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A massless supermultiplet is a positive-energy unitary irreducible representation of the Super-Poincaré algebra with null four-momentum. For finite helicity states in four dimensions, half the supercharges act trivially and the others form a fermionic Fock space.

The CPT theorem reverses helicity and conjugates internal charges. If a supermultiplet is not mapped into itself, a physical spectrum must also include its conjugate supermultiplet; this can double the irreducible state count.

A supersymmetric vector multiplet contains a gauge boson and its gaugino partners, with additional scalar fields for extended supersymmetry. Its gauge group representation is the adjoint group representation.

A massive supermultiplet is a unitary irreducible representation of the Super-Poincaré algebra with positive timelike momentum. At rest, its nonzero supercharges become fermionic creation operators and fermionic annihilation operators. Without central charges in supersymmetry, four-dimensional extended supersymmetry has $2\mathcal{N}$ complex oscillators; a spin-$j$ Clifford vacuum gives $(2j+1)2^{2\mathcal{N}}$ states.

The M-sigma relation is an empirical correlation between supermassive black hole mass and the stellar velocity dispersion of its host's spheroidal component. A commonly used rough parametrization has exponent near four, but the normalization, exponent, scatter, and selection effects depend on the observed sample. It is distinct from the black-hole--bulge mass relation.

The Kähler covariant derivative of a superpotential transforms by the same holomorphic factor as the superpotential under a Kähler transformation. This follows by differentiating $e^{-f}W$ and cancelling the extra derivative of $f$ against the shift of $K_i$.

A supersymmetry generator is an odd Weyl spinor generator of the Super-Poincaré algebra. Acting on a physical state reverses fermion parity while preserving its four-momentum.

A central charge in four-dimensional extended supersymmetry is a Lorentz scalar even generator appearing in $\{Q_{\alpha }^A,Q_{\beta }^B\}={ϵ}_{\alpha \beta }{Z}^{AB}$. It commutes with the displayed translation, Lorentz algebra, and supercharge generators. An added R-symmetry automorphism can act on the central charges.

The subdifferential of the L1 norm consists of vectors $z$ with $z_j=\mathrm{sgn}({\beta }_j)$ when ${\beta }_j\ne 0$ and $z_j\in [-1,1]$ when ${\beta }_j=0$. These coordinate conditions give the Karush-Kuhn-Tucker conditions for Lasso and Square-root Lasso.

For a finite convex function $h:\mathbb{R}\to \mathbb{R}$, the function $f(x)=h(c^{T}x+b)$ is convex. The subgradient inequality gives $c\partial h(u)\subseteq \partial f(x)$, where $u=c^{T}x+b$. Conversely, every subgradient $g$ of $f$ annihilates $\ker c^T$, because $f$ is constant on lines in those directions. For $c\ne 0$, write $g=cv$ and test the subgradient inequality at $y=x+(t-u)c/\Vert c{\Vert }_2^2$; this proves $v\in \partial h(u)$. For $c=0$, $f$ is constant and both sides are $\{0\}$ since a finite convex function on the real line has a nonempty subdifferential everywhere.

A spacetime is future-distinguishing when $I^{+}(p)=I^{+}(q)$ implies $p=q$. Every strongly causal spacetime is future-distinguishing.

The area of a string worldsheet is computed from its induced worldsheet metric. The Nambu–Goto action is minus string tension times Lorentzian worldsheet area.

For a Neumann-Dirichlet open-string boundary condition, the transverse modes have frequencies $r\in \mathbb{Z}+1/2$. The canonical commutation relations give $[{\alpha }_r^i,{\alpha }_s^j]=r{\delta }^{ij}{\delta }_{r+s,0}$ and ${\alpha }_r^{†}={\alpha }_{-r}$. One derivation expands $X-y={\sum }_{r>0}{q}_r\cos (r\sigma )$: spatial orthogonality reduces the Polyakov action to $(8{\alpha }^{\prime }{)}^{-1}\int {\sum }_r({\dot{q}}_r^2-r^2{q}_r^2)d\tau$. These are harmonic oscillators of mass $1/(4{\alpha }^{\prime })$, whose normalized creation operators satisfy ${\alpha }_{-r}=i\sqrt{r}{a}_r^{†}$ in the conventional phased expansion.

Traction is the force per unit area transmitted across an oriented surface. For unit normal vector $\mathbf{n}$, the stress tensor gives $\mathbf{t}=\mathbf{\sigma }\mathbf{n}$. Its normal component is normal stress, and its tangential component is shear stress.

For a straight, unloaded inextensible filament with small transverse displacement, resistive-force theory gives viscous force density $-c_{\perp }{y}_t$. Variation of the bending energy gives $-B{y}_{xxxx}$, while the induced filament tension is higher order. Their instantaneous balance yields $c_{\perp }{y}_t=-B{y}_{xxxx}$. It is an overdamped fourth-order diffusion equation: short-wavelength bends relax much faster than long ones. Wiggins and Goldstein's flexive-propulsion paper develops the elastic-wave mechanism for low-Reynolds-number propulsion.

Let the ambient velocity be $\mathbf{Ex}$, with $\mathbf{E}$ symmetric and trace free, and centre a rigid sphere of radius $a$ at the origin. It is force-free and torque-free, with zero translation and rotation. Its disturbance isThe total velocity vanishes at $r=a$, and the leading disturbance is a stresslet. Harmonic dipole and quadrupole potentials in the Papkovich–Neuber representation give the complete field.

Spot volatility is the instantaneous diffusion coefficient multiplying relative price increments in a stochastic differential equation for an asset. In a stochastic volatility model it is itself a stochastic process.

Langevin dynamics describes a dynamical variable driven by deterministic forces and random noise. Underdamped Langevin dynamics retains inertia, whereas overdamped Langevin dynamics neglects it. At thermal equilibrium, the fluctuation-dissipation relation for a Langevin particle relates Gaussian white noise strength to drag and temperature.

A path probability is a probability measure, or a density relative to a specified reference measure, on histories of a stochastic process. A formal Onsager--Machlup path probability must be interpreted with its discretization and initial-state conditioning specified; conditioning on both endpoints produces a separately normalized bridge measure.

A linear fluctuating interface smooths height gradients by diffusion and receives additive Gaussian white noise. With periodic or Neumann boundary conditions, its nonzero Fourier modes are independent Ornstein-Uhlenbeck processes if the noise is diagonal in that basis. The unconstrained uniform height is a Brownian zero mode of a fluctuating interface.