure. Thus the periodic stability theorem is not a theorem of convergence to the heat equation or of stability for an unspecified boundary treatment.

For a uniformly bounded two-level Fourier companion matrix $C(\theta )$ with distinct roots $g_1,g_2$ of modulus at most one, its powers satisfyA positive frequency-uniform gap therefore gives a uniform bound on every power. The Parseval identity then supplies spatial L2 norm stability. When the gap closes at a repeated unit root, a Jordan block can produce growth proportional to the number of steps despite both roots having modulus one.

Integrating the Keplerian accretion disk diffusion equation gives $dJ/dt=6\pi \sqrt{G{M}_{\star }}[r^{3/2}{\partial }_r(\bar{\nu }\Sigma ){]}_{\mathrm{inner}}^{\mathrm{outer}}$ when the angular-momentum integral and boundary limits are legitimate. An inner zero-torque inner boundary condition alone does not eliminate the outer angular-momentum flux. For divergent total $J$, use finite-radius cutoffs before making a conservation claim.

This approximation studies the limit of a viscous scalar conservation law as its positive diffusion coefficient tends to zero. Under hypotheses giving compactness and appropriate convergence, the viscous entropy dissipation selects an entropy solution of the inviscid scalar conservation law. Fixed-phase decreasing travelling waves for a strictly convex flux converge to an entropy shock. Translating each profile differently can change or destroy the limit, so phase or initial data must specify the family.

In a characteristic vertical model of a radiation-pressure-dominated alpha disk with constant opacity, $\mu \sim \alpha p/\Omega \sim c/\kappa$ and $p\sim \Sigma {\Omega }^{2}H$ imply $H\propto {\Sigma }^{-1}$ and $\bar{\nu }\Sigma \propto {\Sigma }^{-1}$. The resulting $d(\bar{\nu }\Sigma )/d\Sigma <0$ gives a viscous instability of an accretion disk, distinct from a thermal instability of a radiation-pressure-dominated alpha disk.

A valuation distribution is regular when its virtual valuation is a nondecreasing function of value. This is an auction-theory condition, not the unrelated regularity notions used elsewhere in mathematics.

For an independent private values model, let $X_i(v)$ and $P_i(v)$ be interim allocation and payment. The interim payment identity gives $P_i(v)=v{X}_i(v)-U_i({\underline{v}}_i)-{\int }_{{\underline{v}}_i}^v{X}_i(t)dt$. Applying Fubini's theorem to reverse the order of integration yields $\mathbb{E}{\int }_{{\underline{v}}_i}^{V_i}{X}_i(t)dt=\int X_i(t)(1-F_i(t))dt$. Subtracting this term from $\int v{X}_i(v)f_i(v)dv$ proves the identity. Interim individual rationality bounds the lowest-type utilities below by zero.

Virtual surplus weights allocation amounts by the corresponding virtual valuations. Under the hypotheses of the virtual-surplus revenue identity, maximizing it subject to implementability and feasibility yields a revenue-optimal mechanism when lowest-type utilities can be normalized to zero.

For a fixed mass under a spherical-overdensity definition, $R_h\propto ({\Delta }_c{H}^2{)}^{-1/3}$ and $v_h^2=G{M}_h/R_h$. Thus equivalent virial mass scales at two epochs have the displayed velocity ratio. With fixed overdensity during matter domination, $v_h\propto (1+z{)}^{1/2}$. This converts an epoch-specific mass–velocity calibration; it does not require a previously collapsed bound halo to physically expand with the later background density.

In units ${\alpha }^{\prime }=1$, generic fixed $t$ gives simple poles at $s=n-1$, $n\ge 0$, with residue $-{\prod }_{j=2}^{n+1}(t+j)/n!$. The empty product is one. A residue zero removes that pole, and fixed $t$ must be away from its own channel singularities.

The Helmholtz decomposition separates a vector field into a gradient field and a field with zero divergence. The precise function spaces, boundary conditions and decay hypotheses determine existence and uniqueness. In the square-integrable bounded-domain formulation, the solenoidal component is selected by the Leray-Helmholtz projection.

A frame over an open set $U$ is a list of smooth sections $e_1,\dots ,e_r$ of a rank-$r$ vector bundle whose values form a basis in every fibre over $U$. It is equivalent to a vector bundle trivialization by $(x,v)\mapsto {\sum }_a{e}_a(x)v^a$. Two local frames differ by a smooth map $g:U\to \mathrm{GL}(r)$, written $e^{\prime }=eg$.

Every $C^{\infty }(M)$-linear map between the modules of smooth sections of finite-rank smooth vector bundles is induced by a unique vector bundle morphism. A smooth bump function shows that the map is local. A bumped local frame then proves that a section vanishing at $x$ has image vanishing at $x$. Evaluating the image therefore defines a well-defined fiberwise linear map. The images of the bumped frame give smooth matrix columns, proving the resulting morphism is smooth. This elementary argument is also described in Brian Conrad's bundle notes.

For complex numbers $|u_n|\le 1$, $0\le n<N$, and $1\le H\le N$, put $C_h={\sum }_{n=0}^{N-h-1}{u}_{n+h}\overline{u_n}$. ThenExtend the sequence by zero, count each summand in $H$ consecutive windows, and apply the Cauchy-Schwarz inequality to their sum. Expansion of the squared window sums gives the stated coefficients. Fixing $H$ and making each normalized correlation tend to zero bounds the limiting squared average by $1/H$; then let $H$ tend to infinity. This is the scalar finite form underlying the Van der Corput lemma.

A finite open coordinate set witnesses an increasing event when prescribing those coordinates open forces the event, regardless of the remaining configuration. The disjoint occurrence of increasing events $A\square B$ means that $A$ and $B$ have disjoint finite witnesses. For finite-coordinate events this is the usual disjoint-occurrence definition. The Van den Berg-Kesten inequality bounds its probability by $\mathbb{P}(A)\mathbb{P}(B)$ under an independent Bernoulli distribution product measure.