The condition on the potential ensuring the discreteness of the spectrum of infinite quantum graph Hamiltonian is considered. The corresponding necessary and sufficient condition was obtained by Molchanov 50 years ago. In the present paper, the analogous condition is obtained for a quantum graph. A quantum graph with infinite leads (edges) or/and infinite chains of vertices such that neighbor o...

We describe all real points of the parameter space of two-generator Kleinian groups with a parabolic generator, that is, we describe a certain twodimensional slice through this space. In order to do this we gather together known discreteness criteria for two-generator groups and present them in the form of conditions on parameters. We complete the description by giving discreteness criteria for...

algebraic frames are generalizations of fourier transforms on locally compact abelian groups in the sense that the family of vectors forming the frame are replaced by a family of unbounded linear functionals. the paper studies the indexing measure space of the algebraic frames; as the investigation narrows down to the class of lower semiframes, more relations are revealed between the discret...

The causal set approach to quantum gravity embodies the concepts of causality and discreteness. This article explores some foundational and conceptual issues within causal set theory.

We propose a model of inflation driven by the relaxation an initially Planckian cosmological constant due to diffusion. The can generate (approximately) scale invariant spectrum (adiabatic) primordial perturbations with correct amplitudes and red tilt without inflaton. inhomogeneities observable in cosmic microwave background (CMB) arise from those associated fundamental granularity that are im...

We discuss mathematical and physical arguments against continuity and in favor of discreteness, with particular emphasis on the ideas of Emile Borel (1871–1956).

We discuss mathematical and physical arguments against continuity and in favor of discreteness, with particular emphasis on the ideas of Emile Borel (1871–1956).