The sphere $\rho =k^{\prime }/(2\pi Gr)$ for $r<r_0$ has $M(r)=k^{\prime }{r}^2/G$ and $\Phi (r)=-k^{\prime }(2r_0-r)$, with zero potential at infinity. Its finite central potential contrasts with the logarithmic central divergence of a truncated inverse-square density.

A thin uniform spherical shell of fixed mass $M_s$ and radius $R$ has Newtonian gravitational potential energyThis follows from the spherical binding-energy mass integral: the enclosed mass is zero inside $R$ and $M_s$ outside. The radial generalized force, positive outward, is consequently $-{\partial }_R{W}_{\mathrm{self}}=-G{M}_s^2/(2R^2)$, taking the derivative at fixed $M_s$. It is an inward resistance to uniform expansion; individual surface forces cancel as vectors by spherical symmetry.

If an interior spherical component has mass $M_c$, the spherical shell theorem adds inward load $G{M}_c{M}_s/R^2$. The total inward load isFor a swept gas fraction $f$ of an original enclosed mass $M$, with $M_s=fM$ and $M_c=(1-f)M$, this becomes $G{M}^{2}f(1-f/2)/R^2$. Omitting shell self-gravity would give $f(1-f)$ instead. In an accreting shell, evaluate the instantaneous load at its current mass; differentiating an assigned $M_s(R)$ in the fixed-mass energy formula would also include the energy of newly acquired material.

The sum of nuclear coherent scattering lengths per unit volume, ${\rho }_b={\sum }_j{n}_j{b}_j$. Matching it between media suppresses their ideal interface contrast.

A Kiessig fringe is an oscillation in reflected intensity caused by wave interference between contributions from the two sides of a thin film. For specular neutron scattering at grazing angle $\theta$ and wavelength $\lambda$, the normal wave vector transfer is $q_z=4\pi \sin \theta /\lambda$. Well above the critical edge, a film of thickness $d$ contributes a relative wave phase approximately $q_{z}d$, giving fringe spacingThus closely spaced fringes indicate a thicker film. As a calculated example, spacing $0.0628\mathrm{n}{\mathrm{m}}^{-1}$ corresponds to about $100\mathrm{nm}$. Interface surface roughness, contrast, additional layers and instrumental resolution affect the fringe pattern; thickness is inferred with an appropriate reflectivity model, especially near the critical edge.^[ref]

Small-angle neutron scattering probes structure using a neutron beam deflected through small scattering angles. For elastic scattering at wavelength $\lambda$, if $\theta$ is the full angle between incident and scattered beams, the transferred wave vector has magnitudeThe corresponding structural length scale is roughly $2\pi /q$, so a small angle accesses larger structures than an atomic-scale diffraction angle at the same wavelength.^[ref]

A collimated neutron beam with selected wavelength passes through the sample; a distant position-sensitive detector records the angular distribution. Comparing different sample-to-detector distances accesses different $q$ ranges. The measured intensity depends on neutron scattering length density contrast, shape and spatial correlations, rather than giving a direct photograph. Subtracting background and specifying the measured $q$ range are essential to interpreting a fitted size or correlation length.^[ref]

Instantaneous neutrino decoupling followed by electron-positron annihilation conserves electromagnetic entropy. The electromagnetic entropy count falls from $11/2$ to $2$, giving $T_{\nu }/T_{\gamma }=(4/11{)}^{1/3}$ after annihilation.

A neutral fermion without the ordinary active-neutrino weak gauge couplings. Mixing can still let it interact or decay. Its possible dark-matter role depends on lifetime, production and free streaming, rather than on Majorana character alone.

For a massless incident neutrino and a stationary target electron, the invariant squared center-of-mass energy is $s=m_e^2+2m_e{E}_{\nu }$ in natural units. Production of a charged lepton of mass $m_{\ell }$ and a massless neutrino requires $s\ge m_{\ell }^2$, giving the displayed threshold. For the electron flavour the threshold is zero; for heavier charged leptons it is enhanced by the small stationary-target mass.

A neutrino participating in the weak interactions of the Standard Model, in contrast to a sterile state. Its thermal cosmological abundance and momentum distribution depend on weak decoupling and subsequent cosmic history.

In the zero-pseudomomentum sector, a planar relative circular orbit has two signed angular-frequency branches. Here $\mu =m_1{m}_2/(m_1+m_2)$, $e^{\ast }=e(m_1-m_2)/[2(m_1+m_2)]$, $R$ is separation, and the Coulomb constant is absorbed into $e^2$. The centre of mass also circles: ${\mathbf{R}}_{\mathrm{cm}}={\mathbf{R}}_0+eB\mathbf{r}/[(m_1+m_2)\omega ]$. This distinction is essential when reconstructing individual particle orbits.

A convolutional neural network uses local connections and weight sharing across input positions. Each learned filter detects the same local feature at multiple positions; successive activation functions and layers combine those features. Shared filters impose translation equivariance away from boundary and sampling effects, while a pooling layer can reduce spatial size.

For a standard dense-channel two-dimensional convolution with a $k_h$ by $k_w$ kernel, $C_{\mathrm{in}}$ input channels and $C_{\mathrm{out}}$ output channels, the parameter count including one bias per output channel isIt is independent of the image's spatial dimensions. For example, a three-by-three convolution from three channels to sixteen channels has $3\cdot 3\cdot 3\cdot 16+16=448$ parameters. Grouped or depthwise convolutions use different connectivity and therefore different counts.

Reuse of one learnable parameter in multiple computational positions. In a convolution this reduces parameter count and imposes translation equivariance away from boundary effects.

The learned additive constant in a unit preactivation, $a=w^{T}z+b$. It can be represented by weight $w_0=b$ on a constant input $z_0=1$.

For slender hard rods of length $L$ and diameter $D\ll L$, the leading pair excluded volume is the displayed function of their mutual angle $\gamma$. Alignment lowers mutual obstruction and increases positional entropy, while narrowing the orientation distribution lowers orientational entropy. Their competition can drive an isotropic-to-nematic transition without attractive forces. The natural concentration scale is $c{L}^{2}D$ of order one. Finite rod diameter, flexibility and interactions beyond the hard-core second-virial approximation change quantitative predictions.

A cholesteric liquid crystal is a chiral nematic liquid crystal whose local nematic director rotates helically in space. The pitch is the distance over which the director completes a full turn. It retains locally nematic orientational order without smectic density layering; the helical modulation should not be mistaken for a layer lattice.

A surface boundary preference for the nematic director to align normal to the local surface. Around a sphere it favors radial orientation near the surface, which must match the far-field texture.

A surface preference for the nematic director to lie tangent to the local boundary. Degenerate planar anchoring leaves the in-plane direction unselected; a directionally treated surface can instead select an azimuth. Uniform axial alignment satisfies degenerate planar anchoring on every side point of a cylindrical channel.

A point-defect texture that can compensate the radial boundary condition of a sphere with homeotropic anchoring, giving a dipolar particle–defect configuration in a uniform far field. It carries the opposite hedgehog charge to the radial particle boundary under a consistent convention.

A comparison receiving the solvent or delivery formulation without the active compound. It separates compound-associated effects from effects of the formulation.

Circumferential momentum quantization of a graphene Dirac cone produces nanotube dispersion $E_{\pm }(q_{\parallel })=\pm \sqrt{(\hslash v_F{q}_{\parallel }{)}^2+(\hslash v_F{q}_{\perp }{)}^2}$. Comparing with a massive one-dimensional Dirac Hamiltonian identifies the displayed effective mass and gap $2m_{\mathrm{eff}}{v}_F^2$. This is an emergent band mass, not a changed vacuum electron rest mass. A quantization line through the valley gives zero mass in the ideal zone-folding model; curvature, strain and interactions can change that idealization.