We show that a suitably chosen position-momentum commutator can elegantly describe many features of gravity, including the IR/UV correspondence and dimensional reduction (‘holography’). Using the most simplistic example based on dimensional analysis of black holes, we construct a commutator which qualitatively exhibits these novel properties of gravity. Dimensional reduction occurs because the ...

This note presents purely mechanical proofs of the Levi commutator problem in group theory. The problem was solved first by using the theorem prover EQP, developed by William McCune at the Argonne National Laboratory. The fastest proof was found by using Peers-mcd, the Clause-Diffusion parallelization of EQP, developed by the author at the University of Iowa. 1 The Levi commutator problem The L...

In this paper we prove a nontrivial lower bound for the determinant of the covariance matrix of quantum mechanical observables, which was conjectured by Gibilisco and Isola. The lower bound is given in terms of the commutator of the state and the observables and their scalar product, which is generated by an arbitrary symmetric operator monotone function. Introduction The basic object in the st...

We give an error analysis of Strang-type splitting integrators for nonlinear Schrödinger equations. For Schrödinger-Poisson equations with an H4-regular solution, a first-order error bound in the H1 norm is shown and used to derive a second-order error bound in the L2 norm. For the cubic Schrödinger equation with an H4-regular solution, first-order convergence in the H2 norm is used to obtain s...

in this paper, we generalize some transfer theorems.~in particular, we derive one of the main results of gagola(contemp math 524:49--60, 2010) from our results.

The class of commutator-free Magnus integrators is known to provide a favourable alternative to standard Magnus integrators, in particular for large-scale applications arising in the time integration of non-autonomous linear evolution equations. A high-order commutator-free Magnus integrator is given by a composition of several exponentials that comprise certain linear combinations of the value...

We suggest a programming realization of an algorithm for a verification of a given set of algebraic relations in the form of a supercommutator multiplication table for the Verma module, which is constructed according to a generalized Cartan procedure for a quadratic superalgebra and whose elements are realized as a formal power series with respect to noncommuting elements. To this end, we propo...

Charles Sims has asked whether or not, for a free group F , the lower central subgroup γn(F ) coincides with the normal closure in F of the set of basic commutators of weight n. In this paper, we will show that Sims’ question has a positive answer when n = 5. The longest part of the proof is showing that this is true when F has rank 3. Once this is known, the results for higher ranks follow fai...

Let Mb be the operator of pointwise multiplication by b, that is Mb f = bf . Set [A,B] = AB−BA. The Reisz potentials are the operators Rα f(x) = ∫ f(x− y) dy |y|α , 0 < α < 1. They map L 7→ L, for 1 − α + 1 q = 1 p , a fact we shall take for granted in this paper. A Theorem of Chanillo MR84j:42027 [6] states that one has the equivalence ‖[Mb,Rα]‖p→q ≃ ‖b‖BMO with the later norm being that of th...

Two new measures to assess spatial ability are presented: the mental cutting test “Schnitte” (Fay & Quaiser-Pohl, 1999; English version: Fay, Quaiser-Pohl, & Rönicke, 2003), a test for selecting people with extraordinary spatial abilities, and the Picture Rotation Test (Hinze, 2002; Hinze & Quaiser-Pohl, 2003), a mental rotation test for preschool children. Both overcome some of the limits of t...