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For a bounded self-adjoint operator $L$, a spectral Weyl sequence at $\lambda$ consists of unit vectors $f_n$ with $\Vert (L-\lambda I)f_n\Vert \to 0$. Such a sequence exists exactly at points of the spectrum of a bounded operator. If no sequence exists, the shifted operator is bounded below, with closed range and zero kernel. Self-adjointness then makes its range dense, so it is invertible. This spectral criterion is different from the equidistribution Weyl criterion.

The local axisymmetric density/velocity system of a pressured dust layer with linear gas drag and a fixed Keplerian shearing sheet gas flow has this cubic dispersion relation. The $\Omega /2$ azimuthal velocity coefficient includes advection of the Keplerian shear. The razor-thin disk Poisson kernel supplies the $|k|$ self-gravity term. Keeping the full amplitude determinant retains the zero-frequency branch and avoids dividing out the secular mode.

The objective $\Vert u-g{\Vert }_2+\lambda {\sum }_i((Gu{)}_i{)}_{+}^2$ has the exact lift $\min t+\lambda {\sum }_i{q}_i$ with $(t,u-g)\in {\mathcal{Q}}_{n+1}$, $((q_i+1)/2,(q_i-1)/2,a_i)\in {\mathcal{Q}}_3$, $a_i\ge 0$ and $a_i\ge (Gu{)}_i$. Every lifted feasible value bounds the original objective, and choosing tight epigraph variables proves equality. The product-cone barrier parameter is $4n+2$: two for each Lorentz block and one for each orthant slack. The fidelity norm remains unsquared, and negative derivatives remain unpenalized.

For a ring torus of radii $a>b>0$, the Gaussian curvature isIt is positive on the outer side, negative on the inner side and zero on the top and bottom circles. The sign contrast expresses two principal curvatures of the same sign outside and opposite signs inside.

If $X_t=c_t{\epsilon }_t$ with a deterministic periodic scale $c_t=c_{t-S}$ and strong white noise of variance ${\sigma }^2$, then ${\Delta }_S{X}_t=c_t({\epsilon }_t-{\epsilon }_{t-S})$. Its variance is $2{\sigma }^2{c}_t^2$, still seasonal when $c_t^2$ varies. A periodic scale model or variance standardization is more appropriate than blindly applying differencing.

Two equally modulated charge sheets in identical full-space electrolyte have charging energy per area ${\alpha }^2[1+e^{-pd}\cos \theta ]/(4\epsilon p)$, where $p=\sqrt{k^2+{\kappa }_D^2}$. Its phase-dependent interaction favors a half-wavelength offset, with opposite charge patches facing each other. For an electrolyte gap with negligible exterior displacement response, the energy is instead ${\alpha }^2[\coth (pd)+\cos \theta /\sinh (pd)]/(2\epsilon p)$, with the same minimizing phase. Absolute energies require specified exterior electrostatic interface boundary conditions.

Take $a=3\pi /2$, $L=10$ and $u(x)=a\exp (-|x{|}^2/L^2)$ on ${\mathbb{R}}^3$. Set $f=-\Delta u+u+\sin u$. Both functions are smooth and belong to L2 space, and $u\in H^1$. The strictly increasing reaction $h(s)=s+\sin s$ has $0\le h(u)\le h(a)=a-1$, whileConsequently $\Vert f{\Vert }_{\infty }\le a-1+6a/L^2<a=\Vert u{\Vert }_{\infty }$. This $u$ is a minimizer of the screened sine-Gordon energy: it solves the Euler-Lagrange equation and that energy is convex. Thus even the minimizing and smooth hypotheses do not justify the stronger source-size bound. The appropriate general estimate is maximum bound for a monotone reaction term.

A Schwartz-Bruhat function on a non-Archimedean local field is a locally constant, compactly supported complex-valued function, or an element of the Schwartz-Bruhat space. A sufficiently small open additive subgroup leaves it invariant under translation, and only finitely many cosets meet its support.

Among Loewner chains with capacity $2t$, continuous real driver and initial driver zero, scale invariance and the centered domain Markov property of a chordal Loewner chain characterize Schramm–Loewner evolution with some $\kappa \ge 0$. Driving-function reconstruction for a chordal Loewner chain turns the hull properties into stationary independent increments and Brownian scaling for the driver. The continuous Lévy process classification then gives $U_t=bt+\sqrt{\kappa }{W}_t$, and scaling eliminates $b$. Regularity and parametrization are essential hypotheses of this characterization.

A scaled Schoenfeld residual rescales a Schoenfeld residual using fitted information so that its trend can diagnose departures from a constant coefficient in a Cox proportional-hazards model. A smooth trend against transformed event time estimates a time-dependent coefficient pattern; a flat trend supports the working proportional hazards assumption test.

If the risk set has two subjects with covariate zero and one with covariate one, its hazard-weighted mean is $e^{\beta }/(2+e^{\beta })$. A zero-covariate event has Schoenfeld function $-e^{\beta }/(2+e^{\beta })$, while a one-covariate event has function $2/(2+e^{\beta })$. At the true coefficient their probabilities are $2/(2+e^{\beta })$ and $e^{\beta }/(2+e^{\beta })$, so the conditional expected function is zero.

For a reflectionless scattering wavefunction whose two asymptotic phases differ by $\delta (k)$, a large periodic box gives $k_{n}L+\delta (k_n)=2\pi n$. Compared with the free value $k_n^{(0)}=2\pi n/L$, $k_n-k_n^{(0)}=-\delta (k_n^{(0)})/L+O(L^{-2})$ away from thresholds. Discrete bound states and phase-branch changes alter mode indexing and must be counted when comparing complete spectra.

For a barrier of radius $a$ and potential ${\hslash }^2{\gamma }^2/(2m)$, matching the regular inner hyperbolic-sine wave to the outer sine wave gives $a_s=a-\tanh (\gamma a)/\gamma$. The low-energy $s$-wave scattering amplitude is $-a_s$.

For a scalar quartic Landau free energy $r_{t}t{m}^2/2+u{m}^4/4-hm$ with $r_t,u>0$, rescaling $m=(r_t/u{)}^{1/2}|t{|}^{1/2}\psi$ gives this minimized scaling form, with $a=r_t^2/u$ and $b=\sqrt{u}/r_t^{3/2}$. The two functions minimize $\pm {\psi }^2/2+{\psi }^4/4-H\psi$. In particular $f_{+}(0)=0$ and $f_{-}(0)=-1/4$. Below the transition the equilibrium field dependence has a cusp at zero; pure-phase derivatives are one-sided. The form gives order-parameter critical exponent $\beta =1/2$ and magnetic-susceptibility critical exponent $\gamma =1$.

A scalar spheroidal harmonic is a regular angular solution of $(\sin \theta S^{\prime }{)}^{\prime }/\sin \theta +[c^2{\cos }^2\theta -m^2/{\sin }^2\theta +\Lambda ]S=0$; with $e^{im\varphi }$ it is smooth at both axes. For real $c$, regularity gives discrete eigenvalues ${\Lambda }_{\ell m}(c)$. At $c=0$ it reduces to the angular part of a spherical harmonic, with $\Lambda =\ell (\ell +1)$. The sign of the $c^2$ term distinguishes the convention used here.

The Gaussian contraction used while integrating out a momentum shell. Its Fourier support lies entirely in that shell. In particular, convolution with a purely low-momentum field is zero. This distinguishes shell Feynman diagrams from unrestricted loop diagrams.

A smooth-cutoff scalar propagator is the inverse of a positive regulated quadratic kernel. It agrees with the unregulated scalar propagator at low momentum and decreases rapidly at high momentum. Smooth suppression is not identical to vanishing support. Its cutoff derivative is the line weight in an exact renormalization-group flow.

For $N_i=a^2{B}_{,i}$, ${\gamma }_{ij}=a^2[(1-2\Phi ){\delta }_{ij}+2E_{,ij}]$ and $K_{ij}=-({\dot{\gamma }}_{ij}-D_i{N}_j-D_j{N}_i)/(2N)$, put ${\chi }_{+}=a^2(B-\dot{E})/\bar{N}$. Then $K^i{}_j=-H{\delta }^i{}_j+(\dot{\Phi }/\bar{N}+H\Psi ){\delta }^i{}_j+{\partial }^i{\partial }_j{\chi }_{+}$ and $\kappa =3(\dot{\Phi }/\bar{N}+H\Psi )+\Delta {\chi }_{+}$. Defining $\chi =-{\chi }_{+}$ instead makes the trace-free tensor coefficient negative and the trace gradient term $-\Delta \chi$. A sign change must affect both pieces.

For smooth compactly supported data in the scalar conservation law $u_t+f(u{)}_x=0$, put $m={\min }_{\xi }{f}^{\prime \prime }(u_0(\xi ))u_0^{\prime }(\xi )$. Its smooth characteristic solution of a scalar conservation law exists until $t_{\ast }=-1/m$ if $m<0$, and for all times if $m\ge 0$. The characteristic flow map $X_t(\xi )=\xi +t{f}^{\prime }(u_0(\xi ))$ has derivative $1+t{f}^{\prime \prime }(u_0)u_0^{\prime }$; its first zero produces gradient blowup. The formula applies to smooth fluxes of either concavity, although it is particularly useful for a concave function flux.

Before characteristic crossing, the scalar conservation law $u_t+f(u{)}_x=0$ has solution $u(t,X_t(\xi ))=u_0(\xi )$ with $X_t(\xi )=\xi +t{f}^{\prime }(u_0(\xi ))$. When this map is a diffeomorphism, the solution is $u_0\circ X_t^{-1}$. Its spatial derivative is $u_0^{\prime }(\xi )/(1+t{f}^{\prime \prime }(u_0(\xi ))u_0^{\prime }(\xi ))$.