For positive rates $a,b$, the mean value theorem in the rate gives a bound by $|a-b|x{e}^{-\min (a,b)x}$. Its maximum is $|a-b|/[e\min (a,b)]$. Combining this estimate with coefficient errors bounds a finite spectral sum uniformly on a half-line, including slow scales much longer than one. Positivity of both rates is essential: unstable exponentials need a different time-window or relative-error analysis.

With fixed mass and homologous scale $L$, adiabatic pressure obeys $p\propto L^{-3\gamma }$. Its support relative to gravitational energy scales as $p{L}^3/|W_g|\propto L^{4-3\gamma }$. Pressure becomes relatively stronger on contraction for $\gamma >4/3$, weaker for $\gamma <4/3$ and has the same scaling at $\gamma =4/3$.

For fixed mass and temperature, $p\propto L^{-3}$, so the pressure-support scale $p{L}^3$ stays constant while gravitational binding scales as $L^{-1}$. Isothermal pressure therefore becomes relatively less important during homologous contraction.

A uniform mass density slab with free surfaces at $z=\pm H$ has ${\rho }_0=\Sigma /(2H)$, internal gravity $4\pi G{\rho }_{0}z$ and hydrostatic pressure $2\pi G{\rho }_0^2(H^2-z^2)$ above ambient pressure. Its surface modes of a self-gravitating incompressible slab separate into thickness and bending parities.

A radiation-supported vertical disk has vertical hydrostatic equilibrium supplied mainly by radiation pressure. In a constant-opacity, optically thick layer in gravity $-{\Omega }^{2}z$, radiative diffusion gives $F=c{\Omega }^{2}z/\kappa$. This statement applies at a fixed radius and does not resolve the optically thin surface.

For a thin axisymmetric astrophysical disk, angular integration of the Newtonian kernel eliminates odd Legendre polynomials. Define $I_n={\int }_0^R\Sigma (s)s^{n+1}ds$ and $J_n={\int }_R^{\infty }\Sigma (s)s^{-n}ds$. The even coefficient is ${\alpha }_n=\pi [n!/(2^n((n/2)!{)}^2){]}^2$. In differentiating the paired inner and outer contributions, their moving-boundary terms cancel. The resulting galaxy rotation curve includes an inward inner-ring contribution and an outward exterior-ring contribution. For smooth surface density, the force at the disk is interpreted with a symmetric Cauchy principal value or a vanishing-thickness regularization.

A Mestel disk is a thin axisymmetric astrophysical disk with surface density inversely proportional to radius. Its galaxy rotation curve is exactly flat, $v_c=v_M$, and $v_c^2=GM(R)/R$ despite its nonspherical geometry. In the Legendre expansion of thin-disk gravity, all positive-degree interior and exterior terms cancel. Its total mass is infinite, so only potential differences, rather than a potential zero at infinity, are defined.

A protoplanetary disk is a gas-and-solid astrophysical disk around a young star in which planet formation occurs. Its evolution combines accretion, radiation, gas dynamics and the growth of solid material. A minimum-mass solar nebula is one reconstruction of the material needed to form the Solar System.

Let a disc contain fraction $f_d$ of total halo mass, take its edge gravity to be $v_d^2=C_{d}G{M}_d/R_d$, and equate its edge specific angular momentum to $q{R}_h{v}_{\mathrm{vir}}$. With $v_{\mathrm{vir}}^2=G{M}_h/R_h$, these conditions give $R_h/R_d=C_d{f}_d/q^2$. The structural coefficient $C_d$ is essential: disc mass alone does not uniquely determine its edge rotation curve.

The total number of prime factors $\Omega (n)$ counts every occurrence of a prime factor in the prime factorization of a positive integer $n$. Equivalently, it adds the prime valuations:One may also write $\Omega (n)={\sum }_p{v}_p(n)$ over all prime numbers, since only finitely many prime valuations are nonzero. The empty sum gives $\Omega (1)=0$. By contrast, the prime omega function has $\omega (n)=r$ and counts distinct prime factors. ^[ref]

Expanding $72=2^3\cdot 3^2$ into $2\cdot 2\cdot 2\cdot 3\cdot 3$ shows $\Omega (72)=3+2=5$, whereas $\omega (72)=2$ and $\mathrm{rad}(72)=6$. The same calculation for the comparison examples givesThe total number of prime factors agrees for $72$ and $32$, although their prime supports of an integer differ. Meanwhile $72$ and $36$ have the same radical of an integer and prime omega function, but different values of the total number of prime factors. These examples show exactly which information each arithmetic function forgets.

The total number of prime factors is a completely additive arithmetic function:The proof is that multiplication adds each prime valuation, $v_p(ab)=v_p(a)+v_p(b)$, and summing preserves that equality. No coprimality condition is needed. For example, $12=2^2\cdot 3$ and $18=2\cdot 3^2$ both have total number of prime factors three; their product $216=2^3\cdot 3^3$ has six. By comparison, the prime omega function is only an additive arithmetic function: here $\omega (216)=2\ne 4=\omega (12)+\omega (18)$.

There is a useful way to see the total number of prime factors as a count of elementary building blocks: any expression of $n$ as a product of prime numbers has exactly $\Omega (n)$ factors, by the fundamental theorem of arithmetic. Equivalently, every factorization tree for $n>1$ has $\Omega (n)$ leaves. Repeated prime factors increase this count without introducing new kinds of prime number. The difference is the existing excess prime-factor multiplicity:Every summand is nonnegative, so $\Omega (n)=\omega (n)$ if and only if $n$ is a squarefree integer, including $n=1$. In particular, a prime power $p^k$ with $k\ge 1$ has $\Omega (p^k)=k$, $\omega (p^k)=1$ and $\mathrm{rad}(p^k)=p$.

Historically, Hardy and Ramanujan's 1917 paper explicitly separated the prime omega function from the total number of prime factors, writing them as $f(n)$ and $F(n)$, respectively. Their example $2^3\cdot 3^2\cdot 5$ has counts three and six. That distinction let them compare two measures of how an integer is built from prime factors. ^[ref]

The radical of an integer $n>0$ is the product of its distinct prime factors, each included exactly once. If the prime factorization is $n=p_1^{a_1}\cdots p_r^{a_r}$ with distinct prime numbers $p_i$ and positive integer exponents $a_i$, thenThus the radical of an integer remembers which prime factors occur and forgets their prime valuations. At $n=1$, the empty product gives $\mathrm{rad}(1)=1$. The domain of a function here is the positive integers; zero is excluded. ^[ref]

For example, $72=2^3\cdot 3^2$ has distinct prime factors $2,3$, so $\mathrm{rad}(72)=2\cdot 3=6$. Similarly,using the prime factorizations $30=2\cdot 3\cdot 5$, $32=2^5$ and $36=2^2\cdot 3^2$. The prime omega function counts the retained prime factors, while the total number of prime factors counts their occurrences before repetitions are removed.

The radical of an integer is its largest squarefree divisor. Indeed, a squarefree divisor chooses at most one copy of each prime factor, so it divides $\mathrm{rad}(n)$; the radical of an integer itself is a squarefree divisor. Consequently,The last statement follows because every nonzero prime valuation is then one. It includes $n=1$, a squarefree integer. Applying the radical of an integer twice also changes nothing: $\mathrm{rad}(\mathrm{rad}(n))=\mathrm{rad}(n)$.

For any positive integers $a,b$, the exact product rule isTo prove it, a prime factor dividing just one input occurs once in the numerator and not in the denominator. A prime factor dividing both occurs twice in the numerator and once in the denominator, since it divides their greatest common divisor. Each prime factor of $ab$ therefore remains exactly once. For coprime integers, the denominator is $\mathrm{rad}(1)=1$, proving the radical of an integer is a multiplicative function. It is not a completely multiplicative function: $\mathrm{rad}(4)=2\ne 4=\mathrm{rad}(2{)}^2$.

For a worked example with shared prime factors, take $a=12$ and $b=18$. Their greatest common divisor is $6$, and $ab=216=2^3\cdot 3^3$, soThis explains why prime support of an integer is useful: questions about the presence of prime factors can discard their multiplicities. For example, the squarefree divisors of $72$ are exactly the integer divisors of $\mathrm{rad}(72)=6$, namely $1,2,3,6$. Also $\gcd (m,n)=1$ if and only if $\gcd (m,\mathrm{rad}(n))=1$, because coprimality depends only on shared prime factors.

A further characterization uses mutual divisibility of powers: for positive integers $a,b$,If their prime supports of an integer agree, sufficiently large $k,\ell$ make all the required prime valuations large enough. Conversely, these two divisibility relations force each prime factor of either input to divide the other. For $72$ and $36$, one can take $k=2$ and $\ell =1$: $72\mid 36^2$ and $36\mid 72$.

For maximization, an approximation ratio $\alpha \in (0,1]$ means the returned feasible value is at least $\alpha$ times the optimal value on every permitted instance. Some conventions instead report its reciprocal; specify the convention.

An aperture stop is the physical opening that limits the angular cone of rays from an on-axis object point. Its images through preceding and following optics are the entrance and exit optical pupils; it differs from a field stop, which limits the extent of the imaged field.

A joint eigenstate $|j,m⟩$ of $J^2$ and $J_z$, with eigenvalues ${\hslash }^{2}j(j+1)$ and $\hslash m$. Its allowed labels satisfy $j\in \{0,1/2,1,\dots \}$ and $m=-j,-j+1,\dots ,j$.

The operator $J_{-}=J_x-i{J}_y$ acts by $J_{-}|j,m⟩=\hslash \sqrt{(j+m)(j-m+1)}|j,m-1⟩$. It generates lower magnetic states and determines Clebsch-Gordan coefficients.

Probabilistic number theory treats an integer chosen uniformly from $1,\dots ,N$ as a random object and studies the distribution of its arithmetic functions. The distribution changes with $N$, and limit theorems describe the behavior as $N$ grows.

For a fixed prime $p$, the probability that $p$ divides the chosen integer is $\lfloor N/p\rfloor /N$, approaching $1/p$. For finitely many distinct primes, the joint divisibility frequencies similarly approach products of these probabilities. This motivates models of the number of prime factors and helps explain why logarithmic scales and normal distributions appear in integer statistics. Such independence is an asymptotic model with specified ranges, not an assumption valid for every growing collection of primes.

A finite sum of complex exponentials, such as $S(\theta )={\sum }_n{a}_{n}e(n\theta )$ with $e(t)=\exp (2\pi it)$, is an exponential sum. Cancellation in an exponential sum between its terms can make it much smaller than the sum of their absolute values.

For every $A>0$ there is $B>0$ such that, for $Q\le x^{1/2}/{\log }^{B}x$,Here $\psi (y;q,a)$ is the Chebyshev function in an arithmetic progression and $\varphi$ is the Euler totient function. Partial summation gives the corresponding result for the prime-counting function and the offset logarithmic integral function. The theorem controls the total error over many moduli; it need not give the same bound for each individual modulus.

A prime sum is a summation indexed by prime numbers, for example ${\sum }_{p\le x}1/p$. The indexing restriction distinguishes it from a sum over all positive integers.

Given finitely many distinct places of a number field $v_1,\dots ,v_r$, the diagonal image of $K$ is dense in ${\prod }_i{K}_{v_i}$. One can therefore simultaneously prescribe finite congruences and real signs. This is useful in the ray class exact sequence and in finding representatives of an ideal class prime to a prescribed finite set of prime ideals.