A unitary irreducible representation is both unitary and irreducible. Such representations are the building blocks in the harmonic analysis of groups and in quantum symmetry classifications.

In ultra-slow-roll inflation, $z=a\sqrt{2ϵ}\propto a^{-2}\propto {\eta }^2$. The comoving curvature mode equation isand its Bunch-Davies mode is proportional to $(1+ik\eta )e^{-ik\eta }/{\eta }^3$. Unlike ordinary slow roll, the superhorizon curvature perturbation grows as $a^3$.

The Eliassen–Palm flux is a wave-activity flux whose divergence forces the transformed mean flow. Its vertical component combines vertical eddy momentum flux with a correction from meridional buoyancy transport.

The torsion subgroup of an abelian group consists of all elements of finite order. For a finitely generated abelian group, it is the finite direct summand complementary to the free abelian part.

A closed orientable surface of genus $g$ is homeomorphic to a sphere with $g$ handles. Its Euler characteristic is $2-2g$.

An $n$-manifold has zero singular homology in every degree greater than $n$. One proof uses a countable exhaustion by subsets that retract onto $n$-dimensional CW complexes and the compact support of every singular cycle.

Quasi-adiabatic continuation implements a gapped path of local Hamiltonians by a quasi-local unitary. It dresses local operators with exponentially decaying tails while preserving their action within the corresponding low-energy spaces.

An algorithm runs in polynomial time when its running time is at most $O(n^k)$ for some constant $k$, where $n$ is the input length.

The Jacobi theta functionsatisfies $\theta (\tau +2)=\theta (\tau )$ and $\theta (-1/\tau )=(-i\tau {)}^{1/2}\theta (\tau )$.

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.