We measure the spectral dimension of universes emerging from nonperturbative quantum gravity, defined through state sums of causal triangulated geometries. While four-dimensional on large scales, the quantum universe appears two-dimensional at short distances. We conclude that quantum gravity may be “selfrenormalizing” at the Planck scale, by virtue of a mechanism of dynamical dimensional reduc...

It is open the possibility of imposing requisites to the quantisation of Spectral Triples in such a way that a critical dimension D=26 appears. From [1] it is known that commutative spectral triples contain the Einstein Hilbert action, which is extracted by using the Wodziski residue over D/ |D/ |, being D/ a Dirac operator. The theorem was initially enunciated [1] with a complicated proportion...

We show that the family of spectral triples for quantum projective spaces introduced by D’Andrea and Da̧browski, which have spectral dimension equal to zero, can be reconsidered as modular spectral triples by taking into account the action of the element K2ρ or its inverse. The spectral dimension computed in this sense coincides with the dimension of the classical projective spaces. The connecti...

We consider the problem of learning regression functions from pairwise data when there exists prior knowledge that the relation to be learned is symmetric or anti-symmetric. Such prior knowledge is commonly enforced by symmetrizing or anti-symmetrizing pairwise kernel functions. Through spectral analysis, we show that these transformations reduce the kernel’s effective dimension. Further, we pr...

It is shown, using methods of inverse-spectral theory, that there exist Schrödinger operators on the line with fairly general spectral features. Thus, for instance, it follows from the main theorem, that if 0 < α < 1 is arbitrary, and if Σ is any perfect subset of (−∞, 0] with Hausdorff dimension α, then there exist potentials q j , j = 1, 2 such that the associated Schrödinger operators H j ar...