Research proof index

Fix a nonempty finite set $A$ in an elementary abelian $2$-group. For any measurable random variables $Y_1,Y_2$, there are a subspace $H$ and a random variable $U$ uniformly distributed on $H$ such that the source's $\rho[\,\cdot\,\#A]$ functional satisfies $2\rho[U\#A]\le\rho[Y_1\#A]+\rho[Y_2\#A]+8d[Y_1;Y_2]$.

If $f$ is integrable on the open annulus $\{y:R_1<d(x,y)<R_2\}$, then varying the inner radius from the right changes the annular integral continuously at $R_1$:

$R\longmapsto\int_{R<d(x,y)<R_2}f(y)\,dy$

is right-continuous at $R=R_1$.

The angular momentum of a rigid body equals its inertia tensor applied to the angular velocity: L = I ω.

The first moment of the mass distribution about its own centre of mass vanishes: for nonzero mass, ρ of the centred j-th coordinate function is zero.

The rotational kinetic energy equals the mass integral of the local rotational speed squared: T = ½ ∫ |ω × r|² dm.

The center of mass of a solid sphere located at the origin is 0.