The cosmic neutrino background is the relic population of neutrinos that thermally decoupled before electron-positron annihilation. Entropy transfer to photons gives

A symplectic manifold is an even-dimensional smooth manifold $M$ equipped with a closed nondegenerate differential two-form $\omega$.

A Lagrangian subspace of a $2n$-dimensional symplectic vector space $(V,\omega )$ is an $n$-dimensional subspace $L$ on which $\omega$ vanishes.

On ${\mathbb{R}}^{2n}$ let $J_0$ be the standard complex structure, let ${\omega }_0$ be the standard symplectic form, and let $g_0(u,v)={\omega }_0(u,J_{0}v)$ be the Euclidean inner product. A linear map preserving any two of $J_0$, ${\omega }_0$, and $g_0$ preserves the third. Equivalently,

A likelihood contribution is the probability mass or density assigned by a statistical model to one observation's recorded information.

For a survival time with an absolutely continuous probability distribution, probability density function $f$, and survival function $S$, the hazard function iswhere $S(t)>0$, with the derivative identity holding almost everywhere. At points where the local limit exists, equivalentlyThe hazard function measures an instantaneous conditional rate. If time is in years, its units are inverse years, and its value can exceed one. A probability for a short interval is approximately $h(t)\Delta$, not $h(t)$; for an absolutely continuous probability distribution, the probability of an occurrence at the single exact time $t$ is zero. The discrete hazard, by contrast, is a conditional probability at a possible event time.

The factorization $f(t)=h(t)S(t)$ distinguishes a probability density function for the original population from a hazard function for those still event-free. With an exponential distribution of rate $0.2$ per year, $h(5)=0.2$ per year but $f(5)=0.2e^{-1}\approx 0.073576$ per year: the latter also accounts for the probability of reaching year five. The exponential survival risk over an interval example computes finite-horizon probabilities from this rate.

In a matter-plus-cosmological-constant universe, changing the independent variable from cosmic time to $a$ gives

When $H$ is constant and the matter source is negligible,so ${\delta }_m=C_1+C_2{a}^{-2}$. The growing matter-era mode therefore approaches a constant after cosmological-constant domination begins.

Every positive-energy supermultiplet contains equally many bosonic and fermionic states. A supertrace of a positive supercharge anticommutator proves the equality, provided supersymmetry is exact and the representation is finite-dimensional at fixed momentum.

Number-phase conjugacy is the canonical relation between a conserved $U(1)$ charge and the phase of its order parameter. A state of sharp number cannot also have a sharply selected broken-symmetry phase.

The superspin Casimir classifies massive supermultiplets analogously to the Pauli-Lubanski Casimir for Poincare representations. One construction contracts the tensor $C_{\mu \nu }=B_{\mu }{P}_{\nu }-B_{\nu }{P}_{\mu }$, where $B_{\mu }$ is the supersymmetric correction of the Pauli-Lubanski pseudovector.

Under standard assumptions on a nontrivial analytic relativistic S-matrix, the Coleman–Mandula theorem says that every continuous bosonic symmetry algebra is a direct sum of the Poincare algebra and an internal symmetry algebra.

For proper convex functions under a standard relative-interior qualification,In particular, if $f$ is differentiable at $x$, then $\partial (f+g)(x)=\nabla f(x)+\partial g(x)$.

A Regge trajectory is an approximately linear relation between angular momentum and squared mass. A classical rotating open string obeys $J={\alpha }^{\prime }{M}^2$, with quantum effects adding an intercept.

A body or swimmer is torque-free when the resultant external torque, including the moment of hydrodynamic traction, vanishes.

A body or swimmer is force-free when the resultant external force, including the integral of hydrodynamic traction, vanishes.

A microswimmer is a microscopic body that propels itself through a fluid. At small Reynolds number its motion is governed by Stokes flow, and self-propulsion must respect the force-free and torque-free conditions.

A sphere of radius $a$ translating with velocity $\mathbf{U}$ through otherwise stationary fluid hasIts surface traction integrates to the Stokes drag law.

A counting process is an integer-valued stochastic process that starts at zero and increases by recording the cumulative number of events up to each time.

Two stochastic processes $X,Y$ are indistinguishable whenThis is stronger than equality almost surely at each fixed time, though the two notions agree for continuous processes after checking equality on a countable dense set.