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An on-site symmetry of an injective MPS acts on virtual indices by matrices defined up to phase. Their projective cohomology class is the one-dimensional SPT invariant.

For charge $e$, defineIn the symmetric gauge,

Darcy law is a linear relation between the bulk Darcy flux and the net driving force per unit volume. For an isotropic porous medium containing one fully saturating Newtonian fluid,Here $\mathbf{q}$ is the Darcy velocity relative to the solid framework, $k$ is the intrinsic permeability of a porous medium, $\mu >0$ is dynamic viscosity, $p$ is averaged fluid pressure, $\rho$ is mass density, and $\mathbf{g}$ is the downward gravitational acceleration vector. The pressure gradient exerts force density $-\nabla p$, while gravity contributes $\rho \mathbf{g}$; the minus sign makes the discharge follow their sum. The symbols $\mathbf{u}$ and $\mathbf{q}$ are both commonly used for Darcy velocity; neither denotes pore velocity here.

With upward coordinate $z$ and positive gravitational acceleration magnitude $g$, $\mathbf{g}=-g{\mathbf{e}}_z$. Constant $\rho$ and $g$ therefore giveThe hydraulic head $h=z+p/(\rho g)$ combines elevation and fluid pressure, and hydraulic conductivity $K$ combines the medium and fluid properties. Hydrostatic pressure satisfies $\nabla p=\rho \mathbf{g}$, so it produces zero Darcy flux even though fluid pressure increases downward. Conversely, equal pressures at the two ends of a vertical column do not imply zero flow: their elevations give different hydraulic heads.

The appeal of Darcy law is that a complicated connected pore geometry enters a macroscopic calculation through a measurable coefficient. It predicts filter discharge and aquifer flow; combined with mass conservation, it gives an equation for the evolving fluid pressure. Its simple linear form requires a small pore-scale Reynolds number, slow adjustment compared with pore-scale inertial relaxation, and a valid continuum approximation. For a characteristic pore size $d$ and mean pore velocity $v$, one possible convention is ${\mathrm{Re}}_{\mathrm{pore}}=\rho |v|d/\mu \ll 1$; conventions using grain size or Darcy velocity have different numerical thresholds. Merely calling flow laminar does not guarantee negligible inertia. Unsaturated pores, several fluid phases, appreciable inertia, or a fluid whose stress is not Newtonian require further modeling. Woessner and Poeter discuss these limits in their authored hydrology text.^[ref]

Henry Darcy's 1856 book Les fontaines publiques de la ville de Dijon was published by Victor Dalmont; the University Grenoble Alpes digital collection verifies this bibliographic record and supplies a digitized copy.^[ref] Woessner and Poeter's account connects Darcy law to experiments on water flowing through sand filters, relating discharge to cross-sectional area, hydraulic head difference and sample length.^[ref] The modern intrinsic permeability of a porous medium notation above should not be read as a quotation of Darcy's original notation.

Shear-induced dilation is the tendency of a dense granular or suspended-particle packing to increase its volume under shear. In a saturated confined system it requires pore-fluid migration and can delay stress adjustment.

The viscous number $I_v=\eta |\dot{\gamma }|/p_s$ compares viscous shear stress with particle pressure in a dense suspension.

The union of $m$ cones on a nonempty space along one common base is homotopy equivalent to a wedge of $m-1$ suspensions of that space.

Suspension shifts reduced homology: ${\tilde{H}}_n(\Sigma X)\cong {\tilde{H}}_{n-1}(X)$.

When the disease-free equilibrium is unstable, the positive equilibrium satisfiesIts Jacobian has negative trace and positive determinant, so the equilibrium is locally asymptotically stable.

For $b>d$, the disease-free equilibrium isIt is linearly unstable precisely when

For a geodesic in the metric $d{u}^2+Y(u{)}^{2}d{v}^2$, the cyclic coordinate $v$ givesEquivalently, the geodesic meets parallels according to Clairaut's relation.

A circular cylinder of radius $R$ has principal curvatures $0$ and $\pm 1/R$, according to orientation. Hence $K=0$ and $|H|=1/(2R)$.

This is the circular generating-curve case of a Cylinder; the curvature formulas here refer to its lateral surface.

For $X(u,v)=(Y(u)\cos v,Y(u)\sin v,Z(u))$ with $Y^{\prime 2}+Z^{\prime 2}=1$, the induced metric is

Consider the examples of a surface of revolutionwith $u\in \mathbb{R}$ and $s>0$. Their first fundamental forms and Gaussian curvatures areThe diffeomorphism $u=\log \sinh s$ matches Gaussian curvature but does not match the metrics.

To rule out every local isometry, compare the squared norm of the gradient of Gaussian curvature. At matching points put $x=e^{2u}={\sinh }^{2}s>0$. Differentiation givesTheir ratio is $x/(1+x)<1$. A local isometry would preserve both Gaussian curvature, which fixes $x$, and this gradient norm, an impossibility. Checking only one particular diffeomorphism would not exclude a different local isometry.

Forthe area density after integrating over $\theta$ is $2\pi \varphi \sqrt{1+{\varphi }^{\prime 2}}$. If every height interval has area $2\pi r$ times its length, continuity forcesWhere $0<\varphi <r$, the sign of ${\varphi }^{\prime }$ is constant and $\sqrt{r^2-{\varphi }^2}$ has derivative $\pm 1$. Hence the profile lies on a circle of radius $r$.

A catenoid is the surface of revolution obtained by rotating a catenary about its directrix. One conformal parametrization is

A banded-unlink diagram consists of an unlink $U$ and disjoint bands whose simultaneous surgery produces another unlink. Capping the lower and upper unlinks by disks and tracing the band surgeries constructs a surface in $S^4$.

A small-amplitude deep-water wave with surface elevation $\eta =A{e}^{i(kx-\omega t)}$ has dispersion relation ${\omega }^2=gk$ and a velocity potential that decays as $e^{kz}$ below the surface.

A superfield is a function on superspace. Its finite Taylor expansion in Grassmann coordinates packages component fields into a supersymmetry representation.

Active-galactic-nucleus feedback is the transfer of energy and momentum from an active galactic nucleus to surrounding gas. Radiation can heat or accelerate the gas, while astrophysical jets and galactic outflows inject mechanical energy. Heating may offset radiative cooling and reduce the cold material available for star formation or further black-hole accretion; compression can instead favour local star formation. Evidence includes cavities in hot X-ray-emitting plasma that coincide with radio wave emission, together with surrounding shock waves. Their sizes and the ambient pressure constrain the mechanical work, while estimated ages convert an energy budget into a mean power. Such measurements test whether nuclear activity can regulate gas on scales much larger than the central black hole.^[ref]

The Polonyi model uses one chiral superfield, a canonical Kähler potential and a linear superpotential to study supersymmetry breaking. In a global theory, $W={\mu }^2\Phi +W_0$ with $\mu \ne 0$ gives a constant nonzero auxiliary field and scalar potential $V=|\mu {|}^4$, leaving the scalar direction flat at this level.

In supergravity, the constant $W_0$ matters and the relevant condition is the vanishing of the Kähler covariant derivative of a superpotential, rather than its ordinary derivative. Consequently, a linear superpotential does not break supersymmetry for every parameter choice. An appropriate constant can instead give a stable zero-energy vacuum with a nonzero supergravity auxiliary field. This simple model separates the questions of vacuum energy, scalar stabilization and supersymmetry breaking.