In a recent paper, M. Green and P. Griffiths used R. Thomas’ works on nodal hypersurfaces to establish the equivalence of the Hodge conjecture and the existence of certain singular admissible normal functions. Inspired by their work, we study normal functions using M. Saito’s mixed Hodge modules and prove that the existence of singularities of the type considered by Griffiths and Green is equiv...

We study the effect of spatial correlations on quenched disorder in random quantum magnets at and near a quantum critical point. In random transverse-field Ising systems disorder correlations that decay algebraically with an exponent r change the universality class of the transition for small enough r, and off-critical Griffiths-McCoy singularities are enhanced. We present exact results for 1D ...

Two lowness notions in the setting of Schnorr randomness have been studied (lowness for Schnorr randomness and tests, by Terwijn and Zambella [19], and by Kjos-Hanssen, Stephan, and Nies [7]; and Schnorr triviality, by Downey, Griffiths and LaForte [3, 4] and Franklin [6]). We introduce lowness for computable machines, which by results of Downey and Griffiths [3] is an analog of lowness for K. ...