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For a compact Hausdorff space $K$, a subset of $C(K)$ is relatively compact in the uniform norm exactly when it is equicontinuous and pointwise bounded. On a compact metric space, this says that every uniformly bounded equicontinuous sequence has a uniformly convergent subsequence.

An equatorial Rossby wave propagates westward and has a Gaussian-weighted Hermite-polynomial meridional structure.

An equatorial Kelvin wave has zero meridional velocity, eastward phase propagation, and Gaussian meridional trapping.

The impact parameter is the perpendicular distance between an incoming particle's asymptotic straight-line trajectory and the centre of a scattering potential. In central-force scattering it determines the deflection angle whenever the scattering map is single-valued.

For a homogeneous extensive system in thermodynamic equilibrium, the entropy density is entropy per physical volume, $s=S/V$. The thermodynamic Euler relation giveswhere $\rho =U/V$ is energy density, $P$ is pressure, $\mu$ is chemical potential, $n=N/V$ is number density and $T$ is absolute temperature. Here $\rho$ denotes energy density, not mass density; in a relativistic application the energy convention must consistently include the rest-energy paired with the chemical potential.

With several conserved particle numbers, replace $\mu n$ by ${\sum }_i{\mu }_i{n}_i$. The Euler form assumes extensivity and negligible surface terms; it is not an automatic identity for a finite interacting system with appreciable boundary contributions. For a photon gas at equilibrium, $\mu =0$ and $P=\rho /3$, yielding $s=4\rho /(3T)$.

A pair $(U,V)$ is $C$-relevant to $(X,Y)$ when

For finitely supported random variables $A,B$ in an abelian group,To prove it, take two conditionally independent copies $(A_1,B_1)$ and $(A_2,B_2)$ of $(A,B)$ given $A+B$. Since $A_1+B_1=A_2+B_2$, entropy submodularity givesThe first two terms are at most $H(A)+H(B)$. The last joint entropy isConsequently $H(A_1-B_2)\le H(A+B)+2I(A;B)$. Subtractingfrom this bound proves the theorem.

For finitely supported random variables,Equivalently, independent $X,Y$ satisfy

The entropic Ruzsa distance satisfiesFor independent representatives this is equivalent to

For finitely supported random variables, the conditional entropic Ruzsa distance isThus the two conditioning values are sampled independently. If only one variable is conditioned, write $d_R(X\mid U;Y)$.

For a fixed-composition closed system, the heat capacity at fixed pressure is $C_p=(\partial H/\partial T{)}_p$, with enthalpy $H$. Under the quasistatic process assumptions in heat capacity, only pressure-volume work is permitted, so $dp=0$ gives $\delta Q=dH=C_{p}dT$. Thus supplied heat is $Q=\Delta H$, or $Q=C_p\Delta T$ within a single thermodynamic phase when $C_p$ is constant. For an ideal gas, the expansion work explains why this exceeds the required heat at fixed volume for the same temperature increase. Its per-unit-mass counterpart is specific heat capacity at constant pressure.

Since $T=(\partial H/\partial S{)}_p$ and $V=(\partial H/\partial p{)}_S$, equality of mixed derivatives gives $(\partial T/\partial p{)}_S=(\partial V/\partial S{)}_p$.

An entanglement area law bounds a region's entanglement entropy by its boundary size rather than its volume, in a specified family of states and with a stated short-distance convention. For a lattice region $A$, a typical form is $S(A)\le c|\partial A|$ with a constant independent of the region's linear size.

The physical intuition is that correlations contributing most strongly across the cut can be concentrated near its boundary. A generic highly entangled state need not satisfy this: independent Bell pairs joining many interior sites to the exterior can produce volume-scaling entropy. Tensor-network area-law bounds make the boundary counting explicit through the number of cut bonds. Area laws are therefore useful structural constraints, not automatic consequences of calling a state quantum or a Hamiltonian local.

For data $(X_i,Y_i)$ and predictions $h_{\beta }(X)={\beta }^{T}X$, it is convenient to writeIts gradient is

The classical Lane-Emden equation is the dimensionless equationfor a spherically symmetric polytrope in three dimensions. The Generalized Lane-Emden equation extends it to arbitrary dimensions and to functions without radial symmetry.

For real modulus $0\le k<1$, the complete elliptic integral of the first kind isThe substitution $t=\sin \theta$ proves equivalence. Here $k$ is the modulus; a parameter convention instead writes $m=k^2$. The endpoint value is $K(0)=\pi /2$, and $K(k)$ increases without bound as $k\uparrow 1$.^[ref]

For a simple pendulum of length $\ell$ with angular amplitude $0<{\theta }_0<\pi$, conservation of mechanical energy gives its periodUse $\sin (\theta /2)=\sin ({\theta }_0/2)\sin \varphi$ in the quarter-period energy integral. The limit $k\to 0$ recovers the small-angle approximation $T\sim 2\pi \sqrt{\ell /g}$. Near the inverted position $k\uparrow 1$, the logarithmic divergence of $K$ describes the long time spent near the unstable equilibrium.

For a period lattice $\Lambda \subset \mathbb{C}$, the Weierstrass elliptic function is the normally convergent lattice sumIt is an even elliptic function with a double pole of principal part $(z-\omega {)}^{-2}$ and zero residue at every point $\omega$ of the period lattice.

For any finite value $a$, apply the argument principle to $f-a$ around a fundamental parallelogram whose boundary avoids zeros and poles. Periodicity cancels the opposite-edge integrals, so the number of solutions of $f(z)=a$ equals the fixed number of poles of $f$, counting multiplicities.

Near zero, the Jacobi elliptic sine is the inverse of the elliptic integral of the first kind:Choose both square roots with value one at zero. The inverse then has $\mathrm{sn}(0,k)=0$ and derivative one there. Differentiation yieldsFor nondegenerate modulus $k\notin \{0,1,-1\}$, continuation gives a meromorphic doubly periodic elliptic function.

The excluded limits are meaningful: $\mathrm{sn}(u,0)=\sin u$ and $\mathrm{sn}(u,1)=\tanh u$. Their inverse integrals and differential equations verify these identities. These elementary limiting functions are not doubly periodic elliptic functions, so the nondegeneracy qualification cannot be dropped.^[ref]

For real $0<k<1$, its periods are generated by $4K(k)$ and $2i{K}^{\prime }(k)$, where $K^{\prime }(k)=K(\sqrt{1-k^2})$. The limiting loss of a period is another way to understand degeneration to trigonometric or hyperbolic functions.

If an elliptic function had no poles, it would be entire. It is bounded on the closure of a fundamental parallelogram, and periodicity then makes it bounded on the whole complex plane. The Liouville theorem would make it constant. Thus every nonconstant elliptic function has a pole.