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A real nonnegative normalized filter $w$ defines $g_Z=\int w(t)e^{itH}{h}_Z{e}^{-itH}dt$. In an energy eigenbasis, it multiplies each matrix element by $\hat{w}(E_i-E_j)$. A Fourier cutoff at the spectral gap removes couplings between a unique ground state and excitations. The filtered terms still sum to $H$, because the total Hamiltonian is unchanged by its own unitary time evolution. Each term commutes with the ground projector, but need not be minimized there.

A stoquastic Hamiltonian is a real Hermitian Hamiltonian operator with nonpositive off-diagonal entries in a specified basis, usually the computational basis. The property depends on that basis. Permutation matrices in a quantum circuit's propagation terms and $|-⟩⟨-|$ input penalties give a useful family of such operators.

The operator support is the set of subsystems on which an operator may act nontrivially; it acts as the identity outside that set. Unitary time evolution can enlarge the support, while a Lieb-Robinson bound controls the error of replacing the evolved operator by one with smaller support. The support diameter and the number of sites in the support are different quantities.

An axisymmetric cylindrical membrane deformation is invariant under rotations about the cylinder axis, so its radius depends only on the axial coordinate. Its surface area is $2\pi \int R\sqrt{1+R_z^2}dz$, and its enclosed volume is $\pi \int R^{2}dz$. These scalar shape constraints are the basis of the fixed-volume capillary spectrum of a cylindrical membrane.

For a tension-dominated cylindrical lipid vesicle of fixed length $L$ and volume, a real axisymmetric sine amplitude $u_q$ has stiffness $K_q=\pi \sigma L[(q{R}_0{)}^2-1]/R_0$. Fixed volume requires a mean-radius shift $-u_q^2/(4R_0)$. Stable modes have $⟨u_q^2⟩=k_{B}T/K_q$ by the equipartition theorem. Modes with $q{R}_0<1$ have the Rayleigh–Plateau instability, rather than a negative equilibrium variance. Bending modifies very short wavelengths and near-marginal modes.

The exterior of a link is the complement of the interior of a closed tubular neighborhood of the entire link. Its boundary consists of one torus per component; its first homology has the component meridians as a free basis.

For a link with at least two components, this polynomial is the greatest common divisor of the appropriate maximal minors of an abelianized Alexander matrix, with one variable per oriented component meridian. It is defined up to multiplication by a signed monomial. For a deficiency-one presentation with $g$ generators and $g-1$ relators, use its $(g-1)$-row minors. This normalization gives one for a Hopf link.

For a perturbation $h_{xx}=2a(t-z)$, $h_{yy}=2b(t-z)$ with all other components zero, the linearized Vacuum Einstein equations give $(a+b{)}^{\prime \prime }=0$. Put $s=a+b$ and choose ${\xi }_x=sx/2$, ${\xi }_y=sy/2$, ${\xi }_t=-s^{\prime }(x^2+y^2)/4$, ${\xi }_z=s^{\prime }(x^2+y^2)/4$. Under the linearized coordinate gauge transformation, the mixed time/transverse and longitudinal/transverse components cancel, while the other time/longitudinal changes are proportional to $s^{\prime \prime }=0$. The remaining components are $h_{xx}=a-b$, $h_{yy}=-(a-b)$, $h_{xy}=0$, a transverse-traceless gauge perturbation of plus polarization. This explicitly removes a nonzero affine trace without assuming it vanishes by boundary conditions. If $(a-b{)}^{\prime \prime }=0$ too, all linearized curvature vanishes.

Linear instability is growth of small disturbances in the linearized dynamics. An eigenmode with positive temporal growth rate is sufficient to establish it. In a non-normal system, finite transient amplification is a different issue from the existence of an exponentially growing eigenmode. For normal modes written as $e^{i\alpha (x-ct)}$, the temporal rate is $\alpha \mathrm{Im}c$.

A sideband instability is growth of disturbances with wavenumbers slightly displaced from a carrier pattern's wavenumber. A real physical perturbation uses the displaced mode together with its complex conjugate; a complex amplitude equation can couple those two sidebands through its cubic term. The Eckhaus instability is a longitudinal phase-modulation example.

A conservative equilibrium with one real pair of linear eigenvalues $\pm \lambda$ and one imaginary pair $\pm i\omega$ has saddle and centre directions. Generic perturbations have an exponentially growing component, even though finely chosen initial conditions can remain in the centre-stable subspace. The local L2 Lagrange point dynamics is an example.

When the transfer function is $(1+\theta z)/((1-rz)(1-sz))$ with distinct $r,s$ of modulus less than one, partial fractions give coefficients ${\psi }_j=a{r}^j+d{s}^j$, where $a+d=1$ and $-as-dr=\theta$. White-noise orthogonality gives covariance at lag $h\ge 0$ as ${\sigma }^2[a^2{r}^h/(1-r^2)+d^2{s}^h/(1-s^2)+ad(r^h+s^h)/(1-rs)]$. This combines stable recursion with a closed geometric-sum covariance.

Nonempty disjoint linear polyhedra $P=\{x:Ax\le b\}$ and $Q=\{y:Cy\le d\}$ admit $h$ with a strictly positive separation gap. A phase-I linear program minimizing common constraint violation has positive attained optimum. Its Lagrangian dual problem gives nonnegative $\lambda ,\mu$ with $A^T\lambda +C^T\mu =0$ and ${\lambda }^{T}b+{\mu }^{T}d<0$. Then $h=A^T\lambda$ satisfies $h^{T}x\le {\lambda }^{T}b<-{\mu }^{T}d\le h^{T}y$. The polyhedral hypothesis matters; arbitrary disjoint closed convex sets need not have a positive separation gap.

The second moment of area of a cross-section about an axis is $I=\int y^{2}dA$, where $y$ is the perpendicular distance from that axis. Its dimensions are length to the fourth power. For a circular section of radius $a$, polar integration gives $I={\int }_0^a{r}^{3}dr{\int }_0^{2\pi }{\sin }^2\theta d\theta =\pi a^4/4$. A slender rod with Young's modulus $E$ has filament bending modulus $B=EI$: the axial strain caused by curvature $\kappa$ is $-y\kappa$, and integrating the linear elasticity energy density $E{y}^2{\kappa }^2/2$ over the section gives $B{\kappa }^2/2$ per unit length.

The Euler-Bernoulli beam equation models transverse linear bending waves in a slender beam. The normalized equation has $c_b=1$ and may be reduced by the Schrodinger factorization of the elastic beam equation. Prescribing displacement and curvature at an endpoint supplies two boundary traces for its fourth spatial derivative.

An elastic membrane is a thin deformable sheet whose transverse restoring force is supplied by in-plane tension. For uniform tension $T$ and mass per area $m$, a two-dimensional small-displacement model has $m{\eta }_{tt}=T{\eta }_{xx}-[p]$, where the pressure jump is defined from the lower fluid to the upper fluid. Fluid loading modifies its traveling modes through the acoustic wave on a tensioned massive membrane.

The dimensional linear friction coefficient relates drag force and velocity, $F_d=-\zeta v$, and has units of force divided by speed. It differs from the dimensionless aerodynamic drag coefficient. For an isolated sphere in Stokes flow, the Stokes drag law gives $\zeta =6\pi \eta a$.

When momentum relaxes much faster than the imposed motion, force balance replaces the inertial equation: $\zeta \dot{x}=F(x,t)$ in a deterministic one-dimensional model. The linear friction coefficient sets the time scale. Thermal forcing, if included, gives overdamped Langevin dynamics.

If $a,b$ are nonzero integers, $d=\gcd (a,b)$, and $(x_0,y_0)$ is one solution of the linear Diophantine equation $ax+by=c$, then all solutions areSubtract two solutions and use the fact that $a/d$ and $b/d$ are coprime integers. The linear Diophantine equation solvability criterion says that a particular solution exists exactly when $d$ divides $c$; the extended Euclidean algorithm constructs one. This parametrization separates existence from the complete family of answers.

When the characteristic exponents of the density-mode Euler-Cauchy equation coincide, the second independent solution gains a logarithm. In the polytropic matter-era example the complete solution at this threshold is $t^{-1/6}(A+B\log t)$.