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A finite Euclidean configuration is edge Ramsey when every finite edge coloring of a suitable finite Euclidean set contains a monochromatic isometric copy. The edge Ramsey configurations are exactly the equidistant sets, equivalently the vertex sets of regular simplices.

Every regular polygon is a Euclidean Ramsey set. Its cyclic rotation acts transitively on the vertices, so this is a special case of the Kriz theorem for cyclic transitive point sets.

Every line segment is a Euclidean Ramsey set.

Every nondegenerate triangle is a Euclidean Ramsey set.

If finite point sets $X$ and $Y$ are Euclidean Ramsey, then their orthogonal Cartesian product $X\times Y$ is Euclidean Ramsey. The proof first chooses a finite Ramsey witness for $X$, colors a witness for $Y$ by the complete color pattern it induces on the first witness, and then applies the two Ramsey properties in succession.

A finite point set is spherical when it lies on a sphere. Every Euclidean Ramsey set is spherical. Whether every finite spherical point set is Euclidean Ramsey is open.

For $p\in C$,

An equivalent local martingale measure makes every discounted asset price a local martingale. In continuous-time markets this is the usual measure appearing in the no-free-lunch-with-vanishing-risk form of the fundamental theorem of asset pricing.

For a hyperbolic equilibrium of a smooth planar system, the eigenvalues of the Jacobian matrix determine local stability. If its determinant and trace are positive, both eigenvalues have positive real part and the equilibrium is a repeller.

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 equilibrium system with energy density $\rho$, pressure $P$, chemical potential $\mu$, and number density $n$, the entropy density is

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.