Failure to meet a financial obligation, modeled by a random default time. A zero-recovery default model sets the affected asset value to zero after this time.

A local volatility model uses a deterministic function of current time and spot as its diffusion volatility, $\sigma (t,S_t)$. It can reproduce a surface of European call option prices through the Dupire equation.

The arithmetic stock model with constant volatility has additive price dynamics $d{S}_t=\mu dt+\sigma d{W}_t$, with constant $\mu \in \mathbb{R}$ and $\sigma >0$. Its absolute volatility is constant, and the model can take negative stock values. This differs from a geometric Brownian motion, whose noise is proportional to its current price.

With a bank account satisfying $d{B}_t=r{B}_{t}dt$, assume a finite-horizon equivalent risk-neutral measure exists. The market price of risk is $(\mu -r{S}_t)/\sigma$, and the pricing dynamics become $d{S}_t=r{S}_{t}dt+\sigma d{W}_t^Q$. For $\tau =T-t$, the conditional Gaussian law hasThe variance is ${\sigma }^2(e^{2r\tau }-1)/(2r)$ when $r\ne 0$ and ${\sigma }^2\tau$ when $r=0$. The zero-rate case is obtained by the continuous limit, not by division by zero.

Observed symptom-onset intensity is the convolution of past infection intensity with the incubation delay density: $\mu (t)={\int }_0^{\infty }h(t-s)f(s)ds$. Back-calculation infers the infection history from that relation. Include pre-observation infections unless their absence is known.

If $\mathbf{A}={\mathbf{t}}_a\times {\mathbf{t}}_{b}dadb$ and each tangent satisfies $D\mathbf{t}/Dt=(\nabla \mathbf{u})\mathbf{t}$, differentiation of the cross product gives $D\mathbf{A}/Dt=(\nabla \cdot \mathbf{u})\mathbf{A}-(\nabla \mathbf{u}{)}^T\mathbf{A}$. Combining this with ideal induction proves pointwise conservation of $\mathbf{B}\cdot \mathbf{A}$.

Convective acceleration is the spatial part $(\mathbf{u}\cdot \nabla )\mathbf{u}$ of the material derivative of fluid velocity. It can remain nonzero in steady flow, because a moving particle passes through positions with different velocities. The identity $(\mathbf{u}\cdot \nabla )\mathbf{u}=\nabla (|\mathbf{u}{|}^2/2)-\mathbf{u}\times (\nabla \times \mathbf{u})$ relates it to vorticity.

The mass loss rate is the positive rate at which a body loses mass. For a steady spherical stellar wind, the continuity equation gives $\dot{M}=4\pi r^2\rho u_r$, constant across spheres enclosing the source. It is positive for outward radial flow; an accretion convention often uses a separate positive inward rate.

A point mass idealizes a localized body's mass density as $\rho (\mathbf{x},t)=m{\delta }^{(3)}(\mathbf{x}-\mathbf{z}(t))$, using the Dirac delta function. Its second mass moment tensor is $I_{ij}=m{z}_i{z}_j$. The approximation ignores its internal size and structure.

In a simply connected current-free region of constant magnetic permeability, Ampère's circuital law makes the magnetic field curl-free, so it admits a scalar potential. The absence of magnetic monopoles then gives Laplace's equation for that potential. Some conventions define the potential for $\mathbf{H}$ instead of $\mathbf{B}$, differing by the constant permeability. Boundary conditions, including imposed fields at infinity, determine the potential up to an additive constant.

For a smooth divergence-free field on ${\mathbb{R}}^3$, the differential form $F={\iota }_{\mathbf{B}}(d{x}^1\wedge d{x}^2\wedge d{x}^3)$ is closed. The radial homotopy formula gives a global magnetic vector potential $\mathbf{A}(\mathbf{x})={\int }_0^{1}s\mathbf{B}(s\mathbf{x})\times \mathbf{x}ds$ with curl $\mathbf{B}$. For constant $\mathbf{B}$, this reduces to $\mathbf{A}=\frac{1}{2}\mathbf{B}\times \mathbf{x}$.

The magnetic field has zero divergence. The ideal magnetohydrodynamic induction equation preserves this constraint from initial data, since the divergence of its curl is zero.

A solenoid is a coil of electrical conductor, usually wire wound into a helix around a circular cylinder. The same electric current $I$ flows through its successive turns. Each turn contributes a magnetic field; the superposition principle makes their axial contributions reinforce inside the coil. Curl the fingers of your right hand in the direction of the electric current around the turns: the thumb gives the interior magnetic field direction by the right-hand rule. Throughout this branch, $N$ is the total number of turns, $\ell$ is the coil length, and $n=N/\ell$ is the solenoid turn density. ^[ref].

The attraction of a long solenoid is a controllable, nearly uniform magnetic field over an interior working region: change the electric current to tune its strength, or reverse the electric current to reverse its direction. This makes it a useful setting for studying the Lorentz force, measuring magnetic flux, and constructing an inductor. Its geometry turns the many contributions from individual turns into the simple long-solenoid approximation. A real finite coil has end fields and leakage; its exterior magnetic field is not zero. The on-axis magnetic field of a finite solenoid below quantifies that difference.

Historically, the coil helped connect magnetic fields to electric currents. In his memoir on the mathematical theory of electrodynamic phenomena, Ampère modeled the interactions of magnets using the ends of solenoids in place of microscopic magnetic constituents. This was a route toward explaining magnetism by electric currents, rather than a modern microscopic account of magnetization. ^[ref]. The historical setting is documented by the ^[ref].

Radiative decay from the hydrogen $2p$ level to $1s$ produces a Lyman-alpha photon. In an optically thin collisionally excited gas, escape of this photon removes the excitation energy deposited by an electron collision. The excitation threshold is responsible for the rapid temperature sensitivity of this cooling channel.

A neutral hydrogen atom absorbs a photon near the Lyman-alpha resonance by excitation from $1s$ to $2p$. Absorption lines in a quasar spectrum can therefore locate hydrogen along the sightline by their redshift, subject to peculiar-velocity and line-profile effects.

A composite operator is a renormalized local product of quantum fields and their derivatives. Operators with compatible quantum numbers can undergo composite operator mixing, and products at coincident insertion points can need contact terms in a correlation function.

A lipid vesicle is a closed lipid bilayer enclosing fluid. Its shape and thermal membrane undulations depend on membrane tension, membrane bending modulus, and constraints such as fixed enclosed volume.

Line tension is energy per unit length of a boundary, such as the edge between two phases of a lipid bilayer. Positive line tension tends to minimize the perimeter at fixed domain area. Its dimensions and geometry differ from surface tension, which is energy per unit area.

In a homogeneous layer under the hydrostatic approximation, the horizontal pressure gradient is independent of depth. The depth derivatives of velocity obey $(u_z{)}_t-f{v}_z=0$, $(v_z{)}_t+f{u}_z=0$. If the motion starts without shear, these derivatives remain zero. Hydrostatic pressure alone does not eliminate a pre-existing freely oscillating shear.

For a depth-independent horizontal velocity in a layer of reference depth $H$, the linear transports are $U=Hu$, $V=Hv$. They have dimensions of area per time and satisfy the surface continuity equation ${\eta }_t+U_x+V_y=0$. Their momentum equations are obtained by multiplying the depth-independent velocity equations by $H$.

On a beta plane with $f=f_0+\beta y$, set ${\mathcal{F}}_y={\partial }_t^2+f(y{)}^2$. Exact elimination of the depth-integrated shallow-water transports givesThe essential commutator is ${\partial }_y({\mathcal{F}}_{y}V)={\mathcal{F}}_y{V}_y+2f\beta V$. Freezing undifferentiated $f$ factors to $f_0$ is a local approximation. The slow-wave limit gives the shallow-water Rossby-wave dispersion relation with a negative zonal phase velocity for $\beta >0$.