For a fixed nonnegative integer $p$, the Faulhaber polynomial $S_p(n)$ is the rational-coefficient polynomial satisfying $S_p(n)={\sum }_{k=1}^n{k}^p$ for nonnegative integers $n$. It is characterized by $S_p(0)=0$ and $S_p(n)-S_p(n-1)=n^p$.

For example, $S_2(n)=n(n+1)(2n+1)/6$ has degree three and leading coefficient $1/3$. In general the degree is $p+1$ and the leading coefficient is $1/(p+1)$, since taking a finite difference lowers the degree by one. Faulhaber's formula constructs these polynomials from the Bernoulli polynomials.

The Good-Thomas algorithm computes a discrete Fourier transform of size $N=ab$ when $a,b$ are coprime integers. The Chinese remainder theorem reindexes samples and frequencies by pairs of residues. With compatible input and output permutations, the transform kernel factors into an $a$-point kernel and a $b$-point kernel, so the computation becomes independent transforms along two array directions.

For $N=15$, one uses lengths three and five. Unlike the usual Cooley-Tukey FFT algorithm factorization, the coprime reindexing eliminates the intervening twiddle-factor multiplications. The required permutations still have computational cost, and a factorization such as $16=4\cdot 4$ does not satisfy the coprimality hypothesis.^[ref]

Factoradic indexing uses the factorial number system to rank or unrank permutations of an ordered $n$-element set. For zero-based rank $r<n!$, write $r={\sum }_{k=1}^{n-1}{d}_{k}k!$ with $0\le d_k\le k$. Read the digits from highest factorial down, selecting the indexed item from the remaining ordered elements and removing it.

For rank $4$ among permutations of $(a,b,c)$, $4=2\cdot 2!+0\cdot 1!$. Select index $2$, giving $c$, then index $0$, giving $a$, then the remaining $b$: the result is $cab$. This produces a bijection between ranks $0,\dots ,n!-1$ and lexicographically ordered permutations.

For positive integers $a,b$ and a prime $p$, their prime-exponent difference is $v_p(a)-v_p(b)$, where $v_p$ is the exponent in prime factorization. This is the exponent of $p$ in the positive rational number $a/b$, and it may be negative.

For $a=12=2^2\cdot 3$ and $b=18=2\cdot 3^2$, the differences are $1$ at $2$ and $-1$ at $3$, giving $a/b=2/3$. The quotient is an integer exactly when every difference is nonnegative. It is a rational $k$th power exactly when every difference is divisible by $k$, making exponent differences useful for perfect-power and divisibility tests.

The difference between star-plus-planet light outside secondary eclipse and star-only light during eclipse isolates the planet's dayside spectrum. Thermal radiation probes its temperature-pressure structure and opacity, while optical reflected light probes scattering and albedo. Contrast and the subtraction of stellar photons limit sensitivity.

Thermal energy comparable with an exciton binding energy promotes dissociation and often washes out narrow bound-exciton resonances. Comparing $k_{B}T$ with binding energy is a scale estimate, not an exact disappearance criterion: densities, screening, oscillator strength and linewidths also matter.

A closed walk in a graph that traverses each edge exactly once. Vertices may be revisited; the restriction concerns edges.

For the harmonic number $H_N$, the upper sequence $u_N=H_N-\ln N$ decreases and the lower sequence $v_N=H_N-\ln (N+1)$ increases. The inequalities $\ln (1+1/N)>1/(N+1)$ and $\ln (1+1/(N+1))<1/(N+1)$ follow by integrating the decreasing function $1/x$ over a unit interval. They give the two monotonicities, while $u_N-v_N=\ln (1+1/N)\to 0$. The monotone convergence theorem for sequences therefore gives the common limit Euler's constant $\gamma$. Strict monotonicity makes each finite-index bound strict. At $N=1$, this gives $0<1-\ln 2<\gamma <1$, since $\ln 2<1$.

An ideal zero-thickness planar loop under tension $T$ is a separatrix of the elastica equation. Its tangent turns from zero to $2\pi$, the same physical orientation at both ends. With $\alpha =\sqrt{T/(EI)}$, integration gives $x=s-2\tanh (\alpha s)/\alpha$ and $y=2\mathrm{sech}(\alpha s)/\alpha$. Its height is $2\sqrt{EI/T}$. Self-contact and finite beam thickness are absent from this ideal crossing solution.

For a nonzero vector $x\in {\mathbb{R}}^n$, put $r=|x|$ and $\hat{x}=x/r$. The gradient of the Euclidean norm is $\nabla r=\hat{x}$. Differentiate its components to obtain the Hessian matrixThe factor $I-\hat{x}{\hat{x}}^T$ is the orthogonal projection matrix onto the orthogonal complement of the radial direction. Thus the Hessian matrix has eigenvalue zero radially and $1/r$ on each tangential direction, explaining the convexity of the Euclidean norm. Taking the trace gives $\Delta r=(n-1)/r$ away from the origin. The Euclidean norm is not differentiable at the origin: its restriction to any line is a nonzero multiple of the absolute value.

Breaking the bond between an ester oxygen and its alkyl substituent. Acid removal of a tert-butyl group can follow this route without requiring water or a tetrahedral acyl-addition intermediate.

Breaking the bond between an acyl carbon and the ester alkoxy oxygen. Ordinary hydroxide-promoted ester saponification follows addition–elimination through a tetrahedral intermediate.

Equating the molecular root-mean-square speed to the surface escape velocity gives $3k_{B}T/m=2GM/R$, or $T=2mGM/(3k_{B}R)$. This compares speed scales; the high-speed tail can escape below that temperature, and molecular dissociation can invalidate the assumed species at high temperature.

Consider a spherical nonrotating asteroid of radius $r$ with the same mean mass density as a spherical reference planet of radius $R$ and surface gravitational acceleration $g$. Its mass scales as $r^3$, givingA specified take-off speed $v_J$ therefore permits escape for $r\le v_J\sqrt{R/(2g)}$.

Suppose that same take-off speed raises a test particle through height $H$ above the reference planet, neglecting atmosphere and rotation. Using the planet's inverse-square gravitational field, conservation of mechanical energy givesFor $H\ll R$, treating gravity as constant gives $v_J^2\simeq 2gH$ and $r_{\max }\simeq \sqrt{RH}$. Thus the square-root height rule is a small-height approximation, while the speed-based asteroid bound remains the direct criterion. The chosen $H$ specifies a performance estimate; it is not a universal property of a person or a jumping mechanism.

On potential-temperature surfaces, $q=({\zeta }_{\theta }+f)/{\sigma }_{\theta }$, where ${\sigma }_{\theta }=-(1/g)\partial p/\partial \theta$ is isentropic mass density. In adiabatic frictionless flow it is materially conserved. Diagnosing a velocity field also requires balance and boundary data.

For dry atmospheric thermodynamics, choose potential temperature as the materially conserved scalar in Ertel's theorem. The displayed density-normalized scalar has units $\mathrm{K}{\mathrm{m}}^2\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-1}$. Its conservation in adiabatic inviscid motion depends on conservative body forces and a composition-fixed equation of state. It differs in normalization from the Boussinesq approximation convention using buoyancy without the variable density factor. Large-scale balanced weather can be organized by its distribution, with potential-vorticity inversion additionally requiring balance and boundary information.

Translating an input translates its output under a map rather than necessarily leaving the output unchanged. Convolution has this property under consistent domain and boundary conventions.

A body of mass $m$, heat capacity $c$, temperature $T_i$ and speed $v$ that stops in a large reservoir at $T_L$ produces entropy $mc[\log (T_L/T_i)+T_i/T_L-1]+m{v}^2/(2T_L)$, assuming constant heat capacity and no phase changes. The first term follows by adding the body's logarithmic entropy change to the reservoir's heat gain; the second is kinetic-energy dissipation. A numerical answer needs the reservoir temperature.