The approximate fifth-power scaling of an allowed beta decay rate with available energy when electron mass and Coulomb distortion are negligible and the nuclear matrix element is held fixed. It follows from ${\int }_0^Q{p}^2(Q-p{)}^{2}dp=Q^5/30$. Different nuclear matrix elements or nonrelativistic electron kinematics spoil an exact universal law.

$\mathrm{C}{\mathrm{H}}_3{\mathrm{O}}^{-}$, the conjugate base of methanol.

$\mathrm{C}{\mathrm{H}}_3\mathrm{C}{\mathrm{H}}_2{\mathrm{O}}^{-}$, the conjugate base of ethanol.

$(\mathrm{C}{\mathrm{H}}_3{)}_3\mathrm{C}{\mathrm{O}}^{-}$, a bulky alkoxide base, often used in elimination reactions.

A lithium salt of ethoxide. Its reactive oxygen donor and lithium-centred acceptor can participate in complementary contacts with an ester; solvation and aggregation affect a detailed orbital description.

The conjugate base of a propanol; the straight-chain form is CH3CH2CH2O−. A specified isomer matters in product structures.

For $a>0$, this algebraic curve can be constructed using a circle of radius $a$ centred at $(a,0)$ and its tangent line $x=2a$. On a ray from the origin, subtract the distance to its nonzero circle intersection from the distance to the line. The resulting radial distance is $r=2a{\sin }^2\theta /\cos \theta$, for $-\pi /2<\theta <\pi /2$. Thus a parametric curve for it isEliminating $t$ gives the displayed cubic equation. It has a cusp at the origin, horizontal tangent direction there, and vertical asymptote $x=2a$. Its name honours Diocles.

For a finite-dimensional complex group representation, combining inverse and conjugate transpose gives another representation: both operations reverse multiplication order. This is the complex conjugate of the algebraic dual representation in matrices. If $W$ is invariant under $D^{\#}$, its orthogonal complement is invariant under $D$, because $⟨w,D(g)v⟩=⟨D^{\#}(g^{-1})w,v⟩=0$ for $w\in W$, $v\in W^{\perp }$. Thus irreducibility is preserved. Applying the construction twice returns $D$, so the equivalence of irreducibility works in both directions for arbitrary groups in finite dimension.

The occupied-to-vacant probability ratio of the adsorption-site grand partition function is $\theta /(1-\theta )=z_s{e}^{\beta ({\mu }_s+ϵ)}$. Taking its logarithm yields the displayed chemical potential. binding energy and vibrational energies must share a consistent reference so that zero-point energy is not counted twice.

When the entropy of independent magnetic moments dominates, a reversible isolated change fixes $B/T$. Lowering the field therefore lowers temperature as $T_f/T_i=B_f/B_i$. Other heat capacities, internal interactions and residual fields limit this ideal relation.

An addition chain restricted to updates $a_{t+1}=a_t+a_i$ with $0\le i\le t$. Each new power uses the most recently computed power as one factor. This restriction reduces the search space, but a shortest star chain need not be a shortest unrestricted addition chain for every target.

Time-dependent inference of an emitting region's response to a variable illuminating continuum. For an active nucleus, a broad-line delay estimates a light-travel scale and a line width estimates a velocity scale. A virial mass estimate then requires a geometrical and kinematic factor $f_{\mathrm{geom}}$; the lag alone neither fixes the geometry nor establishes purely gravitational motion.

A strong Achilles number is a positive integer $n$ for which both $n$ and its Euler totient $\phi (n)$ are Achilles numbers. For example,In each prime factorization, every prime exponent is at least two and their greatest common divisor is one. Thus both numbers are powerful without being perfect powers. The extra condition matters: $72$ is an Achilles number, but $\phi (72)=24=2^{3}3$ is not powerful, so $72$ is not a strong Achilles number.

An irradiated accretion disk can produce iron fluorescence with a known local energy scale. relativistic Doppler effect, gravitational redshift, beaming and light bending redistribute those photons into a broadened asymmetric line. The red wing is sensitive to the deepest emitting radii, while inclination and orbital motion influence the profile. Coronal illumination, ionization, absorption and the continuum model must be controlled before inferring an inner radius or testing a spacetime geometry.

A geometrically thin, optically thick disk has a radially varying effective temperature. Relativistic transport and an atmosphere model convert its thermal spectrum into an inferred emitting area and inner radius. Black-hole mass, distance, inclination and spectral hardening are important external constraints. Selecting thermal-state observations with moderate luminosity limits contamination and the breakdown of the thin-disk approximation. Relating the fitted inner radius to the innermost stable circular orbit remains a physical modeling assumption.

A cis-regulatory messenger RNA element mediating Nanos-dependent regulation through associated RNA-binding proteins. mutation and reporter transfer can test necessity and sufficiency.

A family of zirconium-based alloys developed for nuclear applications. Specific alloy and heat treatment determine corrosion, hydrogen uptake and irradiation response.