To Split or Not to Split, That Is the Question in Some Shallow Water Equations

In this paper we analyze the use of time splitting techniques for solving shallow water equation. We discuss some properties that these schemes should satisfy so that interactions between the source term and the shock waves are controlled. This paper shows that these schemes must be well balanced in the meaning expressed by Greenberg and Leroux [5]. More specifically, we analyze in what cases it is enough to verify an Approximate C-property and in which cases it is required to verify an Exact C-property (see [1], [2]). We also include some numerical tests in order to justify our reasoning

Spin(7)-instantons, stable bundles and the Bogomolov inequality for complex 4-tori

Using gauge theory for Spin(7)-manifolds of dimension 8, we develop a procedure, called Spin-rotation, which transforms a (stable) holomorphic structure on a vector bundle over a complex torus of dimension 4 into a new holomorphic structure over a different complex torus. We show non-trivial examples of this procedure by rotating a decomposable Weil abelian variety into a non-decomposable one. As a byproduct, we obtain a Bogomolov type inequality, which gives restrictions for the existence of stable bundles on an abelian variety of dimension 4, and show examples in which this is stronger than the usual Bogomolov inequality.Comment: 40 pages, no figures; v2. To appear in Journal de math\'ematiques pures et appliqu\'ee

Glueball-Meson Mixing

Calculations in unquenched QCD for the scalar glueball spectrum have confirmed previous results of Gluodynamics finding a glueball at ~ 1750 MeV. I analyze the implications of this discovery from the point of view of glueball-meson mixing at the light of the experimental scalar sprectrum.Comment: 7 pages, 5 figure

AdS gravity and glueball spectrum

The glueball spectrum has attracted much attention since the formulation of Quantum Chromodynamics. Different approaches give very different results for their masses. We revisit the problem from the perspective of the AdS/CFT correspondence.Comment: 4 pages, no figures, 5 table

Topology in the SU(Nf) chiral symmetry restored phase of unquenched QCD and axion cosmology

We investigate the topological properties of unquenched $QCD$ on the basis of numerical results of simulations at fixed topological charge, recently reported by Borsanyi et al.. We demonstrate that their results for the mean value of the chiral condensate at fixed topological charge are inconsistent with the analytical prediction of the large volume expansion around the saddle point, and argue that the most plausible explanation for the failure of the saddle point expansion is a vacuum energy density $\theta$-independent at high temperatures, but surprisingly not too high $(T\sim 2T_c)$, a result which would imply a vanishing topological susceptibility, and the absence of all physical effects of the $U(1)$ axial anomaly at these temperatures. We also show that under a general assumption concerning the high temperature phase of $QCD$, where the $SU(N_f)_A$ symmetry is restored, the analytical prediction for the chiral condensate at fixed topological charge is in very good agreement with the numerical results of Borsanyi et al., all effects of the axial anomaly should disappear, the topological susceptibility and all the $\theta$-derivatives of the vacuum energy density vanish and the theory becomes $\theta$-independent at any $T>T_c$ in the infinite volume limit.Comment: 18 pages, no figures, typos added in section 4, small changes in the text and some references adde