Return a fraction of the output to the inverting input so that it opposes the input difference. Large stable loop gain makes the closed-loop gain depend mainly on the external feedback ratio rather than the uncertain open-loop gain. Frequency-dependent phase lag can destabilize the loop; saturation and finite slew rate also limit the approximation.^[ref]

In amplifier saturation, the requested output lies outside the amplifier's linear operating range, such as when supply rails limit output voltage. The negative feedback virtual short approximation then need not hold. Output swing, current drive and recovery behaviour depend on the device and load; finite slew rate is a separate dynamic limit even when the output has not reached a rail.

An isolated spin-degenerate orbital coupled to a particle reservoir has Quantum Hamiltonian $H=ϵ(n_{\uparrow }+n_{\downarrow })+U{n}_{\uparrow }{n}_{\downarrow }$. The four occupation microstates have weights $1,z,z,q{z}^2$, where $z=e^{\beta (\mu -ϵ)}$ and $q=e^{-\beta U}$. Thus $\Xi =1+2z+q{z}^2$ and $\bar{n}=(2z+2q{z}^2)/\Xi$. The factor two counts distinct spin states. At $U=0$ the grand partition function factorizes into two independent fermionic orbitals; positive U separates the two electron-addition thresholds.

A nonrelativistic particle accepted only at a fixed flight time through an ideal uniform-field tube has fixed parallel speed $v_{\parallel }=l/t_0$. On-axis entry and wall clearance bound the independent perpendicular speed by $|q|Ba/(2m)$, provided a full gyration occurs before exit. Adding parallel and perpendicular kinetic energies gives the displayed ideal energy window, where $K_0=m{v}_{\parallel }^2/2$. The upper endpoint is a grazing limit; strict wall exclusion makes it a supremum.

For a hot initial plate cooled from above, Fourier's law gives the displayed upward heat flux. At old ages only the steady value $k\Delta T/a$ remains. At young ages the sum approaches a Gaussian integral, giving $q_s\sim k\Delta T/\sqrt{\pi \kappa t}$. This divergence is an idealized sudden surface cooling, not an infinite observed ridge heat flux.

With $X={\sum }_s{a}_s^{†}{b}_s$, $Y=X^{†}$ and a real coefficient $f$, the fixed-occupancy second-order operator is $t[f(\Delta )XY-f(\Delta +2)YX]$. Normal ordering gives $XY=N_a+P$ and $YX=N_b+P$, where $P={\sum }_{s{s}^{\prime }}{a}_s^{†}{a}_{s^{\prime }}{b}_{s^{\prime }}^{†}{b}_s$ exchanges internal states without changing site totals.

A number whose representation contains every numerical digit from a specified alphabet. The convention must specify the base, whether zero is required, and whether repetitions are allowed. In many counting problems, “0-to-9 pandigital” means that all ten decimal numerical digits occur exactly once; other problems require only that each occur at least once.

The digital root of a nonnegative integer in integer radix $b\ge 2$ is the single numerical digit obtained by repeatedly taking its digit sum. For example, in decimal representation, $347\mapsto 14\mapsto 5$, while $999\mapsto 27\mapsto 9$.

The process terminates: if $n\ge b$, some nonzero digit occupies a position above the units, so $n={\sum }_j{d}_j{b}^j>{\sum }_j{d}_j=s_b(n)$. The positive integers therefore decrease until a single numerical digit remains. Each step preserves the integer congruence class modulo $b-1$.

For positive inputs, every digit sum stays positive. The final digit lies in $1,\dots ,b-1$, rather than $0,\dots ,b-2$. Choosing the representative in this range giveswith ${\mathrm{dr}}_b(0)=0$. This explains why a positive multiple of $9$ has decimal digital root $9$, not zero. The formula combines modular arithmetic with the positivity of the terminating representative.

A step number is a positive integer whose adjacent decimal digits differ by exactly $1$. For example, $3210$ and $101$ are step numbers, while $112$ is not: its first two numerical digits have difference zero. Repetition is allowed; only neighboring differences are constrained.

The numerical digits form a walk on the path graph $0-1-\cdots -9$, starting at a nonzero digit. From a last digit $d$, the possible extensions are $d-1,d+1$ within the range $0,\dots ,9$. At endpoints there is only one extension.

For digit dynamic programming, record the last digit and a bit mask encoding the set of digits already visited. Appending $d^{\prime }$ updates the visited set by set union with $\{d^{\prime }\}$. Here pandigital means that every decimal digit appears at least once; repetitions are allowed. Accept only final states whose visited set contains all ten numerical digits. This separates local adjacency from the global requirement to have seen every digit.

A balanced number is a positive integer whose first $\lceil k/2\rceil$ decimal digits have the same digit sum as its last $\lceil k/2\rceil$ decimal digits, where $k$ is the number of numerical digits. For odd $k$, the middle numerical digit appears on both sides and cancels, so only the equally long outer halves need to be compared.

For example, $123006$ is balanced because $1+2+3=0+0+6$. It is not a palindromic number: equality of two digit sums does not require reversed digits. At odd length, $12321$ is balanced because the outer pairs $12$ and $21$ both sum to three; its middle digit is irrelevant to the equality.

Digit dynamic programming can tabulate counts and value sums of fixed-length halves indexed by digit sum, then match equal indices. The first half must begin with a nonzero numerical digit, while the last half may contain initial zeros, as in $123006$. Preserving the half length is essential when assembling the positional value.

One of the ten numerical digits $0,1,\dots ,9$ used in decimal representation.

For a hopping operator $X$ with $[\Delta ,X]=2X$, spectral functions obey $Xf(\Delta )=f(\Delta -2)X$. The coefficient on the left of the hop is evaluated on the final state. This ordering determines the shifted energy denominator in virtual-hopping calculations.

The electrostatic self-energy of a uniformly charged sphere of sharp radius $R$ is $3Z^2\alpha \hslash c/(5R)$. For mirror nuclei, subtract the changed neutron/proton rest-mass contribution from their nuclear mass difference before identifying the residual with a nuclear Coulomb energy difference. The inferred $R$ is a sharp-sphere radius; the corresponding nuclear charge radius defined as an rms radius is $\sqrt{3/5}R$. Replacing $Z^2$ by $Z(Z-1)$ removes a proton self-interaction approximation and shifts the estimate.

This correlation counts distinct detection events without the delta-function self-counting term in an un-ordered product of densities. It is nonnegative as the expectation of $A^{†}A$ with $A={\psi }_n\cdots {\psi }_1$. For a single occupied bosonic mode with sharp occupation $N$, the coincident $n$-point value contains the falling factorial $N(N-1)\cdots (N-n+1)$, rather than $N^n$.

Consider a straight chain with a fixed wall, spring constants $k_1,k_2,k_3>0$, and consecutive point masses $m_1,m_2,m_3>0$, with the third mass freely ending the chain. Displacements are measured from equilibrium. If the middle mass stays at equilibrium, the two adjacent spring forces must cancel: $k_2{q}_1+k_3{q}_3=0$. The outer equations of motion then reduce to $m_1{\ddot{q}}_1=-(k_1+k_2)q_1$ and $m_3{\ddot{q}}_3=-k_3{q}_3$.

A nonzero oscillatory normal mode with $q_2=0$ therefore exists exactly when the two outer squared angular frequencies agree, as displayed. Its mode shape is proportional to $(1,0,-k_2/k_3)$ and its frequency does not depend on $m_2$. Appropriate initial displacements and velocities confined to this shape keep the middle mass still. For $k_1=2k$, $k_2=k_3=k$, $m_1=3m$, $m_3=m$, the common frequency is $\sqrt{k/m}$ and the outer displacements are equal and opposite.

Three particles are attached to a light taut string between fixed endpoints, with four equal equilibrium segment lengths $a$. The outer two masses are $m_o$ and the central mass is $m_c$. For small transverse displacement and tension $T$, set $k=T/a$. In coordinate order $(x,y,z)$ for the two outer particles and the central one, the potential is $V=\frac{1}{2}k[x^2+(x-z{)}^2+(y-z{)}^2+y^2]$ and the mass matrix is $M=\mathrm{diag}(m_o,m_o,m_c)$.

The antisymmetric normal mode is $(x,y,z)\propto (1,-1,0)$ with ${\omega }_a^2=2k/m_o$. In a symmetric mode $x=y=q$, the equations give $z/q=2-m_o{\omega }^2/k$ and $(2k-m_c{\omega }^2)z=2kq$. Their determinant is $m_o{m}_c{\omega }^4-2k(m_o+m_c){\omega }^2+2k^2$, whose roots give the displayed symmetric frequencies. Both are positive for positive masses. The physical order of points along the string is $x,z,y$; keeping it separate from the coordinate-vector order prevents a misleading mode sketch.

A breathing mode is a collective normal mode in which a structure expands and contracts radially while preserving its overall symmetry. For a uniform scaling pattern, the displacement of point $i$ is proportional to its position relative to the center of mass, as displayed above. In a central spring framework this pattern is a normal mode only when applying the stiffness matrix returns a multiple of the same pattern with the appropriate mass matrix; radial motion alone does not establish an eigenvector.

For diagonal kinetic energy, $q_i=\sqrt{m_i}{x}_i$ changes the kinetic quadratic form to $\frac{1}{2}{\sum }_i{\dot{q}}_i^2$. The potential Hessian in these coordinates is $M^{-1/2}K{M}^{-1/2}$.

The mass-weighted force-constant matrix $D=M^{-1/2}K{M}^{-1/2}$ for small oscillations. Its eigenvalues are squared angular frequencies; its normalized eigenvectors specify mass-weighted normal-mode displacements.

The Hessian of potential energy at a mechanical equilibrium, expressed in specified displacement coordinates. It determines restoring forces to first order. Combining it with the mass matrix gives the dynamical matrix whose eigenvalues are squared mode angular frequencies.