On the packing densities of superballs and other bodies

Unitary Issues in Some Higher Derivative Field Theories

“…The breaking of this test of reflection positivity can be explicitly shown by numerical analysis (see Figure 1) and rigorously proved by analytic arguments [21,22]. When 0 < s 1 the exponential term e −p 2s /Λ 2s can be expanded as a non-negative convex combination of Gaussian terms…”

“…Thus, the positivity of S 2 (θx, x) follows from that of the Gaussian with s = 1 Equation (7). However, for s > 1 the exponential term does not admit such a representation and the reflection positivity condition Equation (6) fails (see e.g., Reference [22]). Thus, even if the theory has no ghosts, it does not mean that it is unitary, as we have shown for s > 1.…”

Compositions of sums of absolute powers

“…The methods of this section are laid out in a more general context in [ 15] and [3], so we do not go into detail. The lower bound on lattice-packing density which we state is obtained by applying Construction A of Leech and Sloane [7] to error-correcting codes for the author's G-metric.…”

“…A principle employed several times now, in various forms ( [3], [11], [12], [15]), is that bounded, convex, O-symmetric bodies have unusually high lattice-packing densities, if they contain unusually few lattice points. A body of volume V, centered at a point of the lattice A, is expected to contain about E -V/det A lattice points.…”

Lattice packing of nearly-euclidean balls in spaces of even dimension

“…In [4], an essential improvement to the Minkowski-Hlawka density was obtained for the ball for each real a greater than 2. The present paper follows the procedure used there for the case a = 2 + £, insofar as this is permitted by the new complications which arise.…”