A divisibility sequence is an integer sequence satisfying $m\mid n⟹a_m\mid a_n$, with any conventions about zero terms stated. For example $a_n=2^n-1$ has this property because if $n=km$, thenThe Fibonacci numbers also form a divisibility sequence. The stronger identity $\gcd (a_m,a_n)=|a_{\gcd (m,n)}|$ defines a strong divisibility property and does not follow from the basic implication alone. Such sequences organize common-factor questions by relationships among their indices.

For ideal zero-torque equatorial thin discs, efficiency is $1-E_{\mathrm{ISCO}}$. The formal extremal retrograde, Schwarzschild and extremal prograde limiting radii are respectively $9,6,1$ in units $GM/c^2$, giving efficiencies $1-\sqrt{25/27}$, $1-\sqrt{8/9}$ and $1-1/\sqrt{3}$. Photon capture and non-ideal accretion can change the realized efficiency.

These invertible matrices, for $n\in \mathbb{Z}$, form an infinite cyclic group. The displayed multiplication rule, $B_0=I$, and $B_n^{-1}=B_{-n}$ give a group isomorphism from the additive group of integers via $n\mapsto B_n$. Its generator is $B_1$. The subgroups of an infinite cyclic group become $\{B_{mk}:k\in \mathbb{Z}\}$ for integers $m\ge 0$, and their subgroup indices are $m$ when $m>0$. The case $m=0$ is the trivial subgroup and has infinite index. As transformations, these matrices send $(x,y)$ to $(x+ny,y)$.

The integers form an infinite cyclic group under addition, with identity element zero and inverse element $-n$ for $n$. Its two generators are $1$ and $-1$. Every subgroup is $m\mathbb{Z}$ for a unique nonnegative integer $m$: if it is nonzero, take its least positive element and use division with remainder to show that every element is a multiple of it. The zero subgroup corresponds to $m=0$.

For $G=⟨g⟩$ infinite, every subgroup is $⟨g^m⟩$ for a unique integer $m\ge 0$. This is the classification of subgroups of the additive group of integers transported through a group isomorphism. For $m>0$ the subgroup index is $m$ and the subgroup is again an infinite cyclic group; for $m=0$ it is trivial and has infinite subgroup index.

The complete state of identical bosons is symmetric under exchange, while that of identical fermions is antisymmetric. The condition includes spin and other internal variables as well as spatial coordinates.

The operator ${\hat{P}}_{12}$ interchanges all one-particle variables for particles 1 and 2. Since ${\hat{P}}_{12}^2=\hat{I}$, a state of definite exchange symmetry has eigenvalue $+1$ or $-1$.

With test-parcel escape work $B(r)=-\Phi (r)$, outer gas in a spherical isolated attractive potential is less tightly bound and quenches first under identical local specific heating. A truncated inverse-square density gives $r_q=r_0\exp [1-q(e^{\nu t}-1)/(2k)]$ after edge onset. The result is a fixed-potential closure, not a self-consistent potential update after mass loss.

The local shell-energy prescription sets $B(r)=GM(<r)/r$ rather than the full escape work. It predicts simultaneous quenching for $M(<r)\propto r$, but a centre-to-edge front for $M(<r)\propto r^2$. Exterior-shell contributions explain why this ordering can differ from a true escape-potential calculation.

For an attractive force $-\beta /y^2$ and an otherwise linear vertical spring, differentiation about the charged equilibrium $y_e$ reduces the spring stiffness by $2\beta /y_e^3$. If the uncharged height is $y_0$ and the shift is small, the fractional frequency change is $-\beta /(k{y}_0^3)$. Using only the force value, without its derivative, would miss this frequency shift.

The electrostatic bisector force is laterally anti-restoring, with the positive stiffness coefficient on the right. A spring of tension $T$ and length $\ell$ adds transverse restoring stiffness $T/\ell$ when its fixed support is directly above the charge. The displayed inequality therefore distinguishes actual transverse stability from the saddle of a model with only a vertical spring force. Axial spring stiffness alone does not specify the transverse verdict. At fixed finite support geometry and positive gravitational load, sufficiently small electric charge preserves lateral stability: the destabilizing coefficient tends to zero as $q^2$, while the tension term approaches $mg/\ell >0$. The vertical equilibrium must separately satisfy its own positive-stiffness condition.

An idempotent matrix is a square matrix $P$ with $P^2=P$. Its linear map fixes every vector in its image and sends its kernel to zero. Every vector splits uniquely as $v=Pv+(v-Pv)$, giving a direct sum of the image and the kernel.

Consequently $P$ is diagonalizable over any field, with only the eigenvalues zero and one. For example, $\left(\begin{array}{cc}1& 1\\ 0& 0\end{array}\right)$ is idempotent but not symmetric. An idempotent matrix is an orthogonal projection in a real Euclidean inner product precisely when it is also symmetric; idempotence alone gives an oblique projection.

The enthalpy change on transferring a specified gas-phase ion to the specified aqueous standard state. Charge density strongly affects the stabilization; single-ion values require a consistent convention.

A real four-vector vacuum ${\Phi }_0=v{e}_4$ leaves the rotations of its perpendicular three-space unbroken. In a Hermitian generator basis split into two commuting SU(2) triples with $(T^a+T^{a+3})e_4=0$, define $A^{a,\pm }=(A^a\pm A^{a+3})/\sqrt{2}$. For scalar kinetic term $|D\Phi {|}^2$ and canonical vector kinetic terms, its vacuum value is $g^2{v}^2{\sum }_a(A^{a,-}{)}^2/2$. Comparison with $m^2{A}^2/2$ gives three massless and three mass-$|gv|$ gauge bosons.

The factor multiplying the scalar kinetic term matters: replacing it by $|D\Phi {|}^2/2$ at fixed numerical $v$ changes the massive value to $|gv|/\sqrt{2}$. The unbroken global subgroup is SO(3), rather than its simply connected SU(2) cover. The Spin(4) double cover accounts for that central identification.

Expanding the Higgs doublet kinetic term about its vacuum expectation value gives $\mathcal{L}\supset (2m_W^2/v)h{W}_{\mu }^{+}{W}^{-\mu }$. Its vertex is proportional to $g_{\mu \nu }$, so scalar exchange contracts the external particle polarization vectors.

A muon and antimuon can form a Higgs boson directly in the annihilation channel. Their Higgs Yukawa coupling is proportional to the muon mass, making the corresponding partial width much larger than for an electron pair. The observable resonant cross section also depends on the collider's energy distribution: a beam spread broader than the resonance smears its ideal peak.

Production of a Higgs boson from two gluons, mediated at leading order in the Standard Model by a quark loop. The large top quark Yukawa coupling makes the top loop important. There is no direct renormalizable tree-level coupling of the colorless Higgs field to two gluons.

Production through exchange of weak gauge bosons emitted by incoming fermions. A characteristic Higgs boson process at a hadron collider has two outgoing quark jets and tests the Higgs coupling to weak vectors. Predictions for a specified measured final state must include the consistent set of interfering amplitudes rather than treating a diagram label as a separately measurable event.

On a bar of length $L$, let the initial temperature increase linearly from zero at either endpoint to height $H$ at the midpoint. With thermal diffusivity $D>0$ and insulated Neumann boundary conditions, a Fourier cosine series gives the displayed solution. Its constant term is the initial integral average $H/2$. Integration by parts on the two straight segments gives zero coefficients for odd modes and modes divisible by four; the remaining coefficients are $-4H/[{\pi }^2(2j+1{)}^2]$. Nonconstant modes decay, leaving the uniform temperature $H/2$, consistent with conservation of the total heat. This contrasts with the zero-temperature boundary solution in heat flow from a triangular initial temperature, where heat can leave the bar and the limit is zero. The initial endpoint slopes do not satisfy the insulated condition, so derivatives need not converge to their initial endpoint values as $t\downarrow 0$; the series satisfies the Neumann condition for every $t>0$ and recovers the continuous initial profile.