Sharply o-minimal structures and sharp cellular decomposition

Sharply o-minimal structures (denoted \so-minimal) are a strict subclass of the o-minimal structures, aimed at capturing some finer features of structures arising from algebraic geometry and Hodge theory. Sharp o-minimality associates to each definable set a pair of integers known as \emph{format} and \emph{degree}, similar to the ambient dimension and degree in the algebraic case; gives bounds on the growth of these quantities under the logical operations; and allows one to control the geometric complexity of a set in terms of its format and degree. These axioms have significant implications on arithmetic properties of definable sets -- for example, \so-minimality was recently used by the authors to settle Wilkie's conjecture on rational points in $\mathbb{R}_{{\exp}}$-definable sets. In this paper we develop some basic theory of sharply o-minimal structures. We introduce the notions of reduction and equivalence on the class of \so-minimal structures. We give three variants of the definition of \so-minimality, of increasing strength, and show that they all agree up to reduction. We also consider the problem of ``sharp cell decomposition'', i.e. cell decomposition with good control on the number of the cells and their formats and degrees. We show that every \so-minimal structure can be reduced to one admitting sharp cell decomposition, and use this to prove bounds on the Betti numbers of definable sets in terms of format and degree

Can one control systematic errors of QCD sum rule predictions for bound states?

We study the possibility to control systematic errors of the ground-state parameters obtained by Shifman-Vainshtein-Zakharov (SVZ) sum rules, making use of the harmonic-oscillator potential model as an example. In this case, one knows the exact solution for the polarization operator, which allows one to obtain both the OPE to any order and the parameters (masses and decay constants) of the bound states. We determine the parameters of the ground state making use of the standard procedures of the method of QCD sum rules, and compare the obtained results with the known exact values. We show that in the situation when the continuum contribution to the polarization operator is not known and is modelled by an effective continuum, the method of sum rules does not allow to control the systematic errors of the extracted ground-state parameters.Comment: RevTex, 7 pages, figure 4 modified, version to be published in Phys. Lett.

Hadron form factors from sum rules for vacuum-to-hadron correlators

We analyse the extraction of the bound-state form factor from vacuum-to-hadron correlator, which is the basic object for the calculation of hadron form factors in the method of light-cone sum rules in QCD. We study this correlator in quantum mechanics, calculate it exactly, and derive the corresponding OPE. We then apply the standard procedures of QCD sum rules to isolate the ground-state form factor from this correlator. We demonstrate that fixing the effective continuum threshold, one of the key ingredients of the sum-rule calculation of bound-state parameters, poses a serious problem for sum rules based on vacuum-to-hadron correlators.Comment: 8 page

Self-stabilization of the equilibrium state in ferroelectric thin films

(K,Na)NbO3 is a lead-free and sustainable ferroelectric material with electromechanical parameters comparable to Pb(Zr,Ti)O3 (PZT) and other lead-based solid solutions. It is therefore a promising candidate for caloric cooling and energy harvesting applications. Specifically, the structural transition from the low-temperature Mc- to the high-temperature c-phase displays a rich hierarchical order of domains and superdomains, that forms at specific strain conditions. The relevant length scales are few tens of nanometers for the domain and few micrometers for the superdomain size, respectively. Phase-field calculations show that this hierarchical order adds to the total free energy of the solid. Thus, domains and their formation has a strong impact on the functional properties relevant for electrocaloric cooling or energy harvesting applications. However, monitoring the formation of domains and superdomains is difficult and requires both, high spatial and high temporal resolution of the experiment. Synchrotron-based time-resolved X-ray diffraction methods in combination with scanning imaging X-ray microscopy is applied to resolve the local dynamics of the domain morphology with sub-micrometer spatial and nanosecond temporal resolution. In this regime, the material displays a novel self-stabilization mechanism of the domain morphology, which may be a general property of first-order phase transitions

Cluster Winds Blow along Supercluster Axes

Within Abell galaxy clusters containing Wide-Angle Tailed radio sources, there is evidence of a ``prevailing wind'' which directs the WAT jets. We study the alignment of nine WAT jets and nearby clusters to test the idea that this wind may be a fossil of drainage along large-scale filaments. We also test this idea with a study of the alignment of WAT jets and filament axes. Statistical tests indicate no significant alignment of WAT jets towards nearest neighbour clusters, but a highly significant alignment with the long axis of the supercluster in which the cluster lies

Application of the Large-N_c limit to a Chiral Lagrangian with Resonances

It is shown that the implementation of the Large--$N_c$ approximation helps to get insight into the structure of, in principle, any QCD-like theory. As an example, we will compute the NLO corrections to $L_{10}$ in the chiral limit with a Lagrangian with Resonances.Comment: 9 pages, 1 figure. Talk given at the International School of Subnuclear Physics (Erice 2002). To be published in the Proceeding