A proposed sterile neutrino component of dark matter must survive for cosmological times. If its decay rate is $\Gamma$, its lifetime $\tau =1/\Gamma$ must greatly exceed the time over which the required abundance is to persist. Otherwise the assumed present-day dark-matter density would be depleted and the decay products could alter cosmological observations.

Mixing with active neutrinos can permit decays even when the sterile state has no direct ordinary weak gauge coupling. The same mixing can therefore affect both its production and its survival. A lifetime check must use the allowed channels for the stated mass and model, rather than inferring stability from electrical neutrality or Majorana fermion character. Cosmological abundance and constraints on observable decay products impose additional requirements beyond longevity alone.

Let a fraction $R$ of gross stellar birth mass return immediately, and let the new metal source be $bR\Psi$. The yield per permanently locked mass is then $y=bR/(1-R)$. Metal conservation in an initially pristine closed box gives $Z=y\ln (M_0/M_g)$. A coefficient defined by the metal source must not be confused with a yield normalized per locked stellar mass; the source convention must be stated, especially when taking $R\to 0$ or $R\to 1$ limits.

A square lamina is a frame obtained by removing a concentric square hole from a larger square, with corresponding sides parallel. If their side lengths are $a>b>0$, the uniform border thickness is $t=(a-b)/2$ and the frame has areaFor positive integer sides $a,b$ with the same integer parity, $t$ is a positive integer, and a unit-grid frame contains exactly $4t(b+t)$ unit squares. For example, outer side five and inner side three give thickness one and sixteen unit squares. Concentricity alone would allow a rotated inner square; the parallel-side condition is what makes this a uniform lattice border.

Restrictions linking the exchange symmetry of nuclear spin and spatial wavefunctions. For identical nuclei, the total wavefunction must be antisymmetric for fermions and symmetric for bosons. Electronic and vibrational exchange parity must be included before assigning rotational parity.

A spin-one parent cannot decay into two identical spin-zero particles in the same internal state while preserving rotational symmetry and Bose exchange symmetry. Their final total angular momentum is orbital $L=1$, which is antisymmetric under interchange. Bose symmetry requires even $L$ instead. An energetically allowed decay with matching parity can therefore remain forbidden independently of the interaction's coupling strength.

The released 2015 paper is linked in its paper section. Student-answer availability remains unverified; biology portions are deferred.

For holomorphic $N,D$ with $N(z_0)D(z_0)\ne 0$, write the quotient as $h(z)/(z-z_0)$ where $h=N/D$. Its Taylor series immediately gives the displayed Laurent series. The residue is $h(z_0)$ and the constant regular term is $h^{\prime }(z_0)$; the latter requires differentiating both numerator and remaining denominator. Higher regular coefficients follow from the higher Taylor coefficients of $h$.

For a flat layer of thickness $h$ and shear speed ${\beta }_1$ above a faster half-space of speed ${\beta }_2$, unfolding $m$ down-and-up traversals gives the displayed reflected-ray travel time. Its slope is the horizontal ray slowness. At $x=2mh\tan i_c$, with $\sin i_c={\beta }_1/{\beta }_2$, the corresponding head-wave branch $t=x/{\beta }_2+2mh\sqrt{{\beta }_1^{-2}-{\beta }_2^{-2}}$ is tangent to this curve.

Vibrational excitation opens additional tunnelling channels at biases satisfying $|eV|=\hslash \omega$. A step in differential conductance gives paired opposite-sign features in its voltage derivative.

Interference between incident and elastically reflected surface-state waves gives density oscillations proportional to $\cos (2kx+\varphi )$. Their real-space period is $\pi /k$, half the travelling-wave wavelength.

The tensor spectrum is proportional to $H^2$, but the single-field slow-roll curvature spectrum is proportional to $H^2/{ϵ}_H$. Thus $n_s-1\simeq -2{ϵ}_H-d\ln {ϵ}_H/dN$. A decreasing $H$ gives a red leading tensor tilt, while the evolving scalar conversion factor can change the sign of the scalar tilt.

For $D_{\mu }={\partial }_{\mu }+iq{A}_{\mu }$, the constant phase transformation $\delta \varphi =-iqϵ\varphi$ has the displayed Noether current. Varying the gauge field gives ${\partial }_{\mu }{F}^{\mu \nu }=J^{\nu }$. In this convention $J^{\mu }=iq({\varphi }^{\ast }{\partial }^{\mu }\varphi -\varphi {\partial }^{\mu }{\varphi }^{\ast })-2q^2{A}^{\mu }|\varphi {|}^2$. The matter equations imply ${\partial }_{\mu }{J}^{\mu }=0$; the integrated electric charge additionally requires zero boundary flux. Local phase invariance gives a Noether second theorem identity, rather than independent physical charges for arbitrary functions.^[ref]

For unpolarized ultrarelativistic charged spin-half particles annihilating through a photon into minimally coupled unit-charge scalars of mass $m$, set $\beta =\sqrt{1-4m^2/s}$. The scalar vertex contains the difference $k_{-}-k_{+}$ of outgoing four-momenta. Contraction with the spin-averaged initial fermion current gives $\overline{|\mathcal{M}{|}^2}=e^4{\beta }^2{\sin }^2\theta /2$. The two-body final-state flux factor contributes another $\beta$, givingThe ${\sin }^2\theta$ angular dependence and threshold suppression contrast with electron-positron annihilation into a muon pair. These predictions assume the specified scalar electrodynamic coupling, rather than an arbitrary spin-zero interaction.

A reptile taxon whose precise ancestry-based definition should be stated. clade usages include birds; the traditional everyday assemblage excluding birds is paraphyletic.

A reptile lineage with a two-opening temporal skull organization inferred in its ancestry. Modified descendants can lose or close these openings; turtles illustrate why present-day hole counts do not determine ancestry.

A fossil reptile proposed as an early turtle relative. Its skull anatomy, ontogeny and phylogenetic placement have been used to test the history of temporal roof closure.

A member of the coelacanth lineage of sarcopterygians. Living representatives are specialized descendants, not unchanged ancestral fish.

A member of the lungfish lineage, one of the living branches included in Sarcopterygii.

A finned Devonian relative of digit-bearing vertebrates. Its pectoral skeleton combines retained fin rays with jointed internal structures relevant to limb evolution.

Three mutually perpendicular mirrors reverse a ray's direction when it encounters all three. With orthonormal normals $a,b,c$, write the reflections as $I-2P_a$, $I-2P_b$, $I-2P_c$. Their orthogonal projection matrices have pairwise products zero and sum to $I$ by the resolution of the identity in an orthonormal basis. The product of the three reflections is therefore $-I$. It reverses every incident direction independently of the order in which the mirrors are met. Real apertures restrict which incoming rays actually reach all three reflecting faces. Such retroreflectors provide the returning-light geometry in lunar laser ranging.