A saturating consumption response with encounter coefficient $a$ and handling time $h$. In the simple model $f(N)=aN/(1+ahN)$, consumption tends to $1/h$ at high resource density.

Half the full width between the two half-height points of a symmetric line profile. Stating whether a Doppler formula uses HWHM or FWHM avoids a factor-of-two resolution error.

For coprime integers $a,b>1$, the Frobenius number is $ab-a-b$.

First, $ab-a-b$ cannot equal $ax+by$ with nonnegative integers $x,y$. Reducing modulo $a$ gives $b(y+1)\equiv 0(\mathrm{mod}a)$; coprimality forces $y+1$ to be a positive multiple of $a$. Hence $by\ge b(a-1)>ab-a-b$, a contradiction.

Now take any integer $n>ab-a-b$. Multiplication by $b$ permutes the residue classes modulo $a$, so choose $y\in \{0,\dots ,a-1\}$ with $by\equiv n(\mathrm{mod}a)$. The difference $n-by$ is a multiple of $a$ and satisfies $n-by>-a$. It is therefore nonnegative, giving $x=(n-by)/a\ge 0$ and $n=ax+by$. This proves both the largest gap and representability of every greater integer.

A cosmological metric can measure the interval in units of time rather than length. With dimensionless comoving spatial coordinates, its scale factor $R$ then has units of length and a comoving clock has $ds=dt$. Proper spatial length is $c\sqrt{-d{s}^2}$ at constant cosmic time. This convention is equivalent to the more common length-valued interval $d{\ell }^2=c^{2}d{t}^2-R^{2}d{\Sigma }_k^2$, not a different cosmology.

For positive constant cosmological constant, zero cosmological spatial curvature and a separate positive-density fluid with constant equation-of-state parameter $w>-1$, set $H_{\Lambda }=\sqrt{\Lambda /3}$ in the time-unit convention. Eliminating the fluid density gives $\dot{H}=-3(1+w)(H^2-H_{\Lambda }^2)/2$. On the expanding Big Bang branch,The origin is the Big Bang. The fluid's early-time power-law expansion approaches exponential vacuum domination at late times.

An antiresonance is a suppression of a specified measured response through cancellation of contributions, often between normal modes. It does not generally mean that the entire system is motionless. In an ideal undamped two-oscillator system, the directly driven oscillator can have zero displacement while its partner still moves.

At equilibrium separation $z_0$, expanding the tip force gives $m_{\mathrm{eff}}\ddot{x}=-[c-F^{\prime }(z_0)]x$. Thus $\omega /{\omega }_0=\sqrt{1-F^{\prime }(z_0)/c}$, and Taylor expansion gives the displayed result for $|F^{\prime }|\ll c$. The displacement convention takes positive $z$ away from the surface. A positive gradient of an attractive force softens the resonance; a large-amplitude oscillation samples more than the local gradient.

natural selection in which a variant’s relative fitness increases with its frequency under a specified ecological setting. Shared warning patterns can benefit from prior predator learning.

Selection in which relative advantage decreases as a variant becomes common, potentially maintaining variation under the stated ecological assumptions.

The squared vibrational overlap $|⟨{\chi }_{v^{\prime }}|{\chi }_{v^{\prime \prime }}⟩{|}^2$ entering a vibronic intensity when the electronic transition moment is treated as constant over nuclear coordinates.

To maximize $A(x)/B(x)$ on a nonempty compact feasible set $X$, assume $A,B$ are continuous and $B>0$. Define the parametric optimal valueIf $r_{\ast }={\max }_{X}A/B$, then $F(r)>0$ for $r<r_{\ast }$, $F(r_{\ast })=0$, and $F(r)<0$ for $r>r_{\ast }$. This follows by comparing each ratio with $r$ and multiplying by its positive denominator.

Starting with a feasible ratio $r_0=A(x_0)/B(x_0)$, choose a global maximizer $x_k$ of $A-r_{k}B$ and updateThe values increase while staying below $r_{\ast }$. Their limit must be $r_{\ast }$: otherwise continuity of $F$ and an upper bound on $B$ would force increments bounded away from zero. A zero residual is an exact certificate. These assumptions give attained subproblems and convergence; a merely local inner optimizer does not provide this certificate. The method replaces a ratio objective by a succession of difference objectives. ^[ref]

For sufficiently regular decaying data, this half-line transform has inverse $f(x)=(2/\pi ){\int }_0^{\infty }C(k)\cos (kx)dk$. To derive it, extend $f$ evenly to the real line. Its Fourier transform with negative exponential is $2C(k)$, and its odd sine contribution vanishes. The Fourier inversion theorem then gives the stated factor $2/\pi$. If the forward transform is instead defined as ${\hat{f}}_c=2C$, the inverse factor is $1/\pi$.

With boundary terms at infinity vanishing, two integrations by parts give the cosine transform of $f^{\prime \prime }$ as $-k^{2}C(k)-f^{\prime }(0)$. Thus a homogeneous Neumann boundary condition makes it a multiplier $-k^{2}C(k)$, paralleling the Fourier sine transform with a homogeneous Dirichlet boundary condition.

Extend $f(x)=\max (0,2x)$ on $(-1,1)$ with period two. Its Fourier coefficients in the basis $\cos (n\pi x),\sin (n\pi x)$ are $a_0=1$, $a_n=2[(-1{)}^n-1]/(n\pi {)}^2$, and $b_n=2(-1{)}^{n+1}/(n\pi )$, by integration by parts on $(0,1)$. ThusIts even part is the periodic extension of $|x|$ and its odd part is the Fourier series of a periodic sawtooth for $x$. The cosine terms converge absolutely and uniformly; the sine terms account for the jump at odd integers. At each jump, the full Fourier series has value $1$, the average of the two one-sided limits $0$ and $2$.

For two three-dimensional real maps, let $F_1,F_2$ be their Fourier transforms and let $S_q$ be a reciprocal-space shell around spatial frequency $q$. The displayed normalized coefficient inner product is the Fourier shell correlation. It is defined only when both shell-energy sums in its denominator are positive.

For a shell containing both $\mathbf{k}$ and $-\mathbf{k}$, Hermitian symmetry makes the numerator real. The Cauchy-Schwarz inequality bounds its magnitude by one. Identical nonzero shell coefficients give one; reversing their sign gives minus one. A random-noise estimate can also fluctuate below zero, so the mathematical range is not restricted to zero through one.

Independent half-map reconstruction tests which frequency components reproduce across separate data subsets. This is how the EMDB validation pipeline interprets its FSC curve. ^[ref]

Shared refinement information, applying an image mask or correlated noise can inflate agreement. The selected resolution crossing depends on an explicitly stated threshold and processing convention; a high coefficient alone does not establish that every local feature in either map is correct.

A lower included wavenumber. A finite periodic patch of size $L$ has smallest nonzero wavenumber of order $2\pi /L$; supports and boundary conditions can change the factor. The cutoff limits large-scale fluctuations.