An amplitude characterizing a near-tip linear-elastic crack field for a specified loading mode. The mode-I opening factor scales as the displayed expression for a suitable crack geometry factor $Y$. It is not itself the local stress, and large-scale plasticity can invalidate a linear-elastic characterization.

A region of irreversible deformation near a crack tip where an elastic singular-field extrapolation would exceed the material's local yielding response. Its size depends on loading and constraint. Plastic work and blunting can raise resistance to crack extension.

The reduction in a mechanical system's potential energy per incremental crack area under specified loading conditions. In small-scale-yielding linear elasticity, mode-I plane strain gives $G=K_I^2/E^{\prime }$ with $E^{\prime }=E/(1-{\nu }^2)$. The critical value includes the dissipative processes associated with crack advance.

A requirement that a crack penetrate a pressure boundary and produce a detectable leak before unstable fracture. A simple screening comparison uses critical crack size versus wall thickness, but the result depends on crack geometry and on whether stress or pressure is held fixed.

The decrease in potential energy per increment of crack area under specified loading constraints. Its critical value $G_c$ is a crack-growth threshold, not energy stored per specimen volume.

Crack growth becomes energetically possible when the energy release rate reaches its critical value. For fixed crack geometry this gives a brittle-failure stress scaling as ${\sigma }_f\propto \sqrt{E{G}_c}$.

In the Landauer formula, $G=(e^2/h)\mathrm{tr}(t^{†}t)$ when the flux-normalized transmission matrix $t$ includes every incident spin channel. The trace already sums both spin states if both are represented; multiplying this expression by another factor of two would count them twice.

For a Hamiltonian independent of spin with two identical, uncoupled spin sectors, one may instead calculate a single sector $t_0$ and write $G=(2e^2/h)\mathrm{tr}(t_0^{†}{t}_0)$. For example, one perfectly transmitted spatial mode gives $2e^2/h$ with both sectors present, and $e^2/h$ when only one sector propagates. With spin-orbit coupling, keep the full matrix even when its eigenchannels have no definite spin projection.

A fluorescence microscope technique that illuminates a thin plane and detects emitted light from another direction, enabling repeated optical sectioning of living specimens.

Biological damage caused by illumination, which can alter development even when an image remains visually clear. Non-imaged and dose-matched controls help identify it.

An illumination-induced change in a fluorescent label that allows a selected cell or region to be distinguished and followed over time.

fluorescence microscopy forms images from spatially resolved fluorescence. Probe partitioning, optical resolution and photophysics constrain what a fluorescent pattern establishes about molecular composition or contact.

Fluorescence measurements resolving individual labelled molecules or molecular events, allowing heterogeneous dynamics to be distinguished from population averages.

Two-photon excitation microscopy excites fluorescence through the nearly simultaneous absorption of two photons. In the weak unsaturated regime, the excitation rate is proportional to the square of the local illumination intensity. Focusing a pulsed laser therefore strongly favors excitation near its focus rather than throughout the illuminated path.

Scanning that focus provides optical sectioning without relying on a detection pinhole to reject all out-of-focus emission. The photons can have lower individual energies than a single excitation photon, but their combined energy must match an allowed transition. High peak intensity, bleaching and phototoxicity remain relevant; longer excitation wavelength alone does not guarantee less damage.^[ref]

Stimulated emission depletion microscopy narrows the region from which detected fluorescence is allowed to originate. After excitation, a second beam drives stimulated emission in the surrounding region; a central minimum of that depletion beam leaves a smaller region able to emit the collected fluorescence.

Scanning this region builds an image with detail beyond an ordinary excitation-only diffraction limit. The depletion light must be rejected from the detection channel, and the central minimum must be well aligned with the excitation focus. Increasing depletion intensity can narrow the effective emitting region, but bleaching, background and specimen changes limit the useful improvement.^[ref]

A floral arrangement with multiple planes of symmetry around the reproductive axis. Its developmental basis can differ among lineages, so a derived radial flower need not restore the ancestral organ identities.

A floral arrangement with one principal symmetry plane. Differential deployment of dorsal regulators can contribute, but a shared appearance does not establish one universal developmental mechanism.

On an interval with $x\ne 0$, the substitution $y=xv$ and the product rule give the displayed identity. An equation $x{y}^{\prime }-y=g(y/x)$ becomes $v^{\prime }=g(v)/x^2$ and permits separation of variables wherever $g(v)\ne 0$. Roots $g(v_0)=0$ also give solutions $y=v_{0}x$ and must be retained if division by $g$ was used. This substitution works for this displayed class even though its explicit factor $1/x$ means that $y^{\prime }=y/x+g(y/x)/x$ need not be a homogeneous equation of the usual form $y^{\prime }=F(y/x)$.

For the heat equation with homogeneous Neumann boundary conditions at both ends of a uniform grid, even reflection supplies ghost values for a centered finite difference method. A fourth-order centered second derivative needs two ghost values at each end, obtained with the displayed rule for $j=1,2$. For a sufficiently smooth compatible solution, differentiating the Neumann boundary conditions in time and using the heat equation makes every odd spatial derivative vanish at the endpoints. Its even extension therefore retains the smoothness required by the fourth-order stencil; arbitrary incompatible initial data need care at the initial boundary corners.

The discrete cosine modes diagonalize this reflected spatial operator, with eigenvalues $-4h^{-2}{\sin }^2(\theta /2)[1+{\sin }^2(\theta /2)/3]$ and $\theta =j\pi /N$. Its most negative eigenvalue is $-16/(3h^2)$, so the forward Euler method is stable in the corresponding trapezoid-weighted norm when $\Delta t/h^2\le 3/8$. The Crank-Nicolson diffusion scheme is stable for every time step for this closure, but its amplification factor approaches $-1$ on highly damped spatial modes, allowing sign-alternating numerical transients. The ghost values are defined here as auxiliary samples outside the physical interval, rather than additional unknown initial data.

For nonzero angular constant $h$, the interior null radial equation gives ${\dot{r}}^2\sim 2\mu h^2/r^3$. Its inward branch integrates to $r^{5/2}\sim \frac{5}{2}\sqrt{2\mu }|h|({\lambda }_{\ast }-\lambda )$. Thus the singularity is at finite affine parameter and $r\propto ({\lambda }_{\ast }-\lambda {)}^{2/5}$. The zero-angular-momentum case has a different affine scaling.

A finite-capacity cache that evicts the oldest inserted item when full. A hit leaves the insertion order unchanged, so its behavior differs from an LRU cache.