An L form of Bernstein's inequality.

Suppose P is a pth degree real polynomial function in n variables and f=PS(n-1) is the restriction of P to the unit sphere S(n-1) in R(n). Bernstein's inequality asserts that ([unk](0) (k)f)(2) + p(2)([unk](0) (k-1)f)(2) /= 1 and differentiation is with respect to arc length theta along any geodesic in S(n-1). We find the constant corresponding to p(2k) when parallelf parallelinfinity is replaced by parallelf parallel(2). One application is a condition on the coefficients of the expansion in surface spherical harmonics of any g: S(n-1) --> R, which condition suffices to assure that g is k times differentiable.

Gross thermodynamics of heat engines in deep interior of Earth.

From the gross conservation laws of thermodynamics in a convecting material we derive a bound on the ratio of the rate of production of mechanical or magnetic energy to the rate of internal radioactive heating which drives the convection. Our bound for this "efficiency" depends on the temperatures in the material, and can exceed unity. Whether the bound can be attained by "efficiencies" in real fluids is not known, but a simple machine shows that "efficiencies" larger than unity are physically realizable. Our bound gives upper limits on the viscous dissipation in the earth's mantle and ohmic heating in the core, but these limits are too large to be physically interesting.