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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 the entropy of a region by a constant times the size of its boundary.

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 $0<k<1$, the complete elliptic integral of the first kind is

It is the complete first-kind member of the elliptic integral family.

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.

The Jacobi elliptic sine is the local inverse of the elliptic integral of the first kind: ifthen $z=\mathrm{sn}(u,k)$. Its analytic continuation is a meromorphic doubly periodic function.

It is the sn member of the Jacobi elliptic 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.

Two real-linearly independent periods ${\omega }_1,{\omega }_2$ generate the period lattice

A smooth function $a(x,\xi )$ belongs to $S_{1,0}^m$ when, on every compact $K\subset X$,

A parametrix for $P$ is an operator $A$ such that $PA-I$ and, when relevant, $AP-I$ are regularizing operators.

The method of continuity joins an invertible operator to a target operator through a continuous family. Uniform a priori estimates make the set of invertible parameters both open and closed.

A degenerate elliptic operator has a positive semidefinite principal form that may lose positive definiteness at some points or jets. The p-Laplacian with $p>2$ degenerates where its solution has zero gradient.