A compact astronomical object proposed as a constituent of a galactic dark matter halo. It can be sought through gravitational microlensing even if its emitted light is undetectable. Stellar objects and remnants consume the cosmological baryon budget; primordial black holes do not obey that particular formation-budget argument. Exclusion bounds depend on the object's mass, the halo model and survey selection.

A search for interactions of passing dark matter with a terrestrial target, for example nuclear recoils from elastic scattering. Predicting a count requires the local density and velocity distribution, interaction cross section, target response and detector acceptance. A background-compatible count gives a conditional interaction bound, rather than a measurement of the cosmological relic density.

Direct detection measures scattering or absorption of incident halo particles in a target. Indirect detection searches for secondary products from annihilation or decay. Recording those secondary particles directly does not turn an indirect search into a direct halo-scattering measurement.

For distinct roots $r_{\pm }=(-\gamma \pm \sqrt{{\gamma }^2-4mk})/(2m)$, the deterministic response is $X=(v_0-r_{-}{x}_0)e^{r_{+}t}/(r_{+}-r_{-})+(r_{+}{x}_0-v_0)e^{r_{-}t}/(r_{+}-r_{-})$. It satisfies $X(0)=x_0$ and $\dot{X}(0)=v_0$. With zero-mean future noise and prescribed initial data it is also the conditional ensemble mean, not a full stochastic trajectory.

For fixed spring constant $k$ and linear damping coefficient $b$, replacing mass $m$ by $\lambda m$ changes the exponential envelope rate from $\gamma =b/(2m)$ to $\gamma /\lambda$. The damped angular frequency becomes ${\omega }_{d,\lambda }^2=({\omega }_d^2+{\gamma }^2)/\lambda -{\gamma }^2/{\lambda }^2$. Thus envelope decay times scale exactly as mass, while the familiar inverse-square-root frequency scaling is a weak-damping approximation. A constant gravitational force shifts equilibrium without changing either rate.

For a smooth surface obtained by translating the plane graph $z=F(y)$ along the $x$ direction, the induced metric is $d{s}^2=d{x}^2+[1+F^{\prime }(y{)}^2]d{y}^2$. The transverse arc length coordinate $u(y)={\int }^y\sqrt{1+F^{\prime }(s{)}^2}ds$ makes it $d{s}^2=d{x}^2+d{u}^2$. This is a local isometry to the Euclidean plane, extending the intrinsic flatness of a circular cylinder to arbitrary smooth generating graphs.

Its geodesics therefore lift to straight lines in $(x,u)$ coordinates. Those that are graphs over $x$ satisfy $u=ax+b$, while the additional family $x=\text{constant}$ cannot be written as $y(x)$. The same graph formula follows from the Beltrami identity for the arc-length integrand $\sqrt{1+[1+F^{\prime }(y{)}^2]y^{\prime 2}}$. These straight lines are locally minimizing; on cylinders with a closed generating curve, global identifications can create several different winding paths between two points.

A retro hetero-Diels-Alder reaction is the reverse of a Diels-Alder reaction involving a heteroatom-containing partner. For a six-membered ring with sequence $O-A-B=C-D-E$, cleavage of $O-A$ and $D-E$ can give $A=B-C=D$ and $E=O$. Whether the cycloreversion is concerted or proceeds through ionic intermediates depends on the substrate and conditions.

The pentamethylcyclopentadienyl ligand is the anion ${\mathrm{C}}_5\mathrm{M}{\mathrm{e}}_5^{-}$, normally a six-electron ${\eta }^5$ donor in ionic electron counting. Its methyl substituents change steric hindrance and electron donation relative to an unsubstituted cyclopentadienyl ligand.