The constant mode of the worldline einbein is not removed by gauge transformations whose parameters vanish at the ends of a fixed interval. This Schwinger proper-time parameter remains integrated over after gauge fixing. For $\phi =(p^2+M^2)/2$, the geometric on-shell proper length is $M\int edt$.

For $u_t=u_{xx}+\beta u_x$ with $u_x(0,t)=g(t)$ and $u_x(L,t)=h(t)$, two applications of integration by parts against a Neumann eigenfunction give the boundary forcing $e^{\beta L}{\varphi }_k(L)h-{\varphi }_k(0)g$. Solving the resulting modal linear differential equation givesThe endpoint derivatives are interior limits: the short-time singularity prevents evaluating each homogeneous boundary derivative before the time integral and infinite sum. The constant eigenfunction gives the weighted-mass law ${\partial }_t{\int }_0^L{e}^{\beta x}udx=e^{\beta L}h-g$.

For a locally uniformly elliptic operator $L=a^{ij}{D}_{ij}+b^i{D}_i+c$ with locally bounded coefficients and $c\le 0$, a $C^2$ function satisfying $Lu\ge 0$ on a connected domain cannot attain a nonnegative global maximum inside unless it is constant. If $c=0$, the maximum can have either sign. If the maximum set were proper, choose an interior ball in its complement tangent to that set. The Hopf boundary point lemma gives a nonzero normal derivative at the contact point, contradicting the vanishing gradient at an interior maximum.

The strong minimum principle for elliptic operators follows by applying this to $-u$. For nonnegative $u$ with $Lu\le 0$, replacing $c$ by $\min (c,0)$ retains the inequality, so the strict positivity-or-zero conclusion does not require an original sign restriction on $c$.

If $c\le 0$, $\Delta u+cu=f$, and $u=\psi$ on the boundary of a bounded domain inside $\{|x_1|<d\}$, let $M={\sup }_{\partial \Omega }|\psi |$ and $F={\sup }_{\Omega }|f|$. The nonnegative barrier $v=M+(e^{2d}-e^{x_1+d})F$ satisfies $(\Delta +c)v\le -F$. Apply the weak maximum principle for elliptic operators to $u-v$ and $-u-v$ to get $\Vert u{\Vert }_{\infty }\le M+e^{2d}F$.

For the massless common vector-axial coupling at both vertices in weak charged-current quark scattering, subtracting opposite directions givesIt is undefined for $P=Q=0$. Overall rate normalization cancels. Pure vector or axial interactions have zero asymmetry, while $P=\pm Q$ gives its maximal coefficient. The inequality $4P^2{Q}^2\le (P^2+Q^2{)}^2$ implies $|A|\le 1$. This unpolarized observable does not distinguish purely left- from purely right-handed couplings applied at both vertices.

A nondispersive wave has constant phase velocity on a propagation branch. For $\omega =ck$, its group velocity is also $c$. A frequency relation with a nonzero offset, such as $\omega =ck+b$, has constant group velocity but a phase velocity $c+b/k$ depending on wavenumber; constant group velocity alone is therefore not sufficient for nondispersion.

This matrix has $H_{sr}=1$ when route $r$ serves source-sink pair $s$, and zero otherwise. Thus $Hx=f$ aggregates route flows into source-sink demands.

For route flows $x\ge 0$, let the link-route incidence matrix $A$ give link loads $y=Ax$. Let the source-sink route incidence matrix $H$ enforce fixed demands $Hx=f$. With continuous nondecreasing link delays $D_j$, the Beckmann potential isIts route derivative is ${\partial }_{r}V={\sum }_j{A}_{jr}{D}_j(y_j)$, precisely that route's delay. Convexity follows from monotonicity of each $D_j$. The Karush-Kuhn-Tucker conditions therefore say that used routes have the minimum delay for their source-sink pair, while unused routes have no smaller delay: the Wardrop equilibrium conditions.^[ref]

This potential generally differs from total travel delay ${\sum }_j{y}_j{D}_j(y_j)$. For two parallel links with delays $D_1(y)=y$, $D_2(y)=1$ and total demand two, minimizing $V=x_1^2/2+x_2$ gives $x_1=x_2=1$. Both used routes have delay one. Minimizing total delay $x_1^2+x_2$ instead gives $x_1=1/2$, $x_2=3/2$. The marginal external cost toll aligns the two optimization problems.

In an elastic-demand Wardrop equilibrium, demand $f_s$ responds to the minimum route delay ${\lambda }_s$ through $f_s=B_s({\lambda }_s)$. An inverse demand function supplies the utility term that must be subtracted from the Beckmann potential.

For $u_t=u_{xx}+\alpha u_x$ on the whole line, apply centered differences in space and the Forward Euler method in time. With $r=\Delta t/(\Delta x{)}^2$ and $c=\alpha \Delta t/\Delta x$, the amplification factor isPutting $s={\sin }^2(\theta /2)$ givesFor $\Delta t>0$, the affine expression in braces is nonpositive on $[0,1]$ exactly when $r\le 1/2$ and $c^2\le 2r$. Hence von Neumann stability analysis givesThese conditions are weaker than requiring every stencil weight to be nonnegative; stability in the discrete L2 norm need not imply monotonicity.

For a two-level Fourier recurrence ${\hat{u}}^{n+1}=a(\theta ){\hat{u}}^n+b(\theta ){\hat{u}}^{n-1}$, the amplification matrix isStability requires its powers to be uniformly bounded in the time index and the mesh frequencies. If its two amplification roots have modulus at most one and their separation has a positive mesh-independent lower bound, the eigenvectors $({\xi }_j,1{)}^T$ give a uniformly bounded diagonalization of a matrix, proving stability. A repeated unit-modulus root of this companion matrix instead produces a Jordan block and linear growth in time. Thus checking only the moduli of the roots is insufficient.

For a translation-invariant difference operator $(Lu{)}_m={\sum }_j{a}_j{u}_{m+j}$, its Fourier symbol is $\ell (\theta )={\sum }_j{a}_j{e}^{ij\theta }$. Applying $L$ to a Fourier mode $e^{im\theta }$ multiplies that mode by $\ell (\theta )$. Symbols turn constant-coefficient stencil equations into scalar algebraic equations and expose their von Neumann stability analysis.

CRIRES is a high-resolution infrared spectrograph. Its upgraded form supports molecular spectroscopy of exoplanet atmospheres and measurements based on the Doppler effect. See ESO's instrument overview.

A vector bundle morphism is an isomorphism if it admits an inverse of the same kind. A smooth fiberwise bijective bundle morphism automatically has a smooth inverse, because inverse matrices in vector bundle trivializations depend smoothly on their entries wherever their determinants are nonzero.

The vector area of an oriented surface is the integral of its unit normal vector times its area element. If its oriented boundary is a closed curve $C$, Stokes theorem givesConsequently it depends only on the oriented boundary, not on the spanning surface. For a planar surface it is signed area times a unit normal vector. A current loop has magnetic dipole moment $\mathbf{m}=I\mathbf{S}$; the scalar area of a nonplanar spanning surface is not interchangeable with this vector.

The total variation norm of a finite signed or complex measure is the total mass of its variation measure. Under the Riesz-Markov-Kakutani representation theorem, it equals the norm of the corresponding member of $C(K{)}^{\ast }$.

The Van der Waals equation becomes independent of its material parameters when pressure, temperature and molecular volume are divided by their critical values: $\bar{p}=8\bar{T}/(3\bar{v}-1)-3/{\bar{v}}^2$. This explains the common reduced isotherms within this model, without claiming exact agreement for all real fluids.

For $0<\alpha <\beta$, choosing global minimizers of $D+\alpha J$ and $D+\beta J$ gives $J(u_{\beta })\le J(u_{\alpha })$ and $D(u_{\beta })\ge D(u_{\alpha })$. The decrease in penalty and increase in data misfit satisfy $\alpha \Delta J\le \Delta D\le \beta \Delta J$.

For nonnegative integer $k$ and a finite-index subgroup $\Gamma$, a nonzero modular form has order at infinity, measured in a genuine periodic cusp parameter, at most $k[S{L}_2(\mathbb{Z}):\Gamma ]/12$. The product of its slash translates over the right cosets is a nonzero level-one form; the translates along the infinity-cusp orbit contribute its full local order to this product. The valence formula for the modular group gives the bound. Initial coefficients through that bound therefore give an injective linear map.

A weight-two modular form obeys $f(i)=i^{2}f(i)=-f(i)$, so it vanishes at $i$. The valence formula for the modular group would then have left side at least $1/2$, but right side $2/12=1/6$. All orders are nonnegative by holomorphy, giving a contradiction for nonzero $f$.